pareto and laplace distributions -...

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Pareto and Laplace Distributions

In this chapter we consider the Pareto distribution and Laplace distribution (double exponential

distribution)

6.1. The Pareto distribution

The Pareto distribution is a Continuous probability distribution named after the economist

Vilfredo Pareto.

The Pareto distribution is defined by the following functions:

PDF :

= 0 ,otherwise

CDF :

The parameter k marks a lower bound on the possible values that a Pareto distributed random

Variable can take on.

The mean and variance of Pareto distribution are:

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The Pareto distribution has been used to represent the income distribution of a society. It is also

used to model many phenomena such as city population sizes, occurrence of natural resources,

stock price fluctuations, size of firms, and brightness of comets.

6.2. Parameter Estimation

We are interested in estimating the parameters of the Pareto distribution from which the

sample comes. Here we present the Method of Moments, Method of Maximum Likelihood

Estimation and the Local frequency ratio method of estimation.

6.2.1 Method of moments

The rth

moment about origin is

Put r = 1, we get the first raw moment

……………(6.1)

Put r =2 , we get the second raw moment

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……….. (6.2)

Solving (6.1) and (6.2) ,we have the following estimators:

and

6.2.2 Maximum Likelihood Estimation

Let be a random sample of n observations from the Pareto population with pdf

The likelihood function of this sample is

;

Taking logarithms on both sides

The likelihood equation is = 0

On simplification, we get

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For this distribution, the shift parameter k cannot be estimated under MLE, because it leads to an absurd

estimate k = .For the shift parameter, this seems to be in general, the case for one sided distributions of

the type that one ordinarily encounters. Hence in these cases one can use sample minimum as an estimate

of the shift parameter.

6.2.3 Frequency Ratio Method of Estimation

We now estimate the parameters by considering local frequency ratio method by

putting x = x1, x2 in the Pareto distribution , we get

and

The ratio of the frequencies is

Taking logarithms on both sides, we get

and

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Illustration:

As explained earlier, we generate a sample of size 1000 from a Pareto (a=3, k=1)

distribution using MATLAB function. For the generated data the following frequency

distribution is obtained.

x

1.2501 1.7501 2.2501 2.7501 3.2501

f

677 193 72 28 0

Using these values in the above formula, the estimated value of a is

The above procedure is repeated for 50 samples. The mean, Standard deviation , , bias of

these 50 estimates were computed. The estimated bias was calculated as the mean minus the true value of

the parameter. The Mean Squared Error (MSE) was calculated as the bias squared plus the variance .

TABLE 6.1

K=1;a=3 ns = 50

Method of

moments

Maximum

Likelihood

Frequency Ratio

Method a k a k a k

Mean 3.2064 1.0278 2.9937 1.0004 3.1496 1.0004

Sd 0.3585 0.0482 0.1010 0.0004 0.2469 0.0004

0.4106 1.1852 0.4220 1.4256 0.3798 1.4256

2.9936 4.3673 3.2226 4.4269 3.3016 4.4269

Bias 0.2064 0.0278 -0.0063 0.0004 0.1496 0.0004

MSE 0.1711 0.0031 0.0102 0.0000 0.0834 0.0000

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From the above table, we notice that the actual values of (a, k) and the mean estimated values of

(a,k) are almost same. Therefore, it can be taken as a good estimator. Similar procedure is

followed for different sample sizes and different values of (a, k) and are listed in the tables 6.2 to

6.9 and the histograms in the figures 6.1 to 6.8 .The MATLAB programs are listed in the

Appendix.

6.3 Comparison of Method of Moments, Method of MLE and Frequency Ratio

Method for Different sample sizes and Different parameters:

TABLE 6.2 SIMULATION STATISTICS FOR PARETO(3,1,100)

TABLE 6.3 SIMULATION STATISTICS FOR PARETO (3,1,200)

