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Chiang Mai J. Sci. 2016; 43(3) : 671-681http://epg.science.cmu.ac.th/ejournal/Contributed Paper

Upper Bounds of Generalized p-values for Testing the Coefficients of Variation of Lognormal DistributionsRada Somkhuean, Suparat Niwitpong and Sa-aat Niwitpong*Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand.*Author for correspondence; e-mail: [email protected]

Received: 3 February 2014Accepted: 24 April 2014

ABSTRACT The coefficient of variation is defined as the ratio of the standard deviation to the mean. It has been widely used to compare the variability between groups of observations. Recently, there have been many researchers proposed tests and confidence intervals for the coefficients of variation of data from normal distribution. However, this paper presents the new generalized p-values for testing the single coefficient of variation and the difference between coefficients of variation for lognormal distribution using the generalized p-value developed by Tsui & Weerahandi [16]. Moreover, for each generalized p-value, we derive a closed form expression of its upper bound. Numerical computations illustrate the theoretical results. Keyword : upper bound, generalized p-value, coefficient of variation 1. INTRODUCTION

Statistical inference for lognormal distributions has been widely used in economics, healthcare, and biological and environmental research (Zou et al. [1], Aitchison et al. [2], Crow et al. [3], Krishnamoorth & Mathew [4]). In particular, inference based upon the difference between lognormal means has been presented by Abdollahnezhad et al. [5]. The confidence intervals and hypotheses testing for parameters of means of two independent lognormal distributions, have been presented by Krishnamoorthy & Mathew [4], Chen & Zhou [6], Gill [7], and Gupta & Li [8]. Pardo & Pardo [9] presented new statistics, based upon Rényi's divergence, in order to test the equality of coefficient variations. Forkman [10] proposed a new test statistic ‘F’, which is approximately ‘F’ test. Curto & Pinto [11] derived a hypothesis test, in order to compare the coefficients of variation based upon asymptotically normal distribution, using the bootstrap method. Amirin [12] used the bootstrap method to test for the comparisons of coefficients of variation under assumed normal populations. Bai et al. [13] used the mean-variance ratio (MVR) test in order to test the ratio of coefficients of variation in small samples. Niwitpong [14], and Buntao & Niwitpong [15], proposed confidence intervals for the coefficients of variation of lognormal distributions with restricted parameter spaces. In this paper, the generalized p-value approach, based upon Tsui and Weerahandi’s approach [16], is used to find new generalized p-values, and that is in order to test: 1) the single coefficient of variation and 2) the difference between coefficients of variation in lognormal distributions. Further, we applied a problem described by Tang & Tsui [17], in order to construct the upper boundaries of the mentioned generalized p-values. In section 2, we provide the paths to some basic steps used to construct the generalized p-values. The processes of deriving the upper boundaries, as previously mentioned, are then presented in section 3. Numerical results are displayed in section 4. Finally, in section 5, our conclusion is drawn.

2. BASIC CONCEPT A GENERALIZED P-VALUES AND PROBABILITY DENSITY FUNCTION FOR LOGNORMAL DISTRIBUTION

In this section, we reviewed the probability density function for lognormal distribution, and their mean, variance and coefficient of variation. Moreover, we reviewed the generalized p-value to test the hypotheses of the single coefficient of variation and the difference between the coefficient of variation for lognormal distributions.

Chiang Mai J. Sci. 2016; 43(3)672

ABSTRACT The coefficient of variation is defined as the ratio of the standard deviation to the mean. It has been widely used to compare the variability between groups of observations. Recently, there have been many researchers proposed tests and confidence intervals for the coefficients of variation of data from normal distribution. However, this paper presents the new generalized p-values for testing the single coefficient of variation and the difference between coefficients of variation for lognormal distribution using the generalized p-value developed by Tsui & Weerahandi [16]. Moreover, for each generalized p-value, we derive a closed form expression of its upper bound. Numerical computations illustrate the theoretical results. Keyword : upper bound, generalized p-value, coefficient of variation 1. INTRODUCTION

