The Islamic University of Gaza

Deanery of Higher Studies

Faculty of Science

Department of Mathematics

Generalized Lambda Distributionand

Estimation Parameters

Presented by

A-NASSER L. ALJAZAR

Supervised by

Professor: Mohammed S. Elatrash

Assistant Professor:Mahmoud k. Okasha

Submitted in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

Islamic University of Gaza

June 2005

Dedicated

To

my son

II

Acknowledgement

All my special thank after thanking God, my mother, to Department

of Mathematics in Islamic University of Gaza, special thanks to

Professor: Mohammed Eltrash

Assistant Professor: Mahmoud Okasha

for many interesting suggestions and remarks on my thesis.

Words can never express how I am grateful to my family and my wife

for their endless love and support in good and bad times. I wish to

thank all my friends for their Cooperations.

Finally, I would like to thank everyone who supported me

III

Abstract

Generalized Lambda Distribution and

Estimation Parameters

.

Generalized lambda distribution (GLD) is a very useful mean to

testing and fitting of data to well known distributions. Since the GLD

is defined by its quantile function, it can provide a simple and effective

algorithm for generating random variates. In this thesis we study the

defection of GLD and plotting the function. The purpose of this thesis

is to estimate the four parameters of the GLD by using three methods,

the method of moments, the method of least squares and percentiles

method. Numerical examples were used to estimate the parameters of

the GLD . Finally we applied one method on real data.

IV

Table of Contents Table of contents v Introduction 1

1. The Generalized lambda Family of Distributions 4 1.1 History and Background…………………………………………….......5 1.2 Definition of the Generalized lambda distributions……………………..9 1.3 The parameter space of the GLD………………………………………..13 1.4 Shape characteristics of the FMKL Parameterization…………………..17 2. The Moments of the GLD 22 2.1 Karian and dudewicz (2000) approach:………………………………...22

2.2 The moments of the FMKL parameterizations of the GLD Approach: 28 3. Estimating the parameters of the generalized lambda distribution 32 3.1 Introduction ………………………………………………………………32 3.2 Fitting the GLD by the Use of Tables……………………………………34 3.3 Example………………………………………………………………….36 3.4 Estimating the parameters of the generalized lambda distribution: the least squares method……………………………………………………………….40 3.5 Example…………………………………………………………………..43 4. Estimating the parameters of the generalized lambda distribution 45 4.1 The Use of percentiles…………………………………………………….47 4.2 Estimation of GLD parameters through a method of percentiles…………51 4.3 Example…………………………………………………………………...53

5. GLD approximations to some well known distributions 56 5.1 GLD approximations to some well known distributions by using a method of moment………………………………………………………..56 5.1.1 The normal distribution………………………………………56 5.1.2 The uniform distribution……………………………………...59 5.1.3 The exponential distribution………………………………….60 5.2 GLD approximations of some well – known distribution by using a method on percentiles……………………………………………………………….62 5.2.1 The normal distribution……………………………………….62 5.2.2 The uniform distribution………………………………………65 5.2.3 The exponential distribution…………………………………..65 5.3 Application………………………………………………………………67 Appendices 69 References 82

Introduction

Fitting a probability distribution to data is an important task in any statistical data

analysis. The data to be modeled may consist of observed events, such as a financial

time series, or it may comprise simulation results. When fitting data, one typically

first selects a general class, or family, of distributions and then finds values for the

distributional parameters that best match the observed data. Rachev and Mittnik

(2000),demonstrated that the usual approach to distribution fitting is to fit as many

distributions as possible and use goodness-of-fit tests to determine the best fit. This

method, the empirical method, is subjective and is not always conclusive. However,

except in the case of data, there is no single accepted rule for selecting one distribution

over another.

The Generalized Lambda Distribution (GLD), originally proposed by Ramberg

and Schmeiser (1974) at this time which is called ”RS” distribution , is a four-

parameter generalization of Tukey’s Lambda family (Hastings et al. 1947) that has

proved useful in a number of different applications. Since it can assume a wide variety

of shapes, the GLD offers risk managers great flexibility in modelling a broad range

of financial data. Due to its versatility, however, obtaining appropriate parameters

for the GLD can be a challenging problem. An excellent synopsis of the GLD, its ap-

plications and parameter estimation methods appear in Karian and Dudewicz (2000).

The initial, and still the most popular, approach for estimating the GLD parameters

is based on matching the first four moments of the empirical data. This is undoubt-

edly due in part to the availability of published tables that provide parameter values

for given levels of skewness and kurtosis (see, e.g., Ramberg et al. (1979); Karian and

1

2

Dudewicz (2000)).

However, different parameter values can give rise to the same moments and so,

while the tabulated parameters may match or closely approximate the first four mo-

ments, they may in fact fail to adequately represent the actual distribution of the data.

Thus, as is well noted in the literature, a goodness-of-fit test should be performed

to establish the validity of the results. If this test fails, or if the levels of skewness

and kurtosis are outside of the tabulated values, it is necessary to use numerical pro-

cedures to find suitable parameters. Such procedures, which typically involve the

downhill simplex method (Nelder and Mead 1965) or some variant thereof, require as

input an initial estimate of the parameters. If multiple local optima exist, the solu-

tion returned is contingent on this estimate. Thus, several attempts may be required

before obtaining parameter values that are acceptable from a goodness-of-fit perspec-

tive. Unlike previous approaches, King and MacGillivray (1999) assess the quality

of the GLD directly by performing goodness-of-fit tests for specified combinations of

parameter values

Instead of matching moments, Ozturk and Dale (1985) minimize the total squared

differences between the data and the expected values of order statistics implied by

the GLD. The NelderMead downhill simplex algorithm is used to find the optimal

parameters. The method, which is called ”least squares method,” successfully fits a

set of data for which tabulated moment-matching values are unavailable. As with

moment matching, the resulting distribution must be assessed using a goodness-of-fit

test, and several trials may be required before finding an acceptable solution. Instead

of matching moments or least squares methods. Karian and Dudewicz (2000) devel-

ops a method for fitting a GLD distribution to data that is based on percentiles rather

than moments. This approach makes a larger portion of the GLD family accessible

for data fitting and eases some of the computational difficulties encountered in the

method of moments. Examples of the use of the proposed system are included.

The generalized lambda distribution (GLD) is very useful in fitting data and approx-

imating many well known distributions. Since the GLD is defined by its a quintiles

3

function, it can provide a simple and effective algorithm for generating random vari-

ates. The purpose of this dissertation is to estimate the four parameters of the GLD

by using three methods; the method of moments , the method of least squares and the

method of percentiles . We use the moment-matching ,least squares or the method

of percentiles to obtain a candidate set of parameters.

This thesis will proceed as follows: we start with introduction to the history

and mathematical background of the GLD. This is followed by a discussion of how to

estimate the unknown parameter with the method of moments, least squares method,

and percentiles method. In particular we shall be looking at the application of the

GLD and approximating some well known probability distributions as well as the

quality of fit and solves some Examples.

Chapter 1

The Generalized lambda Family ofDistributions

Much of modern human endeavor, in wide-ranging fields that include science, technol-

ogy, medicine, engineering, management, and virtually all of the areas that comprise

human knowledge involves the construction of statistical models to describe the fun-

damental variables in those areas. The most basic and widely used model, called

the probability distribution, relates the values of the fundamental variables to their

probability of occurrence.

The problem of this thesis is how to model (or, how to fit) a continuous probabil-

ity distribution to data. The area of fitting distributions to data has seen explosive

growth in recent years. Consequently, few individuals are well versed in the new

results that have become available. In many cases these recent developments have

solved old problems with the fitting process .

The Generalized Lambda Distribution (GLD) is a four-parameter generalization origi-

nally proposed by Ramberg and Schmeiser (1974) of the one-parameter TukeyLambda

distribution introduced by Hastings et al in 1947. Since then the flexibility of the

GLD in assuming a wide variety of shapes has seen it being used extensively to fit and

4

5

model a wide range of differing phenomena to continuous probability distributions,

from applications in meteorology and modeling financial data, to Monte Carlo simu-

lation studies. The GLD is defined by an inverse distribution function or percentile

(quantile) function. This is the function Q(u) where u takes values between 0 and

1, which gives us the value of x such that F(x) = u, where F(x) is the cumulative

distribution function (c.d.f). From this it is easy to derive the probability density

function (p.d.f) for the GLD using differentiation by parts, however the cumulative

distribution function needs to be calculated numerically. The most popular method

for estimating the GLD parameters is to match the first four moments of the empirical

data to that of the GLD. The popularity of this method is partly due to the avail-

ability of extensive tables that provide parameter values for given values of skewness

and kurtosis see Ramberg (1979) and Karian and Dudewicz (2000). In our case we

will use the tables to find the parameter values and will be calculating them directly.

From values of given values of skewness and kurtosis see Ramberg (1979) and Karian

and Dudewicz (2000).

1.1 History and Background

The search for a method of fitting continuous probability distributions to data is quite

old. Pearson (1895) gave a four-parameter system of probability density function and

fitted the parameters by what he called the ”method of moments (Pearson (1894))

6

Tukey’s Lambda family of distributions is defined by the quantile function Q(u) ori-

gins in the one-parameter lambda distribution proposed by John Tukey (1960)

Q(u) =

uλ−(1−u)λ

λ, λ 6= 0

log(u)1−u

, λ = 0 , u 6= 1where 1 ≥ u ≥ 0.

Tukey’s lambda distribution was generalized, for the purpose of generating random

varieties for Monte Carlo simulation studies, to the four-parameter generalized lambda

distribution, or GLD, by Ramberg and Schmeiser (1972 - 1974) subsequently, and

Mykytka (1979).

Since the early 1970s the GLD has been applied to fitting phenomena in many

fields of endeavor with continuous probability density functions.

In an early application of the GLD (at the time called the RS (for Ramberg-

Schmeiser) distribution), Ricer (1980) dealt with construction industry data. His

concern was to correct for the deviations from normality, which occur in construction

data, especially in expectancy pricing in a competitive bidding environment, finding

such quantities as the optimum markup. In another important application area,

meteorology, it is recognized that many variables have used empirical distributions

as an alternative, fitting of solar radiation data with the GLD was successful due to

the flexibility and generality of the GLD, which could successfully be used to fit the

wide variety of curve shapes observed.” In many applications, this means that we

can use the GLD to describe data with a single functional form by specifying its four

parameter values for each case, instead of giving the basic data (which is what the

empirical distribution essentially does) for each case, Karian and Dudewicz (2000).