K=1;a=3 ns = 100

Method of

moments

Maximum

Likelihood

Frequency Ratio

Method a K a k a k

Mean 3.2670 1.0377 3.0017 1.0003 3.1478 1.0003

Sd 0.3981 0.0580 0.0979 0.0003 0.2748 0.0003

0.6866 2.0851 0.1785 1.2986 0.4619 1.2986

3.6152 8.7724 2.9173 4.1652 2.8449 4.1652

Bias 0.2670 0.0327 0.0017 0.0003 0.1478 0.0003

MSE 0.2298 0.0044 0.0096 0.000 0.0974 0.000

K=1;a=3 ns = 200

Method of

moments

Maximum

Likelihood

Frequency Ratio

Method a k a k a k

Mean 3.2657 1.0317 3.0098 1.0003 3.1985 1.0003

Sd 0.3979 0.0528 0.0966 0.0003 0.2415 0.0003

0.5345 1.7159 0.3035 1.6101 0.1710 1.6101

3.1307 6.5251 3.2694 5.6326 2.6308 5.6326

Bias 0.2657 0.0317 0.0098 0.0003 0.1985 0.0003

MSE 0.2289 0.0038 0.0094 0.0000 0.0977 0.0000

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K=10;a=5 ns =50

Method of

moments

Maximum Likelihood Frequency Ratio

Method

a k a k a k

Mean 4.9345 9.9662 4.9835 10.0019 5.4030 10.0019

Sd 0.3443 0.1376 0.1563 0.0019 0.3534 0.0019

0.0087 -0.5740 0.0651 1.8182 0.4410 1.8182

2.5867 2.7132 2.7175 6.8251 2.8721 6.8251

Bias -0.0655

-0.0338 -0.0165 0.0019 0.4358 0.0019

MSE 0.0044 0.0012 0.0003 3.76e-006 0.1901 3.76e-006

K=10;a=5 ns = 100

Method of

moments


Method

a k a k a k

Mean 4.9345 9.9662 4.9835 10.002 5.4030 10.002

Sd 0.3443 0.1376 0.1563 0.0017 0.3534 0.0017

0.0087 -0.5740 0.0651 1.1698 0.4410 1.1698

2.5867 2.7132 2.7175 4.2560 2.8721 4.2560

Bias -0.0655 -0.0338 -0.0165 0.0020 0.4030 0.0020

MSE 0.0044 0.0012 0.0003 3.93e-006 0.1625 3.93e-006

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K=10;a=5

ns = 200

Method of

moments


Method

a k a k a k

Mean 4.9161 9.9523 4.9934 10.0019 5.4156 10.0019

Sd 0.3390 0.1554 0.1618 0.0019 0.3240 0.0019

-0.3448 -0.9485 0.1258 1.5016 0.2530 1.5016

3.4192 4.7027 3.1781 5.1698 3.2370 5.1698

Bias -0.0839 -0.0477 -0.0066 0.0019 0.4156 0.0019

MSE 0.0072 0.0023 0.0001 0.0000 0.1729 0.0000

K=4;a=3 ns = 50

Method of

moments


Method

a k a k a k

Mean 2.9221 3.9224 3.0154 4.0018 3.1798 4.0018

Sd 0.2424 0.1480 0.0849 0.0018 0.2453 0.0018

-0.1334 -0.3781 0.1308 1.1341 0.3441 1.1341

1.9689 2.2147 2.4638 3.3688 2.5521 3.3988

Bias -0.0779 -0.0779 0.0154 0.0018 0.1798 0.0018

MSE 0.0061 0.0061 0.0002 3.34e-006 0.0324 3.34e-006

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K=4;a=3 ns = 100

Method of moments Maximum Likelihood Frequency Ratio

Method

A k a k a k

Mean 2.9445 3.9346 3.0000 4.0014 3.1930 4.0014

Sd 0.2717 0.1683 0.1006 0.0013 0.2626 0.0013

-0.1711 -1.1393 0.0270 1.4220 -0.2190 1.4220

3.0482 5.1894 2.3433 6.0989 2.4703 6.0989

Bias -0.0555 -0.0654 4.15e-005 0.0014 0.1930 0.0014

MSE 0.0032 0.0043 1.01e-005 1.86e-006 0.0373 1.86e-006

K=4;a=3 ns = 200

Method of moments Maximum Likelihood Frequency Ratio

Method

A k a k a k

Mean 2.8898

3.8911

3.0047

4.0013

3.1855

4.0013

Sd 0.3320

0.2376

0.0958 0.0011 0.2590 0.0011

-0.6307

-1.7891

-0.0263 1.1865 0.3217 1.1865

3.4086

7.4399

2.8725 4.0392 3.2658 4.0392

Bias -0.1102

-0.1089

0.0047 0.0013 0.1855 0.0013

MSE 0.0123 0.0119 3.11e-005 1.58e-006 0.0345 1.58e-006

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6.4 GRAPHS FOR DIFFERENT SAMPLE SIZES AND DIFFERENT PARAMETERS

FIGURE 6.1 HISTOGRAM FOR PARETO (3,1,50)