Statistical inference for lognormal distributions has been widely used in economics, healthcare, and biological and environmental research (Zou et al. [1], Aitchison et al. [2], Crow et al. [3], Krishnamoorth & Mathew [4]). In particular, inference based upon the difference between lognormal means has been presented by Abdollahnezhad et al. [5]. The confidence intervals and hypotheses testing for parameters of means of two independent lognormal distributions, have been presented by Krishnamoorthy & Mathew [4], Chen & Zhou [6], Gill [7], and Gupta & Li [8]. Pardo & Pardo [9] presented new statistics, based upon Rényi's divergence, in order to test the equality of coefficient variations. Forkman [10] proposed a new test statistic ‘F’, which is approximately ‘F’ test. Curto & Pinto [11] derived a hypothesis test, in order to compare the coefficients of variation based upon asymptotically normal distribution, using the bootstrap method. Amirin [12] used the bootstrap method to test for the comparisons of coefficients of variation under assumed normal populations. Bai et al. [13] used the mean-variance ratio (MVR) test in order to test the ratio of coefficients of variation in small samples. Niwitpong [14], and Buntao & Niwitpong [15], proposed confidence intervals for the coefficients of variation of lognormal distributions with restricted parameter spaces. In this paper, the generalized p-value approach, based upon Tsui and Weerahandi’s approach [16], is used to find new generalized p-values, and that is in order to test: 1) the single coefficient of variation and 2) the difference between coefficients of variation in lognormal distributions. Further, we applied a problem described by Tang & Tsui [17], in order to construct the upper boundaries of the mentioned generalized p-values. In section 2, we provide the paths to some basic steps used to construct the generalized p-values. The processes of deriving the upper boundaries, as previously mentioned, are then presented in section 3. Numerical results are displayed in section 4. Finally, in section 5, our conclusion is drawn.

2. BASIC CONCEPT A GENERALIZED P-VALUES AND PROBABILITY DENSITY FUNCTION FOR LOGNORMAL DISTRIBUTION

In this section, we reviewed the probability density function for lognormal distribution, and their mean, variance and coefficient of variation. Moreover, we reviewed the generalized p-value to test the hypotheses of the single coefficient of variation and the difference between the coefficient of variation for lognormal distributions.

2.1 Probability Density Function for Longormal Distribution Let X be a random variable having a lognormal distribution, and let 1 and 2

1 denote

the mean and variance of ln X respectively, so that 21 1ln ~ ,Z X N . The

probability density function of X is

2

12 2

1 1 11

ln1exp ; 0

, , 22

0 ; ,

xif x

f x x

otherwise

(1)

where 1 and 21 denote the mean and variance of ln X respectively, so that

21 1ln ~ ,Z X N . In particular, the mean, variance and the coefficient of variation

for lognormal distribution are given by, see e.g., Niwitpong [14],

2

11

2 2

1 1 1

2

1

exp exp2

( ) exp 2 exp 1

exp 1,

E X E Z

Var X

CV

where CV denotes the coefficient of variation of X which is computed from /Var X E X .

2.2 Generalized P-values The concept of the generalized p-value has been introduced by Tsui & Weerahandi [16] and Weerahandi [18]. We first briefly review some basic steps to construct the generalized p-value for testing hypothesis problems. Let X be a random variable with a density function |f X , where , , is the parameter of interest, and is a nuisance parameter. Suppose we have the following hypothesis to test:

0 0:H vs 1 0:H where 0 is a specified quantity. Let x be a particular observed sample. The generalized test

variable, , ,T X x , satisfies the following three conditions:

(i) For fixed x and , , the distribution , ,T X x is free from the nuisance parameter .

(ii) , ,obst T x x is free from any unknown parameter.

(iii) For fixed x and , if , ,T X x is either stochastically increasing or decreasing in for any given t.

Under the above conditions, if , ,T X x is a stochastically increasing test variable, then the subset of space an extreme region C consisting of all the samples X that are as extreme as the observed x . Usually, C is of the form : , , 0 .C X T X x Given the observed sample x, the generalized p-value is defined as

0 0

sup | sup : , , 0 ,H H

p x P X C P X T X x

for further details and for several applications based on the generalized p-value, we refer to the book by Weerahandi [18]. Moreover, Tsui & Weerahandi [16] used the generalized p-value p x for the Behrens-Fisher problem of testing the difference of two independent normal distribution means with


1 denote



2

12 2

1 1 11

ln1exp ; 0

, , 22

0 ; ,

xif x

f x x

otherwise

(1)




2

11

2 2

1 1 1

2

1

exp exp2

( ) exp 2 exp 1

exp 1,

E X E Z

Var X

CV









0 0

sup | sup : , , 0 ,H H

p x P X C P X T X x

for further details and for several applications based on the generalized p-value, we refer to the book by Weerahandi [18]. Moreover, Tsui & Weerahandi [16] used the generalized p-value p x for the Behrens-Fisher problem of testing the difference of two independent normal distribution means with

Chiang Mai J. Sci. 2016; 43(3) 673


1 denote



2

12 2

1 1 11

ln1exp ; 0

, , 22

0 ; ,

xif x

f x x

otherwise

(1)




2

11

2 2

1 1 1

2

1

exp exp2

( ) exp 2 exp 1

exp 1,

E X E Z

Var X

CV









0 0

sup | sup : , , 0 ,H H

p x P X C P X T X x

for further details and for several applications based on the generalized p-value, we refer to the book by Weerahandi [18]. Moreover, Tsui & Weerahandi [16] used the generalized p-value p x for the Behrens-Fisher problem of testing the difference of two independent normal distribution means with possibly unequal variances. Later, Tang & Tusi [17] extended the works of Weerahandi [18], Gamage & Weerahadi [19] to derived the formula of the upper bound r of the generalized p-value p x which is in the form of, see also e.g., Kabaila & Lloyd [20],