Before defining the Generalized Lambda Distribution (GLD) family we review

some basic notions from statistics.

7

Definition 1.1.1. . [2] Probability space

A probability space is triplet (Ω ,= , P [ . ]) where Ω is sample space ,= is a collection

of events( each subset of Ω) ,P[ . ]is a probability function with domain =

Definition 1.1.2. . [2] Random Variable ”r.v”

For a given probability space (Ω ,= , P [ . ]) , a random variable, denoted by X or X(.),

is a function with domain Ω. The function X(.) must be such that the set Ar, defined

by Ar = ω : X(ω) ≤ r belong to = for every real number r.

Definition 1.1.3. . [2] Discrete and continuous random variable:

If X can take on only a few discrete values (such as 0 or 1 for failure or success, or

0,1,2,3,...as the number of occurrences of some event of interest), then X is called a

discrete random variable.

If the outcome of interest X can take on values in a continuous range (such as all

values greater than zero and less than one), then X is called a continuous random

variable.

Definition 1.1.4. . [ 2] Cumulative distribution function :

One way of specifying the chances of occurrence of the various values that are pos-

sible. This is, called cumulative distribution function (c.d.f.) of random variable X

, denoted by FX( . ) ,is defined to be that function with domain the real line which

satisfies

FX(x) = p (X < x) −∞ < x < ∞

Definition 1.1.5. .[2]Probability density function:

A second way of specifying the chances of occurrence of the various values of X is to

8

give what is called the probability density function (p.d.f.) of x . This is a function

fX(x) that is fX(x) > 0 for all x, integrates to 1 over the range −∞ < x < ∞ , and

such that for all x,

FX(x) =

∫ x

−∞fX(t)dt

Definition 1.1.6. . [2] Quantiles function:

A third way of specifying the chances of occurrence of the various values of X is to

give what is called the inverse distribution function, or quantiles function (p.f.), of X.

This is the function QX(y) which, for each y between 0 and 1, tells us the value of x

such that

FX(x) = y : QX(y) = The value of x such that FX(x) = y , 0 < y < 1

We see that there are three ways to specify the chances of occurrence of a r.v. We

give the c.d.f. , p.d.f. , and p.f. for a r.v : with the general normal distribution with

mean µ, and variance σ2, N(µ, σ2). The p.d.f. is

f(x) =1√2πσ

exp−(x−µ)2

2σ2

and the percentile function (p.f.) can be obtained as follows:

FX(x) = p (X ≤ x) = y iff

p (X − µ

σ≤ x− µ

σ) = y = p (Z ≤ x− µ

σ) iff

QZ(y) = x−µσ

iff

x = µ + σ QZ(y)

9

therefor QX(y) = x The value of x such that

FX(x) = y = µ + σQZ(y)

We should note that, in addition to QX(y) , there are several notations in common

use for the p.f. one usually finds the notation F−1X (x).

1.2 Definition of the Generalized Lambda Distri-

butions

Definition 1.2.1. .[7]

The generalized lambda distribution family GLD with parameters λ1, λ2, λ3, λ4,

GLD(λ1, λ2, λ3, λ4), is most easily specified in terms of its percentile function

Q(y) = λ1 +yλ3 − (1− y)λ4

λ2

(1.2.1)

Where 0 < y < 1. The parameters λ1 and λ2 are, respectively, location and scale

parameters, λ3 and λ4 determine the skewness and kurtosis of the GLD(λ1, λ2, λ3, λ4).

Note that

Definition 1.2.2. .[2]Parameter

A parameter is a value, usually unknown (and therefore has to be estimated), used

to represent a certain population characteristic. For example, the population mean

µ is a parameter that is often used to indicate the average value of a quantity

10

Definition 1.2.3. .[2] Location Parameter:

let f(x) be any p.d.f .Then the family of p.d.f f(x − µ) indexed by the parameter

µ ,−∞ < µ < ∞ is called location family and µ is called Location Parameter .

Measures of location give information about the location of the central tendency

within a group of numbers.

Definition 1.2.4. .[2]Skewness

Not all distributions are bell shaped (or normal). In the normal distribution, there

are just as many observations to the right of the mean as there are to the left. The

median and mean are also equal. When this is not the case, we say the distribution

is skewed or asymmetrical. If the tail is drawn out to the left, then the curve is left

skewed If the tail is drawn out to the right, then the curve is right skewed.

Definition 1.2.5. .[2]Kurtosis

Another type of departure from normality is the kurtosis, or ”peakedness” of the

distribution. A leptokurtic curve has more values near the mean and at the tails, with

fewer observations at the intermediate regions relative to the normal distribution. A

platykurtic curve has fewer values at the mean and at the tails than the normal

curve, but more values in the intermediate regions. A bimodal (”double-peaked”)

distribution is an extreme example of a platykurtic distribution

11

The properties of the GLD distribution are studied in detail in Ramberg (1979)

at this time which called Ramberg -Schmeiser”RS distribution ” ) . In addition

to elaborating the richness of the fourth parameter of GLD to fit a wide variety

of frequency distributions, Ramberg (1979). A good summary of the shape GLD

distribution is well defined appears in King and MacGillivray (1999).

Recall that for the normal distribution there are also restrictions on (µ, σ2),

namely, σ > 0. The restrictions on λ1, λ2, λ3, λ4 that yield a valid, GLD distribu-

tion will be discussed . It is relatively easy to find the probability density function

from the percentile function of the GLD, as we now show.

Theorem 1.2.1. .[3]:

For the GLD(λ1, λ2, λ3, λ4), the probability density function is

f(x) =λ2

λ3yλ3−1 + λ4(1− y)λ4−1, at x = Q(y) (1.2.2)

where 0 ≤ y ≤ 1

Proof. :

Using the relationships: x = Q(y) and F (x) = y and differentiating with respect

to x, we get:

dy

dx= f(x) or f(Q(y)) =

dy

d(Q(y))=

1d(Q(y))

dy

(1.2.3)

so by differentiating (1.2.1) and outing it into (1.2.4) we find f(x).

dQ(y)

dy=

d

dy(λ1 +

yλ3 − (1− y)λ4

λ2

) =λ3y

λ3−1 + λ4(1− y)λ4−1

λ2

(1.2.4)

12

f(x) =λ2

λ3yλ3−1 + λ4(1− y)λ4−1, at x = Q(y).

In plotting the function f(x) for a density such as the normal, where f(x)is given as

a specific function of x, we calculate f(x) at x values, then plotting the pairs (x,f(x))

and connecting them with a smooth curve. For the GLD family, plotting f(x) proceeds

differently since (1.2.3) tells us the value of f(x) at x = Q(y). Thus, we take a grid of

y values (such as .01, .02, .03, , .99, that give us the ( points), find x at each of those

points from(1.2.1), and find f(x) at that x from (1.2.3). Then, we plot the pairs (x

,f(x)) and link them with a smooth curve.

Example

plot f(x) for a GLD, consider the GLD(λ1, λ2, λ3, λ4) with

parameters λ1 = 0.0305, λ2 = 1.3673, λ3 = 0.004581, λ4 = 0.01020.

Proof. :

The GLD (λ1, λ2, λ3, λ4)

Q(y) = 0.0305 +y0.004581 − (1− y)0.01020

1.3673(1.2.5)

we find that at y = 0.25 the Q(0.25)=0.0280, from (1.2.6). We have , at x =

0.028, using (1.2.3) with the specified values of λ1, λ2, λ3, λ4, f (0.028) = 43.0399612.

Hence, (0.028, 43.04) will be one of the points on the graph of f (x).

Proceeding in this way for y = 0.01, 0.02,, 0.99, we obtain the graph of f(x) given in

Figure (1.1)

13

Figure 1.1: The p.d.f. Of GLD (0.0305, 1.3673, 0.004581, 0.01020)

.

The generalized lambda distribution, also known as the asymmetric lambda, (

or Turkey lambda distribution) or(Ramberg -Schmeiser”RS distribution ” ), is a

distribution with a wide range of shapes. The distribution is defined by its quantile

function, the inverse of the distribution function.

1.3 The parameter space of the GLD

We noted, following formula (1.2.1), that GLD does not always specifies vialed dis-

tribution. The reason is that one cannot just write down any formula and be assured

it will specify a distribution without checking the conditions needed for that fact to

hold.

14

Theorem 1.3.1. :

The GLD(λ1, λ2, λ3, λ4) specifies a valid distribution if and only if

λ2

λ3yλ3−1 + λ4(1− y)λ4−1≥ 0 (1.3.1)

for all y ∈ [0, 1]

Proof. :

In particular, a function f (x) is a probability density function if and only if it satisfies

the conditions

f(x) ≥ 0 and

∫ ∞

−∞f(x)dx = 1 (1.3.2)

From (1.2.4) we see that for the GLD(λ1, λ2, λ3, λ4), conditions (1.3.1) are satisfied

if and only if

λ2

λ3yλ3−1 + λ4(1− y)λ4+1≥ 0 and

∫ ∞

−∞f(Q(y)) dQ(y) = 1 (1.3.3)

Since from (1.2.4) we know that

f(Q(y))dQ(y) = dy

and y is in the range [0,1], the second condition in (1.3.2) follows. Thus, for any

λ1, λ2, λ3 and λ4 the function f(x) will integrate to 1.

It remains to show that the first condition in (1.3.2) holds. The next theorem

establishes the role of λ1 as a location parameter.

15

Theorem 1.3.2. :

If the random variable X is GLD(0, λ2, λ3, λ4) then the random variable X + λ1 is

GLD(λ1, λ2, λ3, λ4).

Proof. :

Suppose that X is GLD(0, λ2, λ3, λ4)

.Then by (1.2.1)

QX(y) =yλ3 − (1− y)λ4

λ2

Since FX+λ1(x) = p [X + λ1 ≤ x] = p [X ≤ x− λ1] = FX(x− λ1) (1.3.4)

then FX(x− λ1) = y also implies FX+λ1(x) = y , yielding

x− λ1 = QX(y) =yλ3 − (1− y)λ4

λ2

, x = QX+λ1(y) (1.3.5)

whence

QX+λ1(y) = x = λ1 + QX(y) = λ1 +yλ3 − (1− y)λ4

λ2

(1.3.6)

This prove that X + λ1 is GLD(λ1, λ2, λ3, λ4) random variable.