FIGURE 6.2 HISTOGRAM FOR PARETO (3,1,100)

0.5 1 1.5 2 2.5 3 3.50

50

100

150

200

250

300

350

400

450

500

Bins

Fre

quency

0.5 1 1.5 2 2.5 3 3.50

50

100

150

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350

400

450

500

Bins

Fre

quency

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FIGURE 6.3 HISTOGRAM FOR PARETO (3,1, 200)

FIGURE 6.4 HISTOGRAM FOR PARETO (5, 10, 50 )

0.5 1 1.5 2 2.5 3 3.50

50

100

150

200

250

300

350

400

450

500

Bins

Fre

quency

5 10 15 20 25 30 350

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700

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quency

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FIGURE 6.5 HISTOGRAM FOR PARETO (5,10, 100)

FIGURE 6.6. HISTOGRAM FOR PARETO ( 5, 10, 200)

5 10 15 20 25 30 350

100

200

300

400

500

600

700

Bins

Fre

quency

5 10 15 20 25 30 350

100

200

300

400

500

600

700

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Fre

quency

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FIGURE 6.7. HISTOGRAM FOR PARETO (3,4, 50)

FIGURE 6.8. HISTOGRAM FOR PARETO (3, 4, 100)

2 4 6 8 10 12 140

50

100

150

200

250

300

350

400

450

500

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Bins

2 4 6 8 10 12 140

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6.5. Laplace Distribution (Double Exponential Distribution)

The Laplace distribution is a Continuous probability distribution named after

Pierre-Simon Laplace. It is also sometimes called the Double Exponential Distribution because it

consists of two exponential distributions (with an additional location parameter) spliced peak to

peak.

The Laplace distribution is defined by the following functions:

PDF :

= 0 , otherwise

CDF :

Where is the location parameter and is the scale parameter.

The case where = 0 and = 1 is called the standard double exponential

distribution. The equation for the standard double exponential distribution is

,

The Laplace distribution has found uses in fields where an alternative to

the Normal distribution that has longer tails and a more pronounced peak is desired. The three

fields with the most prolific use of the Laplace distribution are navigation (particularly in

modeling nautical errors), finance (particularly in forecasting returns on commodities), and

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speech recognition (particularly in modeling natural acoustic noise).Beyond this, the Laplace

distribution is slowly finding more relevance in fields such as energy engineering though

extended use of the distribution is rare.

The following is the plot of the Laplace (Double Exponential Distribution) probability density

function.

Properties of the distribution are :

1) Mean =Median = Mode =

2) Variance = 2

3) Skewness = 0 and Kurtosis = 3.

6.6. Parameter Estimation

We are interested in estimating the parameters of the Laplace distribution from which the

sample comes. Here we present the Method of Maximum Likelihood Estimation and the Local

frequency ratio method of estimation.

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6.6.1 Maximum Likelihood Estimation

Let be a random sample of n observations from the population with pdf

The likelihood function of this sample is

;


The likelihood equation for estimating

Now, = Least absolute estimator to take the derivative of L.

To estimate consider

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Now,

Therefore,

Which is zero if the same numbers of xi are less than as greater than . Therefore ,if n is

even, we choose to be any value between n/2 th and (n/2 + 1)th of the sorted values of x and

if n is odd we choose to be middle value of x . In other words, the MLE for is median

6.6.2 Frequency Ratio Method of Estimation

In this section ,we explain the local Frequency ratio method of estimation by taking x = x1,x2

in the pdf of Laplace distribution ,we get

(taking


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Illustration:

As explained earlier, we generate a sample of size 1000 from a Laplace (α=2, β=3) distribution

using MATLAB function. For the generated data the following frequency distribution is

obtained.

Since the Laplace distribution is symmetric, the maximum frequency contains median. To estimate the β,

we take the frequencies to right of maximum frequency i.e.., 296 and 108 .Using the above formula, we

get

And the parameter α is estimated in the usual way.