.P p x r r In this paper, we also extend Tang & Tsui [17]’s work to find upper bounds of generalized p-values p x for the testing hypotheses of the coefficients of variation for lognormal distributions of the single coefficient of variation and the difference between coefficients of variation. 3. UPPER BOUNDS OF GENERALIZED P-VALUES FOR COEFFICIENTS OF VARIATION FOR LOGNORMAL DISTRIBUTIONS Let 1 2, , ..., nX X X X and 1 2, , ..., mY Y Y Y be two independent samples from

a lognormal distributions, and let 21 1ln ~ ,Z X N , 2

2 2ln ~ ,G Y N ,

which the coefficients of variation are xk and yk where

2 21 2

1 2, , exp , exp2 2x y

Var X Var Yk k E X E Y

E X E Y

2 2 2 21 1 1 2 2 2exp 2 exp 1 , exp 2 exp 1Var X Var Y

and the interest parameters are

21 1exp 1xk and 2 2

2 1 2exp 1 exp 1x yk k

for test the null hypotheses, 01 1 01:H vs 1 1 01:aH and 02 2 02:H vs 2 2 02:aH ,

the sufficient statistics of involving in this problem are 2 21 2, , ,Z G S S , where

222 21 2

1 1 1 1

1 1 1 1, , ,

1 1

n m n m

i i i ji j i j

Z Z G G S Z Z S G Gn m n m

.

The distributions of the underlying random variables are given by

2 22 22 21 21 2

1 2 1 12 21 2

1 1~ , , ~ , , ~ , ~ ,n m

n S m SZ N G N

n m

(2)

and the randon variables 21, ,Z G S and 2

2S are all independent. We denote 1 2, , ..., nD X X X ,

1 2, , ..., nd x x x and 1 2 1 2, , ..., , , , ...,n mQ X X X Y Y Y , 1 2 1 2, , ..., , , , ..., ,n mq x x x y y y where ,d q are respectively the vector of observed sample vectors of D and .Q Let

2 21 2, , ,z g s s be the observed value of the sufficient statistic 2 2

1 2, , , .Z G S S Following Tang

& Tsui [17], the repeated sampling, 2 21 2, , ,z g s s follows the same probability distributions

as (2). CASE I: The Hypothesis Testing of Single Coefficient of Variation We interested the hypothesis testing of the single coefficient of variation for lognornal distribution: 01 1 01:H vs 1 1 01: .aH (3) The parameter of the single coefficient of variation for lognormal distribution is

21 1exp 1xk .

A generalized test variable for 1 is

Chiang Mai J. Sci. 2016; 43(3)674

possibly unequal variances. Later, Tang & Tusi [17] extended the works of Weerahandi [18], Gamage & Weerahadi [19] to derived the formula of the upper bound r of the generalized p-value p x which is in the form of, see also e.g., Kabaila & Lloyd [20],



2 2ln ~ ,G Y N ,


2 21 2

1 2, , exp , exp2 2x y


E X E Y



21 1exp 1xk and 2 2




222 21 2

1 1 1 1

1 1 1 1, , ,

1 1

n m n m

i i i ji j i j


.


2 22 22 21 21 2

1 2 1 12 21 2

1 1~ , , ~ , , ~ , ~ ,n m

n S m SZ N G N

n m

(2)








21 1exp 1xk .


possibly unequal variances. Later, Tang & Tusi [17] extended the works of Weerahandi [18], Gamage & Weerahadi [19] to derived the formula of the upper bound r of the generalized p-value p x which is in the form of, see also e.g., Kabaila & Lloyd [20],



2 2ln ~ ,G Y N ,


2 21 2

1 2, , exp , exp2 2x y


E X E Y



21 1exp 1xk and 2 2




222 21 2

1 1 1 1

1 1 1 1, , ,

1 1

n m n m

i i i ji j i j


.


2 22 22 21 21 2

1 2 1 12 21 2

1 1~ , , ~ , , ~ , ~ ,n m

n S m SZ N G N

n m

(2)








21 1exp 1xk .


21 1 1, , exp 1T X x

2

2112

1

exp 1sS

2

11exp 1

n sU

, where 21~ nU . (4)

It is easy to see that 1 1, ,T X x in (4) satisfies conditions (i)-(iii) in section 2.2.