In addition to this Ramberg et al (1979) noted that there are certain combinations

of λ3 and λ4 for which the distribution given by (1.2.1) is not a valid probability

distribution. This undefined region is 1 + λ23 < λ4 < 1.8(λ2

3 + 1) see Karian and

Dudewicz (2000). for an in depth study.

The regions 1, 2, 3, 4, 5, 6 in Fig (1.2) are the ones for which the distribution

is valid is as follows: to determine the (λ3, λ4) pairs that lead to a valid GLD, we

16

Figure 1.2: Regions 1, 2, 3, 4, 5, 6 for which the GLD parameterization is valid.

consider (λ3, λ4)-space in the following regions:by using a method on percentiles

R1 = [(λ3, λ4)|λ3 ≤ −1, λ4 ≥ 1] , R2 = [(λ3, λ4)|λ3 ≥ 1, λ4 ≤ −1]

R3 = [(λ3, λ4)|λ3 ≥ 0, λ4 ≥ 0] , R4 = [(λ3, λ4)|λ3 ≤ 0, λ4 ≤ 0]

R5 = [(λ3, λ4)|λ3 < 0, 0 < λ4 < −1], R6 = [(λ3, λ4)|0 < λ3 < 1, λ4 < 0]

R7 = [(λ3, λ4)| − 1 < λ3 < 0, λ4 > 1], R8 = [(λ3, λ4)|λ3 > 1,−1 < λ4 < 0]

The GLD(λ1, λ2, λ3, λ4) is valid in Region 1,2 ,3 and 4 , The GLD(λ1, λ2, λ3, λ4)

is not valid in Region 5 and 6. The situation is quite the same in Region 7,8 . A

point in Region 7, is valid if and only if

(1−λ3)(λ4−λ3)

(λ4 − 1)λ4−1 < λ3

λ4

see Karian and Dudewicz (2000). for an in depth study.

17

1.4 Shape Characteristics of the FMKL Parame-

terization

Freimer .(1988) devise a different parameterizations for the GLD, denoted FMKL,

which is given by

Q(y) = λ1 +1

λ2

[yλ3 − 1

λ3

− (1− y)λ4

λ4

] where 0 ≤ y ≤ 1 (1.4.1)

which is well defined over the entire (λ3, λ4)-plane. The variety of shapes offered by

this distribution classify the density shapes we need to know the role which each of

the parameters play within the GLD.From the defection FMKL we get

λ1 is the location parameter.

λ2 determines the scale.

λ3, λ4 determine the shape characteristics. For asymmetric distribution λ3 = λ4

Freimer (1988) classify the shapes returned by (1.4.1) as follows:

Class I

(λ3 < 1, λ4 < 1) : Unimodal densities with continuous tails. This class

can be subdivided with respect to the finite or infinite slopes of the densi-

ties at the end points.

Class Ia (λ3, λ4 < 12), Class Ib (1

2< λ3 < 1, λ4 ≤ 1

2) , and Class Ic (1

2< λ3 <

1, 12

< λ4 < 1) .

Class II

(λ3 > 1, λ4 < 1): Monotone p.d.f.s similar to those of the exponential or

χ2 distributions. The left tail is truncated.

Class III

18

(1 < λ3 < 2, 1 < λ4 < 2): U-shaped densities with both tails truncated.

Class IV

(λ3 > 2, 1 < λ4 < 2): Rarely occurring S-shaped p.d.fs with one mode and

one antimode. Both tails are truncated.

Class V

(λ3 > 2, λ4 > 2): Unimodal p.d.fs with both tails truncated.

Figures 1.3 to 1.10 show examples of each class of shapes.

see ”SusanneW.M.AuY enug”for an in depth study.

Figure 1.3: Class Ia p.d.fs including the normal distribution

Figure 1.4: Class Ib p.d.fs

19

Figure 1.5: Class Ic p.d.fs.

Figure 1.6: Class II p.d.fs includes the exponential distribution.

Figure 1.7: Class III U-shaped p.d.fs.

20

Figure 1.8: Class IV S-shaped p.d.fs.

Figure 1.9: Class V p.d.fs.

Chapter 2

The Moments of the GLD

Moment come in two formes.They are either raw moment or central moments.

Definition 2.0.1. .[2]The Kth raw moment of a probability density function f(x) of the random variableXis defined by :

E(Xk) =

∫ ∞

−∞Xkf(x)dx where k ≥ 1

.In particular the 1st moment E(X) = µ is the mean.

Definition 2.0.2. .[2]The Kth central moment is defined by

E(X − µ) =

∫ ∞

−∞(X − µ)kf(x)dx wherek > 1

2.1 Karian and Dudewicz (2000) approach :

The moments of the GLD(λ1, λ2, λ3, λ4) , parameterizations of the GLD can be de-

rived as follows : We start by setting λ1 = 0 to simplify this task; next, we obtain

the non-central moments of the GLD(λ1, λ2, λ3, λ4); and finally, we derive the central

GLD(λ1, λ2, λ3, λ4) moments.

Theorem 2.1.1.

If X is a GLD(λ1, λ2, λ3, λ4) random variable then Z = X−λ1 is GLD(0, λ2, λ3, λ4).

21

22

Proof. :

Since X is GLD(λ1, λ2, λ3, λ4)

QX(y) = λ1 +yλ3 − (1− y)λ4

λ2

andFX−λ1(x) = p (X − λ1 ≤ x) = p (X ≤ x + λ1) = FX(x + λ1) (2.1.1)

If we set FX(x + λ1) = y , we obtain

x + λ1 = QX(y) = λ1 +yλ3 − (1− y)λ4

λ2

(2.1.2)

From (2. 2.1) we also have FX−λ1(x) = y which with (2.2.2) yields

QX−λ1(y) = x =yλ3 − (1− y)λ4

λ2

Proving that X − λ1 is GLD(0, λ2, λ3, λ4).

Having established λ1 as a location parameter, we now determine the non-central

moment (when they exist) of the GLD(λ1, λ2, λ3, λ4)

Theorem 2.1.2.

If Z is GLD(0, λ2, λ3, λ4); then E(Zk), the expected value of ZK, is given by

E(Zk) =1

λk2

k∑i=0

[(ki )(−1)iβ(λ3(k − i) + 1 , λ4i + 1) (2.1.3)

where β(a, b) is the beta function defined by :

β(a, b) =

∫ 1

0

xa−1(1− x)b−1dx (2.1.4)

Proof.

E(Zk) =

∫ ∞

−∞Zkf(z)dz =

∫ 1

0

(Q(y))kdy

where

f(z) =dy

dz, Q(y) = z (2.1.5)

23

=

∫ 1

0

(yλ3 − (1− y)λ4

λ2

)kdy =1

λk2

∫ 1

0

(yλ3 − (1− y)λ4)kdy

By the binomial theorem,

(yλ3 − (1− y)λ4)k =k∑

i=0

[(ki )(y

λ3)(k−i)(−(1− y)λ4)i) (2.1.6)

using (2.2.6) in the last expression of (2.2.5), we get

E(ZK) =1

λk2

k∑i=1

[(ki )(−1)i

∫ 1

0

(yλ3)(k−i)(1− y)λ4idy]

=1

λk2

k∑i=1

[(ki )(−1)iβ(λ3(k − i) + 1, λ4i + 1)]

Completing the proof of the theorem.

Before continuing with our investigation of the GLD(λ1, λ2, λ3, λ4) moments,we

note the beta function that will be useful in our subsequent work where β(a, b) is the

beta function defined before.

The integral in (2.1.4) that defines the beta function will converge if and only if a

and b are positive (this can be verified by choosing c from the (0, 1) interval and

considering the integral over the subintervals (0, c) and (c, 1)).

Corollary 2.1.3.

The kth GLD(λ1, λ2, λ3, λ4) moment exists if and only if min (λ3, λ4) > −1/k.

Proof. .From Theorem (2.1.1) , E(Xk) will exist if and only if exists, which, by Theorem

(2.1.2), will exist if and only if E(Zk) = E((X − µ)K) exists, which, by Theorem

(2.1.2), will exist if and only if

24

λ3(k − i) + 1 > 0 and λ4i + 1 > 0, for i = 0, 1, ..., k.

Then λ3k − λ3i > −1 and λ3k > −1 + λ3i

Since λ3i is positive then

λ3k > −1, and λ3 > −1/k

Also we haveλ4i + 1 > 0 and λ4i > −1 for i = 0, 1, ., k

if i = k then λ4 > −1/k

This condition will done if

λ3 > −1/k and λ4 > −1/k.

Since, ultimately, we are going to be interested in the first four moments of the

GLD(λ1, λ2, λ3, λ4), we will need to impose the condition λ3 > −1/4 and λ4 > −1/4

throughout the remainder of this chapter.

The next theorem gives an explicit formulation of the first four centralized

GLD(λ1, λ2, λ3, λ4) moments.

Theorem 2.1.4.

If X is GLD(λ1, λ2, λ3, λ4) with λ3 > −1/4 and λ4 > −1/4, then its first fourmoments, α1 , α2 , α3 , α4 (mean , variance , skewness , and kurtosis, respectively),are given by

α1 = µ = E(X) = λ1 +A

λ2

(2.1.7)

α2 = σ2 = E[(X − µ)2] =B − A

λ22

(2.1.8)

α3 =E[(X − E(X))3]

σ3=

C − 3AB + 2A3

λ42σ

3(2.1.9)

25

α4 =E[(X − E(X))4]

σ4=

D − 4AC + 6A2B − 3A4

λ42σ

4(2.1.10)

where

A =1

1 + λ3

− 1

1 + λ4

(2.1.11)

B =1

1 + 2λ3

+1

1 + 2λ4

− 2β(1 + λ3, 1 + λ4) (2.1.12)

C =1

1 + 3λ3

− 1

1 + 3λ4

− 3β(1 + 2λ3, 1 + λ4) + 3β(1 + λ3, 1 + 2λ4) (2.1.13)

D =1

1 + 4λ3

+1

1 + 4λ4

− 4β(1 + 3λ3, 1 + λ4) + 6β(1 + 2λ3, 1 + 2λ4)

−4β(1 + λ3, 1 + 3λ4) (2.1.14)

and β denotes the Beta function.