The above procedure is repeated for 50 samples. The mean, Standard deviation , , bias

of these 50 estimates were computed. The estimated bias and Mean Squared Error (MSE) was

calculated as discussed in the chapter3.

x

(mid-value) -13.52 -9.952 -6.383 -2.815 0.7537 4.322 7.890 11.459 15.028 18.597

f 3 8 27 117 397 296 108 30 9 5

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From the above table, we notice that the actual values of( ) and the mean estimated values of

( ) are almost same. Therefore, it can be taken as a good estimator. Similar procedure is

followed for different sample sizes and different values of ( ) . The Simulation results are

tabulated in tables 6.10 to 6.15 and the corresponding histograms in the figures 6.9 to 6.16.The MATLAB

programs are listed in the Appendix.

6.7 COMPARISION OF METHOD OF MOMENTS, MAXIMUM LIKELIHOOD ESTIMATION

METHOD AND FREQUENCY RATIO METHOD.

TABLE 6.10: SIMULATION STATISTICS FOR LAPLACE (1,3)

(α=2, β=3)

ns=50

MLE Frequency Ratio method

Mean 3.2210 1.9970 2.9259 2.0153

Sd 0.0812 0.0812 0.3516 0.2022

0.0962 -0.0962 0.3609 0.3609

2.8833 2.8833 2.7260 3.0897

Bias 0.2210 -0.0030 -0.0741 0.0153

MSE 0.0554 0.0066 0.1291 0.0411

(α=1,

β=3)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 3.0945 1.0193 2.9898 1.0012 2.8494 1.006 3.0261 0.9886

Sd 0.0841 0.0841 0.3916 0.2798 0.100 0.100 0.4614 0.3169

-0.0414 0.0414 -0.5173 0.3006 0.0523 0.0523 0.5982 -1.0276

3.6052 3.6052 3.3482 4.3686 2.3514 2.3514 2.7809 4.8522

Bias 0.0945 0.0193 -0.0102 0.0012 -0.1506 0.0061 0.0261 -0.0114

MSE 0.0160 0.0074 0.1535 0.0783 0.0327 0.0100 0.2136 0.1006

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TABLE 6.11: SIMULATION STATISTICS FOR LAPLACE (2,1)

TABLE 6.12: SIMULATION STATISTICS FOR LAPLACE(2,5)

(α=2,

β=1)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 0.9742 2.0031 0.9820 2.0164 0.9991 2.0022 1.0017 2.0082

Sd 0.0394 0.0394 0.1380 0.0880 0.0330 0.0330 0.1101 0.0786

0.3231 -0.3231 0.6411 -0.4466 -0.0947 -0.0947 0.0734 -0.4123

2.7522 2.7522 3.1520 3.3254 2.7857 2.7857 2.3102 3.4942

Bias -0.0258 0.0031 -0.0180

0.0164 -0.0009 0.0022 0.0017 0.0082

MSE 0.0022 0.0016 0.0194 0.0082 0.0011 0.0011 0.0121 0.0062

(α=2,

β=5)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 4.8712 2.0157 4.9098 2.0818

4.9955

2.0112 5.0086 2.0408

Sd 0.1968 0.1968 0.6898 0.4402 0.1651 0.1651 0.5506 0.3930

0.3231 -0.3231 0.6411 -0.4466 -0.0947 0.0947 0.0734 -0.4123

2.7522 2.7522 3.1520 3.3254 2.7857 2.7857 2.3102 3.4942

Bias -0.1288 0.0157 -0.0902 0.0818 -0.0045 0.0112 0.0086 0.0408

MSE 0.0553 0.0390 0.4839 0.2005 0.0273 0.0274 0.3033 0.1561

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TABLE 6.13 : SIMULATION STATISTICS FOR LAPLACE(3,2)


(α=3,

β=2)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 2.1707 2.9939 1.9876 3.0056 2.1552 2.9958

2.0483

2.9905

Sd 0.0693 0.0693 0.2109 0.1381 0.0630 0.0630 0.2392 0.1356

-0.2256 0.2256 0.3115 0.3354 0.0141 -0.0141 0.6487 -0.2359

2.6244 2.6244 2.2903 2.0202 2.7078 2.7078 2.9248 3.9061

Bias 0.1707 -0.0061 -0.0124 0.0056 0.1552 -0.0042 0.0483 -0.0095

MSE 0.0339 0.0048 0.0445 0.0191 0.0281 0.0040 0.0596 0.0185

(α=5,

β=10)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 9.8130 4.9585 10.0074 4.9291 10.0096 4.9820 9.9691 5.0060