The generalized p-value, p d is defined, under the null hypothesis 01H , to be

1 01

1 1 1 1 1 1 1 1sup , , , , , , , ,H

p d P T X x T x x P T X x T x x

(5)

Following (5) the generalized p-value for (3) can be defined as

1 1 1 1, , , ,p d P T X x T x x

2

101

1exp 1

n sP

U

2

2101

1exp 1

n sP

U

22101

1ln 1

n sP

U

21

201

1ln 1n s

P U

21

201

1ln 1n s

P U

21

1 201

1ln 1U n

n sE

, (6)

where .UE is an expectation operator with respect to 2

2112

1

1~ n

n SU

and 1 .n is

a cdf of the chi-square distribution with 1n degrees of freedom. Theorem 1. Let 2

1 1nf u u then f u is a convex function of u.

Proof: We have f u f h u , 21h u u and u be the probability density function

of u. Hence

2f h u f h u h u h u h u h u h u h u

since 0u then 0h u . Hence 0h u and 0h u . Moreover

21 0h u h u . Hence 0f h u , and f u is convex in u .

21 1 1, , exp 1T X x

2

2112

1

exp 1sS

2

11exp 1

n sU

, where 21~ nU . (4)



1 01

1 1 1 1 1 1 1 1sup , , , , , , , ,H


(5)


1 1 1 1, , , ,p d P T X x T x x

2

101

1exp 1

n sP

U

2

2101

1exp 1

n sP

U

22101

1ln 1

n sP

U

21

201

1ln 1n s

P U

21

201

1ln 1n s

P U

21

1 201

1ln 1U n

n sE

, (6)


2112

1

1~ n

n SU

and 1 .n is




of u. Hence




Chiang Mai J. Sci. 2016; 43(3) 675

21 1 1, , exp 1T X x

2

2112

1

exp 1sS

2

11exp 1

n sU

, where 21~ nU . (4)



1 01

1 1 1 1 1 1 1 1sup , , , , , , , ,H


(5)


1 1 1 1, , , ,p d P T X x T x x

2

101

1exp 1

n sP

U

2

2101

1exp 1

n sP

U

22101

1ln 1

n sP

U

21

201

1ln 1n s

P U

21

201

1ln 1n s

P U

21

1 201

1ln 1U n

n sE

, (6)


2112

1

1~ n

n SU

and 1 .n is




of u. Hence




21 1 1, , exp 1T X x

2

2112

1

exp 1sS

2

11exp 1

n sU

, where 21~ nU . (4)



1 01

1 1 1 1 1 1 1 1sup , , , , , , , ,H


(5)


1 1 1 1, , , ,p d P T X x T x x

2

101

1exp 1

n sP

U

2

2101

1exp 1

n sP

U

22101

1ln 1

n sP

U

21

201

1ln 1n s

P U

21

201

1ln 1n s

P U

21

1 201

1ln 1U n

n sE

, (6)


2112

1

1~ n

n SU

and 1 .n is




of u. Hence




21 1 1, , exp 1T X x

2

2112

1

exp 1sS

2

11exp 1

n sU

, where 21~ nU . (4)



1 01

1 1 1 1 1 1 1 1sup , , , , , , , ,H


(5)


1 1 1 1, , , ,p d P T X x T x x

2

101

1exp 1

n sP

U

2

2101

1exp 1

n sP

U

22101

1ln 1

n sP

U

21

201

1ln 1n s

P U

21

201

1ln 1n s

P U

21

1 201

1ln 1U n

n sE

, (6)


2112

1

1~ n

n SU

and 1 .n is




of u. Hence




Theorem 2. The upper bound of p d in (6) takes the form 1

1 1 1n np d k r

where 0 0.5r , 2

011 2

1

ln 1k

and 1

1 .n the inverse function of 1 .n .

Proof: From (6)

21

1 201

1ln 1U n

n sp d E

21

1 201ln 1U n

UE

2

011 1 2

1 1

ln 1,U n

UE k

k

.

For any 0.5r and p d r . Hence, by theorem 1, 11

U n

Uf U E

k

is convex in U .

By Jensen’s inequality 1Up d E f U f E U f n 1 1

1

1n

np d

k

.

For 0 0.5r , we have 1: :d dp d p d r P d p d r

1P p d r

1

1

1n

nP r

k

11

1

1n

nP r

k

11 11 nP n r k

2 2

11 11 12 2

1 1

1 n

s sP n r k

2

1 11 1 2

1n

sP U r k

2

1 11 1 1 2

1n n

sE r k

21 1

1 1 1 21

n n

sr E k

,

by Jensen’s inequality 11 1 1n nk r

where 2

011 2

1

ln 1k

and 01 1e . (7)

CASE II: The Hypothesis Testing of Diference Between Coefficients of Variation The hypothesis testing of the difference between coefficients of variation for lognormal distributions is set in the form:

02 2 02:H vs 2 2 02: .aH (8)

Theorem 2. The upper bound of p d in (6) takes the form 11 1 1n np d k r

where 0 0.5r , 2

011 2

1

ln 1k

and 1


Proof: From (6)

21

1 201

1ln 1U n

n sp d E

21

1 201ln 1U n

UE

2

011 1 2

1 1

ln 1,U n

UE k

k

.