Proof. :Let Z be a GLD(0 , λ2 , λ3 , λ4) random variable By (theorem 2.1.1).

E(Xk) = E((Z + λ1)k)

we first express E(Zi), for i = 1,2,3, and 4, in terms of A, B, C and D, to do this forE(Z), we use ( theorem 2.1.2) to obtain.

E(Z) =1

λ2

β(λ3 + 1, 1)− β(1, λ4 + 1)

and since β(λ + 1, 1) = β(1, λ + 1) Since β(λ3 + 1, 1) = 1λ3+1

and

β(1, λ4 + 1) =1

λ4 + 1

we get

E(Z) =1

λ2

(1

λ3 + 1− 1

λ4 + 1) =

A

λ2

(2.1.15)

for E(Z2) we again use (theorem 2.1.2) and the simplification allowed by β(λ+1 , 1) =β(1 , λ + 1) = 1

1+λto get .

E(Z2) =1

λ22

[β(λ3 + 1, 1)− β(λ3 + 1, λ4 + 1) + β(1, 2λ4 + 1))]

26

=1

λ22

(1

2λ3 + 1− 1

2λ4 + 1− 2β(λ3 + 1, λ4 + 1)) =

B

λ22

(2.1.16)

similar arguments ,with somewhat more complicated algebraic manipulations , E(Z3)and E(Z4) produce .

E(Z3) =C

λ32

(2.1.17)

E(Z4) =D

λ42

(2.1.18)

we now use (2.1.15) to derive (2.1.7):

α1 = E(X) = E(Z + λ1) = λ1 + E(Z) = λ1 +A

λ2

(2.1.19)

next we consider (2.1.8):Since

E(X − µ)2 = E(X2)− (E(X))2

α2 = E(X2)−α21 = E((Z+λ1)

2−α21 = E(Z2)+2λ1E(Z)+λ2

1−α21 (2.1.20)

Substituting A/λ2 for E(Z) and

λ1 +A2

λ22

=B − A

λ22

for in (2.2.20) and using (2.2.16), we get. The derivations of (2.2.9) and (2.2.10) aresimilar but algebraically more involved.

see Karian and Dudewicz (2000). for an in depth study.

27

2.2 The moments of the FMKL parameterizations

of the GLD Approach :

The moments of the FMKL parameterizations of the GLD can be derived as follows

We can rewrite equation (1.4.1) as

Q(u) = [λ1 −1

λ2λ3

+1

λ2λ4

+1

λ2

(uλ3

λ3

− (1− u)λ4

λ4

] (2.2.1)

= b + a Q(u)

where

Q(u) =uλ3

λ3

− (1− u)λ4

λ4

Now, If X represents the random variable with percentile function Q(u) given in (

1.4.1) and Y represents the random variable with percentile function Q(u) given by (

2.2.1) , then we have

E(x) = (aE(y) + b)

E(x− k)k = (akE(y) + b)k

Since x = Q(u) , µ = E(X) ,σ2 = E(Xµ)2We now Letting vk = E(Y )k Hence for the

random variable X with the percentile function Q(u).

The kth raw moment is defined by

E(Xk) =

∫ 1

0

Q(u)kdu (2.2.2)

Letting vK = E(Y )K then from (2.2.2) we find that we need to calculate

vk =

∫ 1

0

(uλ3

λ3

− (1− u)λ4

λ4

)kdu (2.2.3)

28

Expanding the integrand on the right-hand side of Equation (2.2.3) using binomial

expansion gives

vk =

∫ 1

0

k∑j=0

(−1)j (kj )

uλ3(k−j)

λ(k−j)3

× (1− u)λj4

λj4

du (2.2.4)

vk =k∑

j=0

(−1)j

λ(k−j)3 λj

4

(kj ) β [λ3(k − j) + 1 , λ4(j + 1)]

where β is the beta function defined as

β(a, b) =

∫ 1

0

xa−1(1− x)b−1dx

The beta function on the right-hand side of Equation (2.2.4) is defined if both of its

arguments are positive, which essentially means that the following holds:

mink,j λ3(k − j) + 1 > 0 (2.2.5)

minj λ4(j + 1) > 0

From Equation (2.2.5) it is clear that the inequality is only crucial when λ3, λ4 < 0

. Since 0 < j < k , Equation (2.2.5) can be written as

min (λ3, λ4) > −1

k

Using equation ( 2.2.4) we can obtain the first forth moment values

v1 = µ =1

λ3 + 1− 1

λ4 + 1

v2 =1

λ23(3λ3 + 1)

+1

λ24(3λ4 + 1)

− 2

λ3λ4

β(λ3 + 1, λ4 + 1)

29

v3 =1

λ33(3λ3 + 1)

− 1

λ34(3λ4 + 1)

− 3

λ23λ4

β(2λ3 + 1, λ4 + 1) +3

λ3λ24

β(λ3 + 1, λ4 + 1)

v4 =1

λ43(4λ3 + 1)

+1

λ44(4λ4 + 1)

+6

λ23λ

24

β(2λ3 + 1, λ4 + 1)− 4

λ33λ4

β(3λ3 + 1, λ4 + 1)

− 4

λ3λ34

β(λ3 + 1, 3λ4 + 1)

From the above results and putting them into

E(x− k)k = (akE(y) + b)k

, we get the first four central moments of the FMKL parameterizations of the GLD.

E(X − µ)2 =1

λ22

(v2 − v21)

E(X − µ)3 =1

λ32

(v3 − 3v1v2 + 2v31)

E(X − µ)4 =1

λ42

(v4 − 4v1v3 + 6v21v2 − 3v4

1)

So from equations α3 = 1σ3 E(X − µ)3 and α3 = 1

σ3 E(X − µ)3 we get the skewness

and kurtosis of the GLD to be

α3 =v3 − 3v1v2 + 2v3

1

(v2 − v21)

32

(2.2.6)

α4 =v4 − 4v1v3 + 6v2

1 − 3v41

(v2 − v21)

2(2.2.7)

30

So if we are given the meanµ, variance σ2, skewness α3, and kurtosis α4 of the

sample data we can find λ3 and λ4 of the GLD by solving Once we have found λ3and

λ4 we can find λ2 from ( 2.2.7) and then λ1 from E(X) = (aE(Y ) + b) and using

a = 1λ2

, b = λ1 − 1λ2

( 1λ3− 1

λ4) we get:

λ2 =

√v2 − v2

1

σ

λ1 = µ +1

λ2

(1

λ3 + 1− 1

λ4 + 1)

Chapter 3

Estimating the Parameters of thegeneralized lambda Distribution

Fitting the GLD thought the method of moment

3.1 Introduction

The method of moment consists of equating the first few moments of a population to

the corresponding sample moments, thus getting as many equations as are needed to

solve for unknown parameter of the population. Thus, if a population has r parame-

ters, the method of moments consists of solving the system of equations

αk = µk k= 1,2,3,......r , for the r parameters.

where, αk is the kth sample moment.

As stated at the beginning, our intention is to fit a GLD to a data set by equating

σ1, σ2, σ3, σ4 to σ1, σ2, σ3, σ4 the sample statistics corresponding to σ1, σ2, σ3, σ4 and

solving the equations for λ1, λ2, λ3, λ4.

For a data set X1, X2.....Xn, the sample moments corresponding to σ1, σ2, σ3, σ4 are

31

32

denoted by σ1, σ2, σ3, σ4, and are defined by

σ1 = X =n∑

i=1

xi

n, (3.1.1)

σ2 =n∑

i=1

(xi − X)2

n, (3.1.2)

σ3 =n∑

i=1

(xi − X)3

nσ3, (3.1.3)

σ4 =n∑

i=1

(xi − X)4

nσ4, (3.1.4)

Solving the system of equation

σi = σi for i = 1, 2, 3, 4 (3.1.5)

for λ1, λ2, λ3, λ4, the system is simplified some what by observing that A ,B ,C and

D of (2.2.11) through (2.2.14) are free of λ1 , λ2 . Thus σ3, and σ4 depend only on

λ3, and λ4. Hence if λ3, and λ4 can be obtained by solving the system

σ3 = σ3 and σ4 = σ4 (3.1.6)

of two equation in the two variables λ3 and λ4, then using (2.2.8) and (2.2.7) suc-

cessively will yield λ2, and λ1 . Unfortunately, (3.1.5) is complex enough to prevent

an exact solution, forcing us to appeal to numerical methods to obtain approximate

solutions. The values of the parameter λ3 and λ4 may be computed by solving system

of equation (3.1.5) in the region (−1/4,∞)(−1/4,∞) of the (λ3 , λ4) . Algorithms

such as for finding numerical solutions to systems of equations such ”search” for a

33

solution by checking if an initial set of values (λ3 = λ∗3 , λ4 = λ∗4) in the case of

(3.1.6)) can be considered an approximate solution. This determination is made by

checking if

Max(|σ3 − σ3| , σ4 − σ4)|) < ε (3.1.7)

when λ3 = λ∗3 and λ4 = λ∗4 . The positive numbers ε represents the accuracy

associated with the approximation; if it is determined that the initial set of values

λ3 = λ∗3, λ4 = λ∗4 does not provide a sufficiently accurate solution, the algorithm

searches for a better choice of λ3 and λ4 and iterates this process until a suitable

solution is reached (i,e,, one that satisfies (3.1.6)).

In algorithms of this type there is no assurance that the algorithm will terminate

successfully nor that greater accuracy will be attained in successive iterations.

Therefore, such searching algorithms are designed to terminate (unsuccessfully) if

(3.1.6) is not satisfied after a fixed number of iterations.

3.2 Fitting the GLD by the Use of Tables

Some readers may not have sufficient expertise in programming or adequate program-

ming support to use the type of analysis that was illustrated in (3.1.1) a number of

investigators have provided tables for the estimation of λ1, λ2, λ3, λ4. The first of

these was given by Ramberg and Schmeiser (1974); et.al. and Dudewicz and Karian

(1996) provide the most accurate and comprehensive tables to date.

Unless some simplifications are used, tabulated results for determining λ1, λ2, λ3, λ4

from σ1, σ2, σ3, σ4. To make the tabulation manageable, we summarize this process

in the GLD-M algorithm below

34

Algorithm GLD- M: Fitting a GLD distribution to data by the method of

moments.