Sd 0.3226 0.3226 1.1254 0.7442 0.3118 0.3118 1.1415 0.6644

0.2986 -0.2986 0.6854 -0.8880 -0.3165 0.3165 0.7413 -0.0305

2.5252 2.5252 3.3718 4.2571 3.0856 3.0856 3.7075 2.9563

Bias -0.1870

-0.0415 0.0074 -0.0709 0.0096 -0.0180 -0.0309 0.0060

MSE 0.1390 0.1058 1.2665 0.5588 0.0973 0.0976 1.3039 0.4415

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(α=5,

β=5)

ns=50 ns= 100

MLE Frequency

Ratio method

MLE Frequency

Ratio method

Mean 5.0607 4.9999 5.0477 4.9757 4.9789 5.0274 5.0715 4.9991

Sd 0.1652 0.1652 0.4245 0.2905 0.1424 0.1454 0.6687 0.3682

0.5854 -0.5854 -0.4897 0.7658 0.1472 -0.1472 0.6201 -0.3775

3.3466 3.3466 2.4426 2.9670 2.7298 2.7298 3.4584 3.2357

Bias 0.0607 -0.0001 0.0477 -0.0243 -0.0211 0.0274 0.0715 -0.0009

MSE 0.0310 0.0273 0.1825 0.0850 0.0207 0.0210 0.4523 0.1356

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6.8 GRAPHS FOR DIFFERENT SAMPLE SIZES AND DIFFERENT PARAMETERS

FIGURE 6.9 : HISTOGRAM FOR LAPLACE(1,2,50)

FIGURE 6.10 : HISTOGRAM FOR LAPLACE (1,2,100)

-25 -20 -15 -10 -5 0 5 10 150

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-15 -10 -5 0 5 10 15 200

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FIGURE 6.11: HISTOGRAM FOR LAPLACE (2, 3,50)


-20 -15 -10 -5 0 5 10 15 20 250

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-20 -15 -10 -5 0 5 10 15 20 250

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FIGURE 6.13: HISTOGRAM FOR LAPLACE (5, 2, 50)


-10 -5 0 5 10 15 200

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Bins

-10 -5 0 5 10 15 200

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FIGURE 6.15: HISTOGRAM FOR LAPLACE (5, 5, 50)


-30 -20 -10 0 10 20 30 400

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MATLAB PROGRAMS

PARETO DISTRIBUTION

% PROGRAM TO FIND THE ESTIMATES OF PARAMETER BY METHOD OF

MOMENTS,MLE AND FREQUENCY RATIO METHOD.

nnska=input('enter n ns k and a as a 4-Vector.');

n=nnska(1);ns=nnska(2);k=nnska(3);a=nnska(4);

%n=1000;ns=200;a=1;k=1;

for i=1:ns,

x=k./((rand(n,1).^(1/a)));

mx=mean(x);varx=var(x);

k0(i)=min(x);

k2(i)=min(x);

a2(i)=n./(sum(log(x./k2(i))));

h=k0(i)/2;

[f,z]=histc(x,k0(i):h:3*k0(i));

hist(x,k0(i):h:3*k0(i))

f1=f(1);f2=f(2);

lf12=log(f1/f2);

x1=k0(i)+h/2;

x2=x1+h;

lx21=log(x2/x1);

a0(i)=(lf12/lx21)-1;

a1(i)=1+sqrt((1+mx*mx)/varx);

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k1(i)=(mx*(a1(i)-1))/a1(i);

end

a2=a2;k2=k2;k0=k0;k1=k1;a1=a1;a0=a0;

%method of moments

ma1=mean(a1);mk1=mean(k1);

sda1=std(a1);sdk1=std(k1);

a1z=(a1-ma1)/sda1;k1z=(k1-mk1)/sdk1;

rb1a1=mean(a1z.^3);b2a1=mean(a1z.^4);

rb1k1=mean(k1z.^3);b2k1=mean(k1z.^4);

ba1=a1-a;mba1=mean(ba1);

bk1=k1-k;mbk1=mean(bk1);

msea1=sda1^2+ mba1^2;

msek1=sdk1^2+ mbk1^2;

mom = [ma1,sda1,rb1a1,b2a1,mk1,sdk1,rb1k1,b2k1]

mombias = [mba1,msea1,mbk1,msek1]

