Bayes Reliability estimation for Lomax distribution

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Bayes Reliability estimation for Lomax distribution

Assist. Prof. Dr. Awatif R. Mezaal Zinah Ahmed Abbas

Department Of Mathematics AL-Mustansiryah University

خلملا ـــــــــــــــــ ص

ول عيزوتل ةيلوعملا ةلادل زيب ردقم داجيا ثحبلا اذه يف مت ملعملا يذ سكام

ةمولعم ريغلا نيت

دوجو ضارتفا ىلعو نيتملعملل نيقباس نيعيزوت

, ىمسي يتامولعم ريغ لولاا Quasi

يناثلاو

يسلاا عيزوتلا (

يتامولعم )

أطخلا عبرم ةراسخ ةلاد تحتو ,

مادختسأب ةيلوعملا ةلاد داجيا مت دقو

يلدنل بولسا ,

ةيئاوشعلا تاريغتملا ديلوتل ةاكاحملا ءارجا مث نمو ركتب

ةبرجتلا را ةرم 1000

تاردقملا نيب ةنراقملاو أطخلا تاعبرم طسوتم ىلع دامتعلااب

.MSE

ABSTRACT

In this paper we obtain Bayesian estimators of reliability for two unknown parameters Lomax distribution on the assumption there is two prior distributions for the parameters , the first is non-informative called QUASI, where the second is informative called EXPONENTIAL distribution under squared errors loss function used idea of Lindley, then the simulation study used to generate random variables with repeated

experiment 1000 and compared between them depending on MSE.

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Introduction 1- ---:

Lomax distribution is also called pareto distribution of second kind with two parameters ( θ,λ ) , has been received greatest attention from

theoretical and statisticians primarily due to its use in Reliability and lifetime testing studies.[2]

It is introduced and studied by Lomax 1954 used for analyze business failure data,economic,actural,science and traffic modeling[7].

Tadikanalla(1980) related Lomax distribution to the Burr family of distribution[10].

The aim of the this paper is to consider the Bayesian analysis of Reliability for two parameters Lomax distribution under two prior

functions , also we consider the squared error loss function .it is observed that the Bayes estimators can not be expressed in explicit forms and they can be obtained .by two dimensional numerical only.

We used the idea of Lindley to approximate Bayes estimators of the unknown parameters and reliability under informative and non- informative priors.

The cumulative distribution function for Lomax distribution is given by:[7]

……..(1) Then the probability density function is :

x>0 θ,λ>0 ……..(2) Where: θ,λare the shape and scale parameters respectively , the Reliability and Hazard function are given by :

.

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Lindley's approximation2- ---:

Lindley [6] proposed his procedure to approximate the ratio of the two integrals such as in equation(7) ,which is used to obtain the approximate Bayes estimators ,then the approximate Bayes estimator of Reliability is define as:

(x) ᴓ(θ_ML ,λ_ML)+½ [ A+ L₃₀B₁₂+L₀₃B₂₁+L₂₁C₁₂+L₁₂C₂₁] +P₁A₁₂+P₂A₂₁ …………(5) Where:

ᴓ(θML,λML) the function of maximum likelihood for the parameters θ,λ.

A=

A_ij = w_iƮ_ii + w_jƮ_ji …….(6) Bij = ( wiƮii + wjƮij )Ʈii …….(7) Cij = 3wiƮiiƮij + wj(ƮiiƮjj + 2Ʈ²ij ) ……..(8) where :

wij =

……..(9)

.

and where:

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…….(10) I =

Now :

i,j = 0, 1, 2, 3 andi+j = 3 ……..(11) Then:

.

later :

………(12)

3-Bayes method ---:

In this section ,we consider the Bayes estimation of the unknown parameter θ,λ and reliability with assumed that θ and λ have joint function h( θ,λ ).

then the joint posterior density function of θ and λ can be written as:[4]

P(θ,λ|x ) =

…………(13)

Therefore ,the Bayes estimator for the joint function of θ,λ under squared error loss function is:

………(14)

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From equation (2) based on the likelihood function of the observed data :

.

=

.

………..(16)

………..(17) Now

.

……(18)

.

Solve this equation numerical by Newten – Raphson method to obtaine the MLE for λ , then compensated in equation (17) for θMLE as:

1-take initial value for λ as λk . 2-compensated λ_k in equation (17) .

3-taking the derivative for equation (19) , then :

.

We stop when the absolute difference | λk+1 – λk | <𝜖 , where 𝜖is very small value , the compensated in equation (17) to obtain the MLE of θ.

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by Lindley idea ,and equations (16), (18):now

.

Then for equations (10)and (a,b,c) :

.

Where ,Ʈ_ij = -I^-1

Now for Reliability estimation and by

equation(3),(5)and (9) ,we get:

..

.

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.

Now by equation(11) ,(15-a) :

.

..

Later and by equations (6),(7),(8), we find:

.