GLD-M-1. Use (3.1.1) through (3.1.4) to compute ; σ1, σ2, σ3, σ4

GLD-M-2. Find the entry point in a table closest to ( | σ3 | , σ4)

GLD-M-3. Using ( | σ3 | , σ4) from Step GLD-M-2 to extract λ1(0, 1) , λ2(0, 1)

λ3 and λ4 from the table

GLD-M-4. If σ3 < 0 , interchange λ3 and λ4 and change the sign of λ1(0, 1) ;

GLD-M-5. Compute λ1 = λ1(0, 1)√

α2 + σ1 and λ2 = λ2(0, 1)/√

α2.

To illustrate the use of Algorithm GLD- M and the table Appendix A

Suppose that σ1, σ2, σ3, σ4 have been computed to have value

σ1 = 2, σ2 = 3, σ3 = −√

0.025, σ4 = 2 (3.2.1)

Note that σ4 has been taken to be the same as, and σ3 has been taken to be the

negative of Step GLD-M-1 is taken care of since σ1 , σ2 , σ3 , and σ4 have is given.

For Step GLD-M-2, we observe that σ3 = - 0.15811; hence, the closest point to

(| σ3 | , σ4). in the Table of Appendix A is (0.15,2.0), giving us

λ1(0, 1) = −1.3231, λ2(0, 1) = 0.2934, λ3 = 0.03145, λ4 = 0.7203

The instructions on the use of the table in Appendix A indicate that a superscript of

b in a table entry designates a factor of 10−b . In this case, an entry of 0.31451 for

λ3 indicates a value of 0.3145× 10−1 = 0.03145 .

Since σ3 < 0 , step GLD- M- 4 readjusts these to

λ1(0, 1) = 1.3231 , λ2(0, 1) = 0.2934 , λ3 = 0.7203 , λ4 = 0.03145

35

with the computation in , step GLD-M-5 we get

λ1 = 4.2917, λ2 = 0.1694, λ3 = 0.7203, λ4 = 0.03145

3.3 Example

Dudewicz, Levy, Lienhart, and Wehrli (1989) give data on the brain tissue MRI scan

parameter, AD. It should be noted that the term ”parameter” is used differently

in brain scan studies it is used to designate what we would term random variables.

In the cited study the authors show that AD2 has a normal distribution while AD

does not, and report the following 23 observations associated with scans of the left

thalamus.

108.7 107.0 110.3 110.0 113.6 99.2 109.8 104.5

108.1 107.2 112.0 115.5 108.4 107.4 113.4 101.2

98.4 100.9 107.1 107.1 108.7 102.5 103.3

Proof. We compute the σ1, σ2, σ3, σ4 for AD to obtain

σ1 = 106.835 , σ2 = 22.299 , σ3 = −0.162 , σ4 = −2.106

With these σ1 , σ2 , σ3 , σ4 , Karian, Dudewicz and McDonald (1996) fitted this data

by following Algorithm GLD-M of, using the entry at (|σ3|, σ4) = (0.15, 2.1) to obtain

the fit

GLD1(112.1335 , 0.06374 , 0.6387 , 0.05478)

36

with support [96.445, 127.822]. Here we appeal to adequate programming to deter-

mine the more precise fit

GLD2(102 . 8998 , 0.08060 , 0.1475 , 0.8041)

whose support is [ 90.493 , 115.307 ].

Figure 3.1 (a) shows the p.d.f.s of GLD1 (labeled (1)), and GLD2 (labeled (2)), and

a histogram of the data; Figure 3.1(b) shows the e.d.f. of the data with the d.f.s

of GLD1 and GLD2 (the former is not labeled; the latter is labeled by (2)). The

λ1 , λ2 , λ3 , λ4 of GLD1 and GLD2 seem to differ significantly; however, at least

visually, the p.d.f.s and d.f.s of the distributions appear to provide equally valid fits.

With the small sample size of this example, we are not able to perform a chi-square

test but we can partition the data into classes, such as

(0, 103), [103, 107), [107, 110.5), [110.5, 0),

whose respective frequencies are

7, 6, 5, 6,

and calculate the chi-square statistic of the two fits to obtain 1.3122 and 1.6405 for

GLD1 and GLD2 respectively

GLD1 : 9.9332 , 28.9050 , 49.9837 , 52.8583 ,

42.8412 , 29.6866 , 18.2559 , 10.0063 , 7.5298

GLD2 : 9.6062 , 25.1726 , 51.3265 , 58.5142,

37

Figure 3.1: Histogram of AD and GLD1 and GLD2 p.d.f.s, designated by (1) and (2)in (a); e.d.f.s of AD with the d.f.s of GLD1 and GLD2 in (b).

44.7824 , 27.8634 , 15.6573 , 8.34979 , 8.7277.

These give the following chi-square statistics and p-values for the two fits:

GLD1 : χ2statistic = 1.6000,

p-value = 0.8088

GLD2 : χ2statistic = 2.2661,

p-value = 0.6869.

Note that:

1- The p-value, which directly depends on a given sample, attempts to provide a

measure of the strength of the results of a test for the null hypotheses, in contrast to

a simple reject or do not reject in the classical approach to the test of hypotheses. If

the null hypothesis is true and the chance of random variation is the only reason for

sample differences, then the p-value is a quantitative measure to feed into the decision

38

making process as evidence. The following table provides a reasonable interpretation

of p-values:

P-value Interpretation

P < 0.01 very strong evidence against H0

0.01 < P < 0.05 moderate evidence against H0

0.05 < P < 0.10 suggestive evidence against H0

0.10 < P little or no real evidence against H0

39

3.4 Estimating the parameters of the generalized

lambda distribution; the least squares method

One of the problems involved in summarizing a set of data is to find a probabil-

ity distribution model that will fit the data well. The usual way of tackling this

problem is to use a flexible family of distributions. Families of distributions that

have been constructed to cover a wide range of distribution shapes include the Pear-

son system (Johnson and Kotz 1970), the Johnson system (Johnson 1949), and the

generalized lambda distribution (GLD) developed by Ramberg and Schmeiser (1972,

1974). Ramberg and Schmeiser used the GLD, which is a generalization of Tukey s

lambda distribution (Tukey 1960), to generate continuous random variables.

The least squares method for computing the parameters of the GLD (Ozturk and

Dale 1985) can be described as follows let

xi , i = 1, .., n

denote the ith order statistic of the data which is to be represented by the quantile

function Q(u) and let

ui, i = 1, , , ., n

denote the order statistic of the corresponding uniformly distributed random variable

F(X).

Definition 3.4.1. . [2] Order statistics

Let X1, X2, ........, Xn, denote a random sample of size n from a cumulative distribution

function f(.), where Yi are the Xi arranged in order of increasing magnitudes and

defined to the order statistics corresponding to the random sample X1, X2, ........, Xn.

40

The least squares method finds the values of λ for which the differences between

the observed and predicted order statistics are as small as possible, Thus , it minimizes

the function

G(λ) =n∑

i=1

[xi − λ1 +Zi

λ2

]2 (3.4.1)

where

Zi =1

λ3

[E(uλ3

(i))− 1]− 1

λ4

[E(1− u(i))λ4 − 1]

The formula for computing

E(uλ3

(i)) and E(1− u(i))λ4 and

E(uλ3

(i)) =Γ(n + 1) Γ(i + λ3)

Γ(i) Γ(n + λ3 + 1)and

E(1− u(i))λ4 =

Γ(n + 1) Γ(n− i + λ4 + 1)

Γ(n− i + 1)Γ(n + λ + 1)

Note that the function to be minimized in (Equation 3.4.1) depends on all four param-

eters. To avoid solving a computationally demanding minimization problem in four-

dimensional space,(λ1 , λ2) are decoupled from (λ3 , λ4) as in the moment-matching

method. We first assume that λ3 and λ4 are constant and solve the minimization

problem for λ1 and λ2. The values for λ1 and λ2, once obtained, are substituted into

the minimization function and then compute λ3 and λ4 .

We adopt this strategy based on the observation that the expression for G(λ) has λ1

41

and λ2 is in linear form, whereas λ3 and λ4 appear in nonlinear form. Thus, differ-

entiating G(λ) with respect to λ1 and λ2 and setting the resulting function equal to

zero, we obtain

λ1 = µx − bxzµz (3.4.2)

λ2 =1

bxz

where µx and µz denote the mean of sample data and the quantities Zi , respec-

tively, and the regression coefficient bxz is given by

bxz =

∑ni=1(xi − µx)(zi − µz)∑n

i=1(zi − µz)(3.4.3)

Inserting Equations (3.4.2) and (3.4.3) in Equation (3.4.1), we obtain, after some

rearrangement of terms,

G(λ3 , λ4) = (1− rxz(λ3 , λ4)2)

n∑i=1

(xi − µx)2

where rxz is the correlation coefficient between the quantities xi and zi . Thus, in

order to minimize G(λ3 , λ4) we need to maximize the quantity rxz(λ3 , λ4) or,

equivalently, minimize the function

H(λ3 , λ4) = −rxz(λ3 , λ4)2 (3.4.4)

Once λ3 and λ4 have been obtained by minimizing Equation (3.4.4), they can be in-

serted into Equations (3.4.2) and (3.4.5) in order to compute λ1 and λ2 Before leaving

this section, we again emphasize that maximizing the square of the correlation coef-

ficient rxz does not guarantee that the estimated parameters will yield a distribution

42

that closely matches the empirical distribution of the sample data.

We now describe an example that illustrates the least Square method.

3.5 Example

To illustrate the LS method, consider the ordered data set given in the Table.

3.836 3.909 3.935 3.943 3.975 4.008 4.019 4.023 4.051 4.060

4.105 4.109 4.124 4.183 4.207 4.233 4.248 4.307 4.332 4.395

4.405 4.411 4.462 4.486 4.495 4.495 4.509 4.552 4.599 4.606

4.614 4.633 4.645 4.665 4.657 4.683 4.770 4.841 4.921 4.962

4.966 4.993 5.003 5.091 5.118 5.121 5.139 5.244 5.249 5.396

5.445 5.457 5.496 5.539 5.567 5.662 5.659 5.683 5.873 5.932

5.963 5.977 6.041 6.061 6.125 6.159 6.572 6.653 6.762 6.932

7.269 7.391 7.510 7.862 7.863

.

