On Reliability Estimation for the Exponential Distribution Based on ...

IHSCICONF 2017

Ibn Al-Haitham Journal for Pure and Applied science

Special Issue https://doi.org/ 10.30526/2017.IHSCICONF.1814

On Reliability Estimation for the Exponential Distribution Based on Monte Carlo Simulation Abbas Najim Salman College of Education for Pure Science/Ibn-Al-Haitham, University of Baghdad [email protected] Taha AnwarTaha Directorate- General for Education of Anbar Ministry of Education [email protected]

Abstract This Research deals with estimation the reliability function for two-parameters Exponential distribution, using different estimation methods ; Maximum likelihood, Median-First Order Statistics, Ridge Regression, Modified Thompson-Type Shrinkage and Single Stage Shrinkage methods. Comparisons among the estimators were made using Monte Carlo Simulation based on statistical indicter mean squared error (MSE) conclude that the shrinkage method perform better than the other methods . Keywords: The Exponential distribution, Median-First Order Statistics, Ridge regression, Modified Thompson, Shrinkage methods, Mean squared error.

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Introduction The Exponential distribution is generally use for unit failure model that have an constant failure rate, it be able to be with one or two parameters. "Maguire, Pearson and Wynn (1952) studied mine accidents and showed that time intervals between industrial accidents follow exponential distribution";[6]. "Cohen and Helm (1973) used (BLUE), (MLE), (ME), (MVUE) and MME to estimate the parameters of the exponential distribution";[3]. "Peter (1974) used robust M-estimation method for the scale parameter, with application to the exponential distribution";[8]. " Lawless (1977) studied a confidence interval for the scale parameter and obtained a prediction interval for a future observation"; [5] . "Afify (2004) and Muhammad and Ahmed (2011) reviewed and compared several methods for estimating the two-parameter exponential distribution";[1],[7]. "Two new classes of confidence interval for the scale parameter proposed by Petropoulos (2011)"; [9] and "Lai and Augustine (2012) obtained interval estimates for the threshold (location) parameter and derived a predictive function for the two parameter";[4]. The aim of this research is to estimate the reliability function for two-parameters Exponential distribution, using different estimation methods ; Maximum likelihood, Median-First Order Statistics, Ridge Regression, Modified Thompson-Type Shrinkage and Single Stage Shrinkage methods where a location parameter (𝛼) is known. The probability density function of two parameters Exponential distribution is given by 1 (𝑡𝑖 − 𝛼) 𝑒𝑥𝑝 − ( ) 𝛽 𝑓(𝑡𝑖 , 𝛼, 𝛽) = {𝛽 0

𝛼 < 𝑡𝑖 < ∞ }

(1)

𝑜. 𝑤

Ω = {(𝛼, 𝛽); 𝛼 > 0, 𝛽 > 0} Where 𝛼 is a location parameter and 𝛽 is a scale parameter. The mean and the variance of Exponential distribution are: 𝑀𝑡 = 𝐸(𝑡) = 𝛽 + 𝛼 (2) 2 2 𝜎 𝑡 = 𝑣𝑎𝑟(𝑡) = 𝛽 (3) 2) 2 2 𝐸(𝑡 = 2𝛽 + 2 𝛼𝛽 + 𝛼 (4) The distribution, reliability and hazard functions of Exponential distribution respectively as below: (𝑡 − 𝛼) 𝐹(𝑡) = 1 − 𝑒𝑥𝑝 (− ) (5) 𝛽 (𝑡 − 𝛼) (6) 𝑅(𝑡) = 𝑒𝑥𝑝 (− ) 𝛽

ℎ(𝑡) =

1 𝛽

(7)


IHSCICONF 2017



Experimental Estimation Method (Theoretical Part) In this section, we discussed five estimation methods for the parameter and the reliability function of two-parameter Exponential distribution

Maximum likelihood method The likelihood function L(𝑡𝑖 , 𝛼 , 𝛽 ) is defined as below, [3] L 𝑛

= ∏ 𝑓(𝑡𝑖 , 𝛼 , 𝛽)

(8)

𝑖=1

L

𝑛

=∏ 𝑖=1

1 (𝑡𝑖 − 𝛼) 𝑒𝑥𝑝 (− ( )) 𝛽 𝛽2

(9)

Taking the Natural logarithm equation (9) , so we get the following: Ln L = −n Ln 𝛽 n (𝑡𝑖 − 𝛼) + ∑( ) 𝛽

(10)

𝑖=1

The partial derivative for equation (10) with respect to unknown parameter β, is: ∂Ln L ∂𝛽 n −𝑛 (𝑡𝑖 − 𝛼) = + ∑( ) 𝛽 𝛽2 𝑖=1

Equating equation (11) to zero to solve this equation: n −𝑛 (𝑡𝑖 − 𝛼) +∑ =0 𝛽 𝛽2 𝑖=1 −𝑛 𝛽̂ + ∑𝑛𝑖=1(𝑡𝑖 − 𝛼) =0 𝛽̂ 2 𝑛

−𝑛 𝛽̂ + ∑(𝑡𝑖 − 𝛼) = 0 𝑖=1 𝑛

𝑛 𝛽̂ = ∑(𝑡𝑖 − 𝛼) 𝑖=1


(11)

IHSCICONF 2017



∴ 𝛽̂𝑀𝐿 𝑛 1 = ∑(𝑡𝑖 − 𝛼) 𝑛

(12)

𝑖=1

Then the estimation of Reliability function for the two-parameters Exponential distribution using ML technique will be 𝑅̂𝑀𝐿 (𝑡𝑖 − 𝛼) = exp(− ) (13) 𝛽̂𝑀𝐿

Median-First Order Statistics Method "In this modification the second moment is replaced by Met = tme, where Met is the population median and tme is the sample median";[7]. The Exponential distribution median can be found by 0.5 𝐴

= ∫ 𝑓(𝑥) 𝑑𝑡

(14)

0

1 𝐴 (𝑡𝑖 − 𝛼) 1 ∫ exp[− ] 𝑑𝑡 = 𝛽 0 𝛽 2

𝐴 = 𝛼 + 𝛽 (𝑙𝑛2) The C.D.F. of t (1) defined below 𝐹(𝑡(1) ) = 𝑝𝑟[𝑡(1) < 𝑡] = 1 − 𝑝𝑟[𝑡(1) > 𝑡)] = 1 − 𝑝𝑟[𝑡(1) > 𝑡 , 𝑡(2) > 𝑡, … , 𝑡(𝑛) > 𝑡] = 1−[𝑝𝑟(𝑇 > 𝑡)]𝑛 = 1 − [𝑅(𝑡)]𝑛 So, the P.D.F. of 𝑡(1) became 𝑓(𝑡(1) ) = =

𝑛

𝑑𝐹(𝑡(1) ) 𝑑𝑡(1)

exp [−𝑛 ( 𝛽

𝑡(1) −𝛼) 𝛽

)]

; 𝛼 < 𝑡(1) < ∞ , 𝛽 > 0

Then, the mathematical expectation of a random variable 𝑡(1) is ∞

𝐸[𝑡(1) ] = ∫ 𝛼

𝑛(𝑡(1) − 𝛼) 𝑡𝑛 exp [− ( )] 𝑑𝑡 𝛽 𝛽 𝛽

=𝛼+𝑛 𝑊ℎ𝑒𝑟𝑒 𝑡(1) 