U n

Uf U E

k

is convex in U .


1

1n

np d

k

.


1P p d r

1

1

1n

nP r

k

11

1

1n

nP r

k

11 11 nP n r k

2 2

11 11 12 2

1 1

1 n

s sP n r k

2

1 11 1 2

1n

sP U r k

2

1 11 1 1 2

1n n

sE r k

21 1

1 1 1 21

n n

sr E k

,


where 2

011 2

1

ln 1k

and 01 1e . (7)


02 2 02:H vs 2 2 02: .aH (8)

Chiang Mai J. Sci. 2016; 43(3)676


where 0 0.5r , 2

011 2

1

ln 1k

and 1


Proof: From (6)

21

1 201

1ln 1U n

n sp d E

21

1 201ln 1U n

UE

2

011 1 2

1 1

ln 1,U n

UE k

k

.


U n

Uf U E

k

is convex in U .


1

1n

np d

k

.


1P p d r

1

1

1n

nP r

k

11

1

1n

nP r

k

11 11 nP n r k

2 2

11 11 12 2

1 1

1 n

s sP n r k

2

1 11 1 2

1n

sP U r k

2

1 11 1 1 2

1n n

sE r k

21 1

1 1 1 21

n n

sr E k

,


where 2

011 2

1

ln 1k

and 01 1e . (7)


02 2 02:H vs 2 2 02: .aH (8)


where 0 0.5r , 2

011 2

1

ln 1k

and 1


Proof: From (6)

21

1 201

1ln 1U n

n sp d E

21

1 201ln 1U n

UE

2

011 1 2

1 1

ln 1,U n

UE k

k

.


U n

Uf U E

k

is convex in U .


1

1n

np d

k

.


1P p d r

1

1

1n

nP r

k

11

1

1n

nP r

k

11 11 nP n r k

2 2

11 11 12 2

1 1

1 n

s sP n r k

2

1 11 1 2

1n

sP U r k

2

1 11 1 1 2

1n n

sE r k

21 1

1 1 1 21

n n

sr E k

,


where 2

011 2

1

ln 1k

and 01 1e . (7)


02 2 02:H vs 2 2 02: .aH (8)

The parameter of the difference between coefficients of variation for lognormal distributions is

2 22 1 2exp 1 exp 1x yk k .


2 22 2 1 2, , , , exp 1 exp 1T X Y x y

2 2

2 21 21 22 2

1 2

exp 1 exp 1s sS S

2 2

1 21 1exp 1 exp 1

n s m sU V

,

where 2 21 1~ , ~n mU V . (9)

It is easy to see that 2 2, , , ,T X Y x y in (9) satisfies conditions (i)-(iii) in section 2.2.

Without loss of generality, suppose 2 02 0 . The generalized p-value, p q is defined, under the null hypothesis 02H , to be

2 02

2 2 2 2sup , , , , , , , ,H

p q P T X Y x y T x y x y

2 2 2 2, , , , , , , ,P T X Y x y T x y x y (10)

Following (10) the generalized p-value for (8) can be defined as 2 2 2 2, , , , , , , ,p q P T X Y x y T x y x y

2 2

1 21 1exp 1 exp 1 0

n s m sP

U V

2 2

1 21 1exp 1 exp 1

n s m sP

U V

2 2

1 21 1exp 1 exp 1

n s m sP

U V

2 2

1 21 1n s m sP

U V

2122

11

n sUP

V m s

2122

1 11 1

n n sP F

m m s

21

1, 1 22

F n m

sE

s

, (11)

where .FE is an expectation operator with respect to F and 1, 1 .n m is a cdf of the F-

distribution with 1n and 1m degrees of freedom.

Theorem 3. If 21

1, 1 22

n mg f f

then g f is a convex function of f .

Chiang Mai J. Sci. 2016; 43(3) 677

The parameter of the difference between coefficients of variation for lognormal distributions is

2 22 1 2exp 1 exp 1x yk k .


2 22 2 1 2, , , , exp 1 exp 1T X Y x y

2 2

2 21 21 22 2

1 2

exp 1 exp 1s sS S

2 2

1 21 1exp 1 exp 1

n s m sU V

,

where 2 21 1~ , ~n mU V . (9)

It is easy to see that 2 2, , , ,T X Y x y in (9) satisfies conditions (i)-(iii) in section 2.2.