The random sample was generated using the quantile function of the GLD for

which λ1 = 4.114 , λ2 = .1333 , λ3 = .0193 , and λ4 = .1588 to give values of µ = 5.0

, σ = 1.0 , α3 = 1.0 , and α4 = 4.0 for the mean, standard deviation, skewness, and

kurtosis, respectively.

The 75 observations have sample moments, skewness, and kurtosis

X = 5.109 S = 1.013 α3 = 1.009 α4 = 3.344

By using the tables of Ramberg et al. (1979), it can be shown that for the preceding,

combination of coefficients of skewness and kurtosis, corresponding values of λ3 and

λ4 do not exist see Oztrk A. Dale R.F. (1985). Hence the method of moments cannot

be used for this data set. Least squares estimates of λ3 and λ4 based on

Xi = λ1 +Qi

λ2

+ ei

43

Where Qi is defined as

Qi = E(uλ3

(i))− E(1− u(i))λ4

and the random variable ei has mean zero and variance σ2i , are obtained using the

Nelder-Mead simplex procedure for function minimization, where the objective func-

tion

ϕ(λ3, λ4) = 1− r2X Q

is minimized subject to the constraint that λ3 , λ4 > 0 . ( See Olsson 1974 for the

use of subroutine NELMIN for the Nelder-Mead procedure.) Then λ3 and λ4 > 0

are calculated from (4.1.3) . The corresponding estimates are

λ1 = 6.0217 λ2 = −0.4525 , λ3 = 8.2956 , λ4 = 0.9221

Defining the quantity Qi , as in

Xi = λ1 +Qi

λ2

+ ei,

Qi = E(uλ3

(i))− E(1− u(i))λ4

is also used to estimate the parameters following the same minimization procedure.

Parameter estimates are calculated to be

λ1 = 6.0235 λ2 = −0.4519 , λ3 = 7.9080 , λ4 = 0.9221

Chapter 4

Estimating the parameters of the

generalized lambda Distribution

The Percentiles Method for fitting generalized lambda distribution

. to data sets

The generalized lambda distribution, GLD(λ1, λ2, λ3, λ4) , is a four-parameter

family that has been used for fitting distributions to a wide variety of data sets. In

almost all cases the method of moments has been used to determine the parameters of

the GLD that fits a given data set, negating the possibility of applying those members

of the GLD family that do not possess the first four moments and yet may provide

superior fits to the data. In this chapter we develop a method for fitting a GLD

distribution to data that is based on percentiles rather than moments. This approach

makes a larger portion of the GLD family accessible for data fitting and eases some

of the computational difficulties encountered in the method of moments.

We now consider a GLD(λ1, λ2, λ3, λ4) fitting process that is based exclusively on

percentiles. The concept and name ”percentile” (also, ”quartile” and ”decile”) are

44

45

due to Galton (1875), who in his 1875 paper proposed to characterize a distribution

by its location (median), and its dispersion (half the interquartile ranger) and it fits a

GLD(λ1, λ2, λ3, λ4) distribution to a given dataset by specifying four percentile-based

sample statistics and equating them to their corresponding GLD(λ1, λ2, λ3, λ4) statis-

tics. The resulting equations are then solved for (λ1, λ2, λ3, λ4) with the constraint

that the resulting GLD be a valid distribution. To make the percentile approach an

acceptable alternative to the method of moments and to provide the necessary com-

putational support for the use of percentile-based fits, Dudewicz and Karian(2000)

give extensive tables for estimating the parameters of the fitted GLD(λ1, λ2, λ3, λ4)

distribution. These tables are reproduced in Appendix B.

There are three principal advantages to the use of percentiles:

1. There is a large class of GLD(λ1, λ2, λ3, λ4) distributions that have fewer than

four moments and these distributions are excluded from consideration when one uses

parameter estimation methods that require moments. On the occasions when mo-

ments do not exist or may be out of table range, percentiles can still be used to

estimate parameters and obtain GLD(λ1, λ2, λ3, λ4) fits.

2. The equations associated with the percentile method that we will consider are

simpler and the computational techniques required for solving them provide greater

accuracy.

3.The relatively large variability of sample moments of orders 3 and 4 can make it

difficult to obtain accurate GLD(λ1, λ2, λ3, λ4) fits through the method of moments.

46

4.1 The Use of Percentiles

For a given sample of independent observation, Xl, X2......., Xn, let πp denote the

(100p)th percentile of the data. πp is computed by first writing (n + l)p as r + ab

,

where r is a positive integer and ab

is a proper fraction, possibly zero. If Y1, Y2, ., ., Yn

are the order statistics of the observation Xl, X2......., Xn, , then πp can be obtained

from

πp = Yr +a

b( Yr+1 − Yr) (4.1.1)

where Y1 ≤ Y2 ≤ ...... ≤ Yn are the order statistics.

This definition of the (100p)th data percentile differs from the usual definition. Con-

sider, for example, p = 0.5 where the sample median is usually defined as

Mn = Yk if n = 2k

for some integer k and

Mn =(Yk + Yk+1)

2if n = 2k + 1

for some integer k. By contrast, the sample quantile of order 0.5 is usually defined

as Z0.5 = Y[0.5n]+1 where [0.5n] denotes the largest integer less than [0.5n] . Since the

sample quantile can be defined as a function of a single order statistic, it is mathe-

matically somewhat simpler.

The sample statistics that we will use are defined by

47

ρ1 = π0.5 (4.1.2)

ρ2 = π1−u − πu (4.1.3)

ρ3 =π0.5 − πu

π1−u − π0.5

(4.1.4)

ρ4 =π0.75 − π0.25

ρ2

(4.1.5)

where u is an arbitrary number between 0 and 1 . These statistics have the following

interpretations (where for ease of discussion we momentarily assume u = 0.1).

1. ρ1 is the sample median;

2. ρ2 is the inter-decile range, i.e., the range between the 10th percentile and 90th

percentile;

3. ρ3 is the left-right tail-weight ratio, a measure of relative tail weights of the left

tail to the right tail ( distance from median to the 10th percentile in the numerator

and distance from 90 percentile to the median in the denominator);

4. ρ4 is the tail-weight factor or the ratio of the inter-quartile range to the inter-

decile range, which cannot exceed 1 and measures the tail weight (values that are

close to 1 indicate the distribution is not greatly spread out in its tails, while values

close to 0 indicate the distribution has long tails). In the case of N(µ, σ2) , the normal

distribution with mean µ and variance σ2 , we have

48

ρ1 = µ , ρ2 = 2.56 σ , ρ3 = 1 , ρ4 =1.36

2.65= 0.53

This indicates, respectively, that the median of N(µ, σ2) , is µ , the middle 80 of

the probability is in the range of about two-and-a-half standard deviations from the

median, left and right tail weights are equal, and the inter-quartile range is 53 of the

inter-decile range From the definition of the GLD(λ1, λ2, λ3, λ4) inverse distribution

function (1.2.1), Dudewicz and Karian(1999). We now define ρ1 , ρ2 , ρ3 , ρ4 the

GLD counterparts of ρ1 , ρ2 , ρ3 , ρ4 as

ρ1 = Q(1

2) = λ1 +

(12)λ3 − (1

2)λ4

λ2

(4.1.6)

ρ2 = Q(1− u)−Q(u) =(1− u)λ3 − uλ4 + (1− u)λ4 − uλ3

λ2

(4.1.7)

ρ3 =Q(1

2)−Q(u)

Q(1− u)−Q(12)

=(1− u)λ4 − uλ3 + (1

2)λ3 − (1

2)λ4

(1− u)λ3 − uλ4 + (12)λ4 − (1

2)λ3

(4.1.8)

ρ4 =Q(3

4)−Q(1

4)

Q(1− U)−Q(12)

=(3

4)λ3 − (1

2)λ4 + (3

4)λ4 − (1

4)λ3

(1− u)λ3 − uλ4 + (1− u)λ4 − uλ3(4.1.9)

The following are direct consequences of the definitions of ρ1 , ρ2 , ρ3 , ρ4

1- Since λ1 may assume any real value, we can see from (4.1.6) that this is also true

for ρ1.

2- Since 0 < u < 14

, we have 34

< 1− u and from (4.1.7) we see that ρ2 > 0.

3- The numerator and denominator of ρ3 in(4.1.8) are both positive; therefore ρ3 > 0.

49

4- In (4.1.9), because of the restriction on u, the denominator of ρ4 must be greater

than or equal to its numerator, confining ρ4 to the unit interval.

In summary, the definitions of ρ1 , ρ2 , ρ3 , ρ4 lead to the restrictions:

−∞ < ρ1 < ∞ , ρ2 > 0 , ρ3 > 0 , 0 < ρ4 < 1 (4.1.10)

The fitting of a GLD(λ1 , λ2 , λ3 , λ4) to a given data set X1, X2, .., Xn is done

by solving the system of equations

ρi = ρi (i = 1, 2, 3, 4)

for λ1 , λ2 , λ3 , λ4

The definitions of ρ1 , ρ2 , ρ3 , ρ4 in (4.1.2) through (4.1.5) may have seemed strange

or arbitrary to this point. However, we now observe the main advantage of these

definitions.

the subsystem ρ3 = ρ3 and ρ4 = ρ4 involves only λ3 and λ4 , allowing us to first solve

this subsystem for λ3 and λ4 and then substitutes λ3 and λ4 in ρ2 = ρ2 to obtain λ2

from

λ2 =(1− u)λ3 − uλ4 + (1− u)λ4 − uλ3

ρ2

(4.1.11)

and finally, using the values of λ2 , λ3 and λ4 in ρ1 = ρ1 to obtain λ1 from

λ1 = ρ1 +(1

2)λ3 − (1

2)λ4

λ2

(4.1.12)

As we consider solving the system ρ3 = ρ3 and ρ4 = ρ4, it becomes necessary to give

u a specific value. For particular u we must have (n + 1)u ≥ 1 to be able to compute

50

πu ,π1−u and eventually ρ2 , ρ3 and ρ4 . If u is too small , say u = 0.01, then our

method will be restricted to large samples (n > 99 for the u = 0.01 case).