%maximum likelihood estimation






ba2=a2-a;mba2=mean(ba2);

bk2=k2-k;mbk2=mean(bk2);

msea2=sda2^2+ mba2^2;

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msek2=sdk2^2+ mbk2^2;

mle = [ma2,sda2,rb1a2,b2a2,mk2,sdk2,rb1k2,b2k2]

mlebias = [mba2,msea2,mbk2,msek2]

%local frequency ratio method






ba0=a0-a;mba0=mean(ba0);bk0=k0-k;mbk0=mean(bk0);

msea0=sda0^2+mba0^2;msek0=sdk0^2+mbk0^2;

local = [ma0,sda0,rb1a0,b2a0,mk0,sdk0,rb1k0,b2k0]

lobias=[mba0,msea0,mbk0,msek0]

DOUBLE EXPONENTIAL DISTRIBUTION

%PROGRAM TO ESTIMATE THE PARAMETERS BY MLE AND LOCAL FREQUENCY

RATIO METHOD.

nsm=input('enter nsm,ssize,scale and loc params as a 30vector.')

n=nsm(1);s=nsm(2);m=nsm(3);

%ns=50;n=1000;s=5;m=5;

tic,

x0=exprnd(s,n,ns);

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r=rand(n,ns)<.5;

ex=(x0.*r-x0.*(1-r))+m;

k1=0;k2=0;k=0;

for i=1:ns,

x=ex(:,i);

y=sort(x);

medx(i)=median(x);

meanx=mean(x); hist(x),

n2=ceil(n/2);

ris=((mean(y(n2:end))-medx));

% les = (-(mean(y(1:(n2-1))-medx)));

[f0,z]=hist(x);%h=(max(x)-min(x))/10;

[u,v]=max(f0);

f2=u;

x2=z(v);

x1=z(v-1);

x3=z(v+1);

x4=z(v+2);

f1=f0(v-1);

f3=f0(v+1);

f4=f0((v+2));

f1234=[f1 f2 f3 f4];

g(i)=f1*f3*f4==0;

f13=f1/f3;

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f34=f3/f4;

lf13=log(f13);

lf34=log(f34);

x1p3=x1+x3;

x4m3=x4-x3;

sest=x4m3/lf34;

mest=(x1p3-sest*lf13)/2;

s2(i) = sest;m2(i)=mest;

end

%Maximum Likelihood Estimation

s1=ris;m1=medx;

ms1=mean(s1);sds1=std(s1);

mm1=mean(m1);sdm1=std(m1);

s1z=(s1-ms1)/sds1;m1z=(m1-mm1)/sdm1;

rbs1=mean(s1z.^3);b2s1=mean(s1z.^4);

rbm1=mean(m1z.^3);b2m1=mean(m1z.^4);

%bias

bs1=s1-s;mbs1=mean(bs1);

bm1=m1-m;mbm1=mean(bm1);

%mean square error

mses1=sds1^2+ mbs1^2;

msem1=sdm1^2+ mbm1^2;

MLE = [ms1,sds1,rbs1,b2s1,mm1,sdm1,rbm1,b2m1]

mlebias=[mbs1,mses1,mbm1,msem1]

127

%By FREQUENCY RATIO METHOD

s2=s2;m2=m2;

ms2=mean(s2);sds2=std(s2);

mm2=mean(m2);sdm2=std(m2);

s2z=(s2-ms2)/sds2;m2z=(m2-mm2)/sdm2;

rbs2=mean (s2z.^3);b2s2=mean(s2z.^4);

rbm2=mean (m2z.^3);b2m2=mean(m2z.^4);

%bias

bs2=s2-s; mbs2=mean (bs2);bm2=m2-m;mbm2=mean(bm2);

%mean square error

mses2=sds2^2+ mbs2^2; msem2=sdm2^2+ mbm2^2;

Local = [ms2, sds2, rbs2, b2s2,mm2, sdm2,rbm2,b2m2]

Local bias = [mbs2, mses2,mbm2, msem2]

pareto and laplace distributions -...

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