Without loss of generality, suppose 2 02 0 . The generalized p-value, p q is defined, under the null hypothesis 02H , to be

2 02

2 2 2 2sup , , , , , , , ,H

p q P T X Y x y T x y x y

2 2 2 2, , , , , , , ,P T X Y x y T x y x y (10)

Following (10) the generalized p-value for (8) can be defined as 2 2 2 2, , , , , , , ,p q P T X Y x y T x y x y

2 2

1 21 1exp 1 exp 1 0

n s m sP

U V

2 2

1 21 1exp 1 exp 1

n s m sP

U V

2 2

1 21 1exp 1 exp 1

n s m sP

U V

2 2

1 21 1n s m sP

U V

2122

11

n sUP

V m s

2122

1 11 1

n n sP F

m m s

21

1, 1 22

F n m

sE

s

, (11)

where .FE is an expectation operator with respect to F and 1, 1 .n m is a cdf of the F-

distribution with 1n and 1m degrees of freedom.

Theorem 3. If 21

1, 1 22

n mg f f

then g f is a convex function of f .

Proof: We have g f g h f , 2122

h f f

and f be the probability density

function of f . Hence

2g h f g h f h f h f h f h f h f h f

since 0f then 0h f . Hence 0h f and 0h f . Moreover

2122

0h f h f

. Hence 0f h f , and g f is convex in f .

Theorem 4. The upper bound of p q in (11) takes the form 11, 1 2 1, 1m n n mp q k r

where 0 0.5r ,

22

2 21

31

1m

km

, 1, 1 .m n is distribution of F and 1

1, 1 .n m the

inverse function of 1, 1 .n m .

Proof: From (11) 21

1, 1 22

F n m

sp q E

s

21

1, 1 22

F n mE F

21

1, 1 22

,F n mE Fc c

.

For any 0.5r and p q r . Hence by theorem 3, such that. E g F g E F

1, 1F n mp q E Fc 1, 1n m FcE F

1, 1 1

13n m

c mp q

m

For 0 0.5r , we have 1: :q qp q p q r P q p q r 1P p q r

1, 1

13n m

c mP r

m

11, 1

13 n m

c mP r

m

11, 1

31n m

mP c r

m

21 21, 1 2

1

31n m

m sP F r

m s

21 2

1, 1 1, 1 21

31m n n m

m sE r

m s

21 2

1, 1 1, 1 21

31m n n m

m sr E

m s

Chiang Mai J. Sci. 2016; 43(3)678

Proof: We have g f g h f , 2122

h f f

and f be the probability density

function of f . Hence

2g h f g h f h f h f h f h f h f h f

since 0f then 0h f . Hence 0h f and 0h f . Moreover

2122

0h f h f

. Hence 0f h f , and g f is convex in f .

Theorem 4. The upper bound of p q in (11) takes the form 11, 1 2 1, 1m n n mp q k r

where 0 0.5r ,

22

2 21

31

1m

km

, 1, 1 .m n is distribution of F and 1

1, 1 .n m the

inverse function of 1, 1 .n m .

Proof: From (11) 21

1, 1 22

F n m

sp q E

s

21

1, 1 22

F n mE F

21

1, 1 22

,F n mE Fc c

.

For any 0.5r and p q r . Hence by theorem 3, such that. E g F g E F

1, 1F n mp q E Fc 1, 1n m FcE F

1, 1 1

13n m

c mp q

m

For 0 0.5r , we have 1: :q qp q p q r P q p q r 1P p q r

1, 1

13n m

c mP r

m

11, 1

13 n m

c mP r

m

11, 1

31n m

mP c r

m

21 21, 1 2

1

31n m

m sP F r

m s

21 2

1, 1 1, 1 21

31m n n m

m sE r

m s

21 2

1, 1 1, 1 21

31m n n m

m sr E

m s

21 2

1, 1 1, 1 21

31m n n m

mr

m

11, 1 2 1, 1m n n mk r

(12)

where

22

2 21

31

mk

m

.

4. NUMERICAL RESULTS In this section, we used functions written in the R program [21] to compute the values of the upper bounds of the generalized p-values proposed in Theorems 2 and 4. For given values of n, m, 01 , 1k , 2k , r, we computed the upper bounds of p d and p q , by using the results from Theorems 2, 4, shown in Tables 1-2. As we can see in these tables, all results of the upper bounds of the generalized p-values proposed in Theorems 2, 4 are depend mainly on a variety of values of n, m, 01 , ik , 1, 2i , and r. As a result, these upper bounds confirm our proof in Theorems 2, 4. CONCLUSION In this paper, we proposed two new generalized p-values for testing the hypotheses of a single coefficient of variation and the difference between coefficients of variation for lognormal distributions. We also provied new upper bounds for our proposed generalized p-values. We note here that the result for these findings for the hypothses case 1 and case 2, were analogous to the upper bound of the generalized p-value for the Behrens-Fisher problem proposed by Tang & Tsui [17]. Numerical results shown in Tables 1 and 2, confirmed our findings in Theorems 2 and 4. From Figures 1 and 2, we also found that the proposed upper bounds for the hypothses case 1 and case 2 are increasing up on the parameter values of 1k and 2k . ACKNOWLEDGEMENTS The authors would like to thank the editor and the referees for their constructive comments, which have led to substantial improvements in this paper. The authors also thank the partial financial support from King Mongkut’s University of Technology North Bangkok. The third author is grateful to the grant number KMUTNB-GOV-55-06 from King Mongkut’s University of Technology North Bangkok.