4.2 Estimation of GLD Parameters through a Method

of Percentiles

As was the case with the equations of Chapters 2 , ρ3 = ρ3 , ρ4 = ρ4 , cannot be

solved in closed form. We use an algorithm to obtain approximate solutions to these

equations. In this case the equations are simpler and good approximations with

Max(| ρ3 − ρ3 | , |ρ3 − ρ3 |) < 10−6

are generally obtained within 3 or 4 iterations.

Solutions for a given (ρ3 , ρ4) can be found in various regions of (λ3 , λ4) space

Depending on the precise values of ρ3 and ρ4 , as many as four solutions may exist in

just one region.

With the availability of the tables of Appendix B, the algorithm below shows how to

obtain numerical values of λ1 , λ2 , λ3 , and λ4 for a GLD fit.

Algorithm GLD-P: Fitting a GLD distribution to data percentile method.

GLD-P-1. Use (4.1.2) through (4.1.5) to compute ρ1 , ρ2 , ρ3 , ρ4.

GLD-P-2. Find the entry point in one or more of the tables of Appendix B

closest to (ρ3 , ρ4) ; if ρ3 > 1 , use ( 1ρ3

, ρ4) instead of (ρ3 , ρ4) .

GLD-P-3. Using the entry point from Step GLD-P-2, extract λ3 and λ4 , if ρ3 > 1

, interchange λ3 and λ4.

51

GLD-P-4. Substitute λ3 for λ3 and λ4 for λ4 in (4.1.7) to determine λ2.

GLD-P-5. Substitute λ2 for λ2 , λ3 for λ3 and λ4 forλ4 in (4.1.6) to obtain λ1 .

The tables of Appendix B can be used to obtain reasonable starting points for a

search that ultimately is likely to estimate λ1 , λ2 , λ3 , , and λ4 more accurately.

Examples that illustrate this below

Note: for specified ρ1 and ρ2 and we find five tables that give λ3, and λ4 values

for fitting GLD system by method of percentiles , in the regions designated by

T1 , T2 , T3 , T4 and T5 respectively , in Figure (4.1) below

Figure 4.1: (ρ3 , ρ4) -space covered by table 1,2 and 3 denoted by T1,T2 and T3

respectively

52

We now describe an example that illustrates the method of percentiles

4.3 Example

The data for this example (listed below) are given in example (2) of Karian and

Dudewicz (2000) .

1.99 -0.424 5.61 -3.13 -2.24 -0.014 -3.32 -0.837 -1.98 -0.120

7.81 -3.13 1.20 1.54 -0.594 1.05 0.192 -3.83 -0.522 0.605

0.427 0.276 0.784 -1.30 0.542 -0.159 -1.66 -2.46 -1.81 -0.412

-9.67 6.61 -0.598 -3.42 0.036 0.851 -1.34 -1.22 -1.47 -0.592

-0.311 3.85 -4.92 -0.112 4.22 1.89 -0.382 1.20 3.21 -0.648

-0.523 -0.882 0.306 -0.882 -0.635 13.2 0.463 -2.60 0.281 1.00

-0.336 -1.69 -0.484 -1.68 -0.131 -0.166 -0.266 0.511 -0.198 1.55

-1.03 2.15 0.495 6.37 -0.714 -1.35 -1.55 -4.79 4.36 -1.53

-1.51 -0.140 -1.10 -1.87 0.095 48.4 -0.998 -4.05 -37.9 -0.368

5.25 1.09 0.274 0.684 -0.105 20.6 0.311 0.621 3.28 1.56

.

Proof. We first attempt to obtain fits by using the moment-based methods of Chap-

ters 2 and compute σ1 , σ2 , σ3 , σ4 , to get

α1 = 0.346 , α2 = 49.491 , α3 = 1.867 , α4 = 31.392 ,

The (α23 , α4) that we have is well outside the range of the tables in Appendices A

, making it impossible to fit a distribution from the GLD family by the methods

moments. To obtain a percentile-based fit, we compute ρ1 , ρ2 , ρ3 , ρ4 ,

ρ1 = −0.1482 , ρ2 = 7.26 , ρ3 = 1.87 , ρ4 = 31.39 ,

53

and obtain two fits from Tables B-l and B-5, respectively, of Appendix B.

GLD1(−0.3848, 0.1260, 5.2455, 10.2631),

GLD5(−0.2830,−2.4471,−0.9008,−1.0802).

GLD5 turns out to be the superior fit. A assuring us that the, support of the resulting

fit will be (−∞,∞) ; by contrast, the support of GLD1 is [-8.3,7.6].

Although most of the data is concentrated on the interval [-6 , 6] , the range of the

data is [-37.9 , 48.4]. A histogram on [-37.9 , 48.4] would be so compressed that its

main features would not be visible. A slightly distorted histogram of the data (when

the 8 of the 100 observations outside of the interval [-6 , 6] are ignored) and the

GLD1 and GLD2 p.d.f. shown in Figure 4.2 (a) (the p.d.f. of GLD1 rises higher at

the center). Figure 4.2 (b) shows the e.d.f. of the data with d.f.s of GLD1 and GLD2.

When the data is partitioned into the intervals

(−∞,−3], (-3, -1.5], (0, .4], (-1.5, -.7], (-.7, -.4], (.4, .7],(.7, 1.5], (1.5, 3], (-.4,0] ,

(3,∞),we obtain observed frequencies of

10, 12, 11, 11, 14, 8, 8, 7, 6, 13.

and the expected frequencies for these intervals that result from GLD1 are

10.2541, 7.1461, 9.3964, 10.8408, 18.5659, 8.9770, 8.1299, 4.0927, 7.5986, 14.9985.

These lead to the chi-square goodness-of-fit statistic and corresponding p-value

of 9.7352 and 0.08310,

respectively. For GLD2, the expected frequencies are

10.2644, 9.7906, 14.5337, 8.6285, 12.2154,9.8868, 5.4195, 8.9350, 7.4654, 12.8607,

54

Figure 4.2: Histogram of data and the p.d.f.s of the fitted GLD1 and GLD5 (a); thee.d.f. of the data with the d.f.s of the fitted GLD1 and GLD2 (b).

and the resulting chi-square statistic and p-value are

4.5740 and 0.4700,

Justifying our earlier observation that GLD5 is better of the two fits .

Chapter 5

GLD Approximations to SomeWell Known Distributions

5.1 GLD Approximations to Some Well Known

Distributions by using a method of moment

The large variety of shapes that the GLD(λ1 , λ2 , λ3 , λ4) p.d.f. can attain. For the

GLD(λ1 , λ2 , λ3 , λ4)to be useful for fitting distributions to data, it should be able

to provide good fits to many of the distributions the data may come from. In this

section we see that the GLD(λ1 , λ2 , λ3 , λ4) fits well many of the most important

distributions.

5.1.1 The Normal Distribution

The normal distribution, with mean µ and variance σ2 ,(σ > 0) ,

N(µ , σ2) , has p.d.f.

f(x) =1

σ√

2πexp−

(x−µ)2

2σ2 ,−∞ < x < ∞

55

56

Since all normal distributions can be obtained by a location and scale adjustment to

N (0, 1), we consider aGLD(λ1 , λ2 , λ3 , λ4) fit to N (0, 1) for which

α1 = 0 , α2 = 1 , α3 = 0 , α4 = 3 ,

Appendix A suggests (λ3, λ4) = (0.13 , 0.13) as a starting point for our invocation of

adequate programming support which yields

GLD(0 , 0.1975 , 0.1349 , 0.1349)

Figure 5.1:

Fig. 5.2 shows a plot of N(0,1) and the GLD approximation, as you can see the

GLD provides a very good fit with the two distributions almost indistinguishable

except near x = 0 where the GLD rises slightly higher. If we denote the pdf of N(0,1)

by f(x) and that of the GLD by f(x) we get

Max|f(x)− f(x)| = 0.002813

So we can say that the GLD is accurate to within 0.002813. Next we shall look at the

c.d.f of N(0,1) and the corresponding GLD. Unfortunately we cannot find the cdf of

57

the Normal distribution exactly however there are a number of good approximations,

we have implemented the approximation, which returns the c.d.f of N(0,1) ,F (x) to

a minimum accuracy of 7.5× 10−8 . The graph obtained is shown in Fig 5.3

Figure 5.2:

From the graph above we can see that the GLD c.d.f provides a very good fit to

that of the c.d.f of N(0,1), in fact the two curves are so close together you can hardly

tell them apart. We also found

Max|F (x)− F (x)| = 0.00118

Which shows that the GLD provides an even better fit to the c.d.f than did the

p.d.f, these results together show that the GLD provide a very good fit to the normal

distribution.

58

5.1.2 The Uniform Distribution

The pdf of the uniform distribution on the interval [a,b], U(a,b) is as follows

f(x) =

1

b−a, ifa < x < b

0 , otherwise

where

µ =a + b

2, σ2 =

(b− a)2

12, α3 = 0 , α4 =

9

5

Looking at U (0,1) for which µ = 0.5 and σ2 = 0.08333 we get GLD parameter

values of λ1 = 0.5 , λ2 = 2 , λ3 = 1 , λ4 = 1. In this case the GLD will be a perfect

fit since these values of λ1 , λ2 , λ3 , λ3 in (1.2 .2) yield Q(u) = u and thus F(x) =

u and f(x) =1, which are the c.d.f and p.d.f of U(0,1). Fig. 5.4 shows the resulting

graph for the c.d.f which you can see is exact.

Figure 5.3:

59

5.1.3 The Exponential Distribution

The p.d.f of the Exponential distribution is given as

f(x) =

1β

exp−xβ , for x > 0

0 otherwise

µ = β , σ2 = β2 , α3 = 2 , α4 = 6

The c.d.f is defined as:

F (x) = 1− exp−xβ

We can see that the values of α3 , α4 remain unchanged for whatever value of β

Therefore, we can use the values of λ3 and λ4 that we obtain for say β = 1 to all

other exponential distributions. Using β = 1 , we haveµ = 1 and σ2 = 1 which gives

λ1 = 0.155 , λ2 = 1.0300 , λ3 = 6.784 , λ4 = 0.0010.

Figure 5.4:

60

We can see from Fig. 5.5 that the GLD is slightly higher initially but still provides

a good fit, this is illustrated by:

Max|f(x)− f(x)| = 0.038867.

If we now look at how the c.d.fs compare, we find max|F (x) − F (x)| = 0.0118, and

from Fig6.6 we see again that the GLD provides a good approximation.