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i.Bayes Reliability estimator under Quasi prior

The Quasi prior function for θ and λ define as [9]:

. . where :θ,λ>0 and k is positive integer number.

Then the joint function for θ and λ given as:

. and by equation (12) , we get:

.

by equation (21):

.ln h₁(θ,λ) = -klnθ –klnλ then for Quasi prior :

.

now by equations (15) and (21):

.

. Then the joint posterior distribution under Quasi prior is:

.

𝝎

and the Bayes Reliability estimation with squared error loss function under Quasi prior is:

.

.Where

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ii- Bayes Reliability estimator under Exponential prior .

depending on all equations by Lindley approximate which are derived in section (2) ,but under exponential prior function for θ and λ we get:[1]

.

. so, and by equation (12):

. Now by equation (15) and (28):

.

Then the joint posterior distribution under Exponential prior is:

.𝜔

and the Bayes Reliability estimation with squared error loss function under Exponential prior substitute equation (26) is :

.

Simulation results 4-

For compare between the Reliability estimators which derived above , we used simulation study, that generated data distributed Uniform distribution and transformed to data as Lomax distribution with two parameters θ,λ used cumulative distribution function as:

.

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.Let U=

. then:

.

used equation (33) to generate data for different sample sizes

[n=15,25,50,100] with values of parameters ( θ,λ ) in table (1), and [k=2.5 a=1.5 , b=2] the parameters values of prior distributions, then find the reliability estimators from equations (25),(32) to choose the best Bayes estimator for reliability under two priors function with squared error loss function in table (2) ,using Mean Square error ( MSE ) , where:

. ………(34) Where L = 1000

Tables (1) , (2, 3, 4, 5, 6, 7,8, 9, and 10 ) represent the above simulation.

Table (1)

Default values of parameters

0.9 0.7

θ 0.5

1 0.6 λ 0.4

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Table (2)

The (MSE) and best reliability estimatorwith θ=0.5, λ=0.4

MSE

Best n

RE

0.0128 0.0194

15

RE

0.0076 0.0095

25

RE

0.0040 0.0044

50

RE

0.0020 0.0021

100

Table(3)

The (MSE)and the best reliability estimator with θ=0.5 ,λ=0.6

MSE

RE Best RQ

n

RE

0.0126 0.0169

15

RE

0.0071 0.0084

25

R_E 0.0036

0.0039 50

R_E 0.0019

0.0020 100

Table (4)

The (MSE) and the best reliability estimator with θ=0.5 , λ=1

MSE

Best RE

RQ

n

RE

0.0104 0.0113

15

RE

0.0065 0.0067

25

RE

0.0032 0.0033

50

RQ , RE

0.0015 0.0015

100 Table (5)

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The (MSE) and the best reliability estimator with θ=0.7 , λ=0.4

MSE

Best RE

RQ

n

RE

0.0122 0.0214

15

R_E 0.0068

0.0093 25

RE

0.0034 0.0040

50

RE

0.0018 0.0019

100

Table (6)

The (MSE)and the best reliability estimator with θ=0.7 , λ= 0.6

MSE

Best RE

RQ

n

RE

0.0137 0.0206

15

R_E 0.0075

0.0095 25

R_E 0.0036

0.0042 50

RQ , RE

0.0018 0.0018

100

Table (7)

The (MSE) and the best reliability estimator with θ=0.7 , λ= 1

MSE

Best RE

RQ

n

RE

0.0131 0.0167

15

RE

0.0073 0.0081

25

R_E 0.0033

0.0036 50

R_E 0.0017

0.0018 100

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Table (8)

The (MSE) and the best reliability estimator with θ=0.9 , λ=0.4

MSE

RE Best RQ

n

RQ

0.0125 0.0124

15

RE

0.0058 0.0084

25

RE

0.0029 0.0036

50

RE

0.0014 0.0015

100

Table (9)

The (MSE) and the best reliability estimator with θ=0.9 ,λ= 0.6

MSE

RE Best RQ

n

RE

0.0145 0.0229

15

RE

0.0071 0.0097

25

RE

0.0033 0.0038

50

R_E 0.0016

0.0017 100

Table (10)

The (MSE) and the best reliability estimator with θ=0.9 , λ=1

MSE

RE Best RQ

n

RE

0.0154 0.0200

15

RE

0.0069 0.0084

25

RE

0.0037 0.0041

50

R_E 0.0016

0.0017 100

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Now for all tables above and all samples sizes , show that the best Bayes reliability estimator for two unknown parameters Lomax distribution under squared error loss function using Lindley approximate is with Exponential prior and different default values of parameters( θ,λ ) ,accept when [ θ=0.9 , λ=0.4] in [ n=15] the best estimator with Quasi prior, where in [ n=100 ] the two reliability estimators equal increasingly likely. Also the tables show that the value of MSE decreasing in large sample size for two priors.

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