21 2

1, 1 1, 1 21

31m n n m

mr

m

11, 1 2 1, 1m n n mk r

(12)

where

22

2 21

31

mk

m

.

4. NUMERICAL RESULTS In this section, we used functions written in the R program [21] to compute the values of the upper bounds of the generalized p-values proposed in Theorems 2 and 4. For given values of n, m, 01 , 1k , 2k , r, we computed the upper bounds of p d and p q , by using the results from Theorems 2, 4, shown in Tables 1-2. As we can see in these tables, all results of the upper bounds of the generalized p-values proposed in Theorems 2, 4 are depend mainly on a variety of values of n, m, 01 , ik , 1, 2i , and r. As a result, these upper bounds confirm our proof in Theorems 2, 4. CONCLUSION In this paper, we proposed two new generalized p-values for testing the hypotheses of a single coefficient of variation and the difference between coefficients of variation for lognormal distributions. We also provied new upper bounds for our proposed generalized p-values. We note here that the result for these findings for the hypothses case 1 and case 2, were analogous to the upper bound of the generalized p-value for the Behrens-Fisher problem proposed by Tang & Tsui [17]. Numerical results shown in Tables 1 and 2, confirmed our findings in Theorems 2 and 4. From Figures 1 and 2, we also found that the proposed upper bounds for the hypothses case 1 and case 2 are increasing up on the parameter values of 1k and 2k . ACKNOWLEDGEMENTS The authors would like to thank the editor and the referees for their constructive comments, which have led to substantial improvements in this paper. The authors also thank the partial financial support from King Mongkut’s University of Technology North Bangkok. The third author is grateful to the grant number KMUTNB-GOV-55-06 from King Mongkut’s University of Technology North Bangkok.

Chiang Mai J. Sci. 2016; 43(3) 679

21 2

1, 1 1, 1 21

31m n n m

mr

m

11, 1 2 1, 1m n n mk r

(12)

where

22

2 21

31

mk

m

.

4. NUMERICAL RESULTS In this section, we used functions written in the R program [21] to compute the values of the upper bounds of the generalized p-values proposed in Theorems 2 and 4. For given values of n, m, 01 , 1k , 2k , r, we computed the upper bounds of p d and p q , by using the results from Theorems 2, 4, shown in Tables 1-2. As we can see in these tables, all results of the upper bounds of the generalized p-values proposed in Theorems 2, 4 are depend mainly on a variety of values of n, m, 01 , ik , 1, 2i , and r. As a result, these upper bounds confirm our proof in Theorems 2, 4. CONCLUSION In this paper, we proposed two new generalized p-values for testing the hypotheses of a single coefficient of variation and the difference between coefficients of variation for lognormal distributions. We also provied new upper bounds for our proposed generalized p-values. We note here that the result for these findings for the hypothses case 1 and case 2, were analogous to the upper bound of the generalized p-value for the Behrens-Fisher problem proposed by Tang & Tsui [17]. Numerical results shown in Tables 1 and 2, confirmed our findings in Theorems 2 and 4. From Figures 1 and 2, we also found that the proposed upper bounds for the hypothses case 1 and case 2 are increasing up on the parameter values of 1k and 2k . ACKNOWLEDGEMENTS The authors would like to thank the editor and the referees for their constructive comments, which have led to substantial improvements in this paper. The authors also thank the partial financial support from King Mongkut’s University of Technology North Bangkok. The third author is grateful to the grant number KMUTNB-GOV-55-06 from King Mongkut’s University of Technology North Bangkok. REFERENCES

[1] Zou G.Y., Huo C.Y. and Taleban J., Environmetrics, 2009; 20: 172-180. DOI 10.1002/env.919.[2] Aitchison J. and Brown I.A.C., The Lognormal Distribution, Cambridge, UK, 1957.[3] Crow E.L. and Shimizu K., Lognormal Distributions: Theory and Application, Environmetrics, New York,

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Ranked Set Sampling, PhD Thesis, Uppsala University, Sweden, 2011.