Figure 5.5:

In the next section we approximate the above three distribution with a percentile

method

61

5.2 GLD Approximations of Some Well-Known Dis-

tribution by using a method on percentiles

In this section we use the percentile-based method described in chapter5 fit GLD

distributions to some of the important distributions. That in most cases we will be

able to fit several GLD fits to a given distribution, Of course, this does not mean that

all the fits will be good ones. In the following sections as we consider two distribu-

tions. We will discover that, generally, the percentile method produces three fits of

which one is clearly superior to the others. In our first example we fit the N (0,1)

distribution, obtain three fits, and give all the details associated with each fit.

5.2.1 The Normal Distribution

for the normal distribution the percentile statistics, ρ1 , ρ2 , ρ3 , ρ4 are

ρ1 = µ , ρ2 = 2.563σ , ρ3 = 1 , ρ4 = 0.526

Locating (ρ3 , ρ4) in Figure 6.8, we see that three fits are available from Tables B-l,

B-2, and B-3. Through the use of these table entries, we get the three fits GLD1 ,

GLD2 and GLD3 to N(0,1), associated with Tables B-l, B-2, and B-3, respectively.

These fits are

GLD1(−0.858 , 0.397 , 1.554 , 15.477)

GLD2(0 , 0.546 , 3.389 , 3.390)

GLD3(0 , 0.214 , 0.149 , 0.149)

62

with respective supports

[−3.38 , 1.66] , [−1.83 , 1.83] , and [−4.67 , 4.67]

. GLD1 seems to be asymmetric (even though it has ρ3 = 1 and hence, as measured by

the left-right tail-weight is ρ3 -symmetric) because ; therefore, may not be a suitable

fit for N (0, 1), depending on why N (0, 1) is chosen in a particular application. GLD2

is symmetric but its support is much too confined to be a suitable fit for N (0, 1).

GLD1 , GLD2 , GLD3 , and the N (0, 1) p.d.f.s are shown in Figure 5.7 and the GLD

s is marked by ”(1)”, ”(2)”

Figure 5.6: The GLD1, GLD2, and GLD3 fits to N(0,1) marked by ”(1),”(2),”and”(3),” respectively; the N(0,1) and GLD3 p.d.f.s cannot distinguished.

and ”(3).” In the figure above N(0,1) p.d.f. cannot be seen as a distinct curve

because it coincides (visually) with GLD3. The support of the moment-based GLD

fit of N (0, 1) obtained was [-5.06, 5.06] however, this may not be a problem in many

applications (see discussion at the beginning of Chapter 2) because the N(0,1) tail

probability outside of the GLD3 support is approximately 3×10−6. We complete our

63

first check for the N (0, 1) fits by noting that

Sup |f1(x)− f(x)| = 0.1547

Sup |f1(x)− f(x)| = 0.0862|

Sup |f1(x)− f(x)| = 0.00022

where fi(x) are the GLDi p.d.f.s and f(x) is the p.d.f. of N(0,1). There are perceptible

differences between the graphs of the GLD1 and GLD2 d.f.s and the d.f. of N(0, 1).

However, the graphs of the N(0,1) and GLD3 d.f.s appear to be identical and to

complete our second check, we note that

Sup |F1(x)− F (x)| = 0.0478

Sup |F1(x)− F (x)| = 0.0330

Sup |F1(x)− F (x)| = 0.0005

where Fi(x) are the GLDi estimated commutative distribution function F (x) and

F(x) is the c.d.f. of computed from the normal distribution N(0,1).

The result is, as indicated in the introductory paragraph of this section, we obtained

several fits, with one of the fits, GLD3 , clearly superior to the others.

64

5.2.2 The Uniform Distribution

The values of ρ1 , ρ2 , ρ3 , ρ4 for the uniform distribution on the interval (a, b) are

ρ1 =1

2(a + b) , ρ2 =

4

5(a + b) , ρ3 = 1 , ρ4 =

5

8

We can see from Figure 4.1 that it is possible to find fits from Tables B-l, B-2,

B-3, and B-4. Computations associated with the four entries, when a = 0 and b = 1.

Obtained from Tables B-l through B-4, yield the following fits, respectively.

GLD1(0.5309× 10−7 , 1.0000 , 1.0000 , 154.5288),

GLD2(0.5000, 2.0000, 2.0000, 2.0000),

GLD3(0.5000, 2.0000, 1.0000, 1.0000),

GLD4(−0.2552× 10−12 , 1.0000 , −0.1112× 10−12 , 1.0000).

We observe that GLD1 lacks symmetry, as did one of the fits to N (0, 1) . For

either GLD2 or GLD3, the substitution of λ1 , λ2 , λ3 , λ4 into Q(y), the inverse

distribution function that defines the GLD(λ1 , λ2 , λ3 , λ4) (see (1.2.1)), yields

Q(y) = y This not only establishes that GLD2 and GLD3 are the same, but that

this common fit is a perfect one that matches the inverse distribution function of the

uniform distribution on (0,1). See the latter part for a discussion of how to obtain

a GLD fit to the general uniform distribution on the interval (a, b) from a fit of the

uniform distribution on (0,1).

5.2.3 The Exponential Distribution

The d.f. of the exponential distribution is

F (x) =

∫ x

0

f(t)dt = 1− exp−xβ , for x > 0

65

and 0 for x < 0, where f(t) is the p.d.f. of the exponential distribution . The percentile

function, Q(x), of this distribution is Q(x) = −β(1− x) .

Since for 0 < p < 1 , πp , the (100p)-th percentile, is characterized by πp= Q(p), ”we

can easily compute π0.I , π0.25, π0.5, π0.75; and π0.9 to obtain the ρ1 , ρ2 , ρ3 , ρ4 the

exponential distribution with parameter β. These are

ρ1 = β ln 2 , ρ2 = 2β ln 3 , ρ3 =ln 9

ln 5− 1 = 0.3652 , ρ4 =

1

2

Since for all values of β, (ρ3, ρ4), rounded to two decimals, is (0.37,0.50), Figure 6.8

indicates the possibility of solutions from Tables B-l and B-2. When β = 3, these

lead, through the use of Algorithm GLD-P, to the two fits

GLD1(1.0498 , 0.1250 , 2.9578 , 22.5624),

GLD2(5.0180 , 0.1967 , 5.6153 , 0.7407),

with respective supports [-6.95,9.05] and [-0.066,10.10]. Figure 4.1 shows the p.d.f.s

of GLD1, GLD2, and the exponential distribution with β = 3.

Figure 5.7: The p.d.f.s of the exponential distribution (with θ = 3), GLD1 markedby ”(I),” and GLD2 marked by ”(2)”.

66

5.3 Application

Example

Data giving shows the dose of cytoxan (CTX) which used to treat various types of

cancer in the oncology department in European Gaza Hospital. The dose calculated

by body mass area multiplies in prescribed dose of that drug according to the patient.

The total dose unit is measured by mg. The data illustrated the prescribed dose for

40 patients.

370 390 420 470 560 590 600 630 650 700

720 750 790 800 820 840 850 850 850 860

870 880 890 920 930 950 960 970 980 1000

1020 1030 1050 1050 1080 1100 1250 1300 1370 1450

Proof. We compute α1 , α2 , α3 , α4 for (CTX) dose

α1 = 864 , α2 = 63137.44 , α3 = 0.081 , α4 = 0.132

with these α1 , α2 , α3 , α4 , we cant fitted this data by Algorithm GLD-M since

( |α3| , α4 ) out of given tables by Karian and Dudewicz (1999).

So we compute ρ1 , ρ2 , ρ3 , ρ4 , by Algorithm GLD-P

ρ1 = 865 , ρ2 = 765 , ρ3 = 0.275 , ρ4 = 0.41

From the location of ( 0.270, 0.41) in Figure5.1 we should expect two solution to arise

from entries of Tables B-1, and B-5 of Appendix B . These solutions, extracted from

67

the tables, have, respectively

(λ3 , λ4) = (4.1503 , 22.1414) (−0.842 , −0.5111)

and then by using

ρ1 = Q(1

2) = λ1 +

(12)λ3 − (1

2)λ4

λ2

ρ2 = Q(1− u)−Q(u) =(1− u)λ3 − uλ4 + (1− u)λ4 − uλ3

λ2

we get GLD1 ( 864.923 , 1.11× 10−3 , 4.1503 , 22.1414)

With support [ 863.555 , 866.290 ]

And GLD5 ( 899.99 , -0.0105 , -0.842 , 0.5111)

With support [ 804.75 , 995.228 ]

GLD5 in spite of it’s support is the better of two fits

Appendices

Table A-l for GLD Fits: Method of Moments

For specified α3 and α4, Table A-l gives the λ1(0, 1), λ2(0, 1),λ3 , and λ4 for GLD

fits. In order to give a sufficient number of significant digits, superscripts are used to

designate factors of .

Thus, an entry of as designates a×10s. For example, for (α3, α4) close to (0.15,4.1),

the table gives (λ1(0, 1), λ2(0, 1)), λ3, λ4) = (−0.72681, 0.16031, 0.83782, 0.95642).

The proper interpretation of these table entries is (λ1(0, 1), λ2(0, 1)), λ3, λ4) = (-

0.07268,0.01603,0.008378,0.009564). With few exceptions, Table A-l provides values

of λ1, λ2, λ3, λ4 for which

Max |αi − αi| < 10−5

The exceptions occur when very small changes in λ3 or λ4 cause large variations in

α3 and α3. a situation that arises when λ3 or λ4 gets close to 0. When |λ3| < 10−2

or |λ4| < 10−2, we generally have

Max |αi − αi| < 10−3

68

69

In the rare instances where |λ3| < 10−4 or |λ4| < 10−4, we can only be assured of

Max |αi − αi| < 10−2

The entries of this part of Table A-l are from ”The Extended Generalized Lambda

Distribution (EGLD) System for Fittin g Distributions to Data with Moments, II:

Tables” by E.J. Dudewicz and Z.A. Kari-an, American Journal of Mathematical and

Management Sciences, V. 16, 3 and 4 (1996), pp. 287-307, copyright 1996 by Ameri-

can Sciences Press, Inc., 20 Cross Road, Syracuse, New York 13224-2104. Reprinted

with permission.

71

72

73

74

75

76

77

78

79

80

81

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