Chiang Mai J. Sci. 2016; 43(3)680

Table 2. Upper bounds of generalized p-values for the hypothesis case 2.

n, m k2 r=0.01 r=0.02 r=0.05 r=0.10 r=0.20

5,5 1.011.051.10

0.010185080.010939330.01191297

0.020356690.021806670.02367003

0.050823640.054154750.05839659

0.10148930.10747620.1150189

0.2025120.21251960.2249272

10, 10 1.011.051.10

0.010339690.011769160.0137173

0.020635430.023291190.02686708

0.051405490.057212040.06487221

0.10245610.11248710.125455

0.20400890.22013510.2404344

15, 15 1.011.051.10

0.010452220.012398870.01515336

0.020836760.024403040.02936196

0.051823890.059473540.06981487

0.10315270.11617520.1333152

0.20509400.22572990.2519804

20, 20 1.011.051.10

0.010544530.012932040.01641324

0.021001890.025340410.03153038

0.052167610.061371470.07405984

0.10372630.11926150.140004

0.20599010.23039640.261702

25, 25 1.011.051.10

0.010624670.013407320.01756953

0.021145340.026173530.03350628

0.052466580.063051910.07788970

0.10422590.12198560.1459886

0.20677160.23449960.2703128

30, 30 1.011.051.10

0.010696530.013843380.01865735

0.021274010.026935900.03535354

0.052734970.064584070.08143821

0.10467470.12446160.1514907

0.20747410.23821410.278155

Table 1. Upper bounds of generalized p-values for the hypothesis case 1 when 21 1σ = .

n 01θ k1 r=0.01 r=0.02 r=0.05 r=0.10 r=0.20

5 1.32 1.0088331.33 1.0184501.34 1.028047

0.010168660.010353710.01053986

0.020329650.020691100.02105447

0.050784100.051642740.05250477

0.10147090.10307900.1046907

0.20263450.20550700.2083779

10 1.32 1.0088331.33 1.0184501.34 1.028047

0.010327990.010693140.01106601

0.020625320.021320060.02202796

0.051430380.053013740.05462096

0.10258940.10544460.1083311

0.20444290.20931450.2142110

15 1.32 1.0088331.33 1.0184501.34 1.028047

0.010452470.010961780.01148770

0.020853180.021810380.02279548

0.051919670.054061490.05625303

0.10342530.10722560.1110914

0.20577790.21213890.2185577

20 1.32 1.0088331.33 1.0184501.34 1.028047

0.010558100.011192120.01185296

0.021045580.022228290.02345551

0.052330210.054947250.05764265

0.10412340.10872170.1134230

0.20688760.21449560.2221971

25 1.32 1.0088331.33 1.0184501.34 1.028047

0.010651690.011397980.01218217

0.021215610.022600400.02404758

0.052691710.055731860.05888081

0.10473640.11004140.1154887

0.20785910.21656480.2254013

30 1.32 1.0088331.33 1.0184501.34 1.028047

0.010736760.011586540.01248596

0.021369850.022940260.02459193

0.053018810.056445680.06001310

0.10528990.11123810.1173694

0.20873460.21843410.2283029

[13] Bai Z., Wang K. and Wong W.-K., Stat. Prob. Lett., 2011; 81: 1078-1085. DOI 10.1016/j.spl.2011.02.035.[14] Niwitpong S., Appl. Math. Sci., 2013; 7: 3805-3810.[15] Buntao N. and Niwitpong S., Appl. Math. Sci., 2012; 6: 6691-6704.[16] Tsui K.-W. and Weerahandi S., J. Am. Stat. Assoc., 1989; 84: 602-607. DOI 10.1080/01621459.1989.10478810.[17] Tang S. and Tsui K.-W., Stat. Prob. Lett., 2007; 77: 1-8. DOI 10.1016/j.spl.2006.05.005.[18] Weerahandi S., Exact Statistical Methods for Data Analysis, Springer, New York, 1995.[19] Gamage J. and Weerahandi S., Comm. Stat., 1998; 27(3): 625-640. DOI 10.1080/03610919808813500.[20] Kabaila P. and Lloyd C.J., J. Aus. Stat. Assoc., 1997; 39(2): 193-204. DOI 10.1111/j.1467-842X.1997.

tb00535.x.[21] The R Development Core Team, An introduction to R, Vienna, http://R-project.org., 2010.

Chiang Mai J. Sci. 2016; 43(3) 681

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

Figure 1. Upper bounds of generalized p-values for the hypothesis case 1.

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

r = 0.01 r = 0.02

r = 0.05 r = 0.10

r = 0.20

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k1=1.33 k1=1.34

k1=1.32 k2=1.33 k2=1.34

k1=1.32 k2=1.33 k2=1.34

Figure 2. Upper bounds of generalized p-values for the hypothesis case 2.

upper bounds of generalized p-values for testing the ... · chiang mai j. sci. 2016; 43(3) :...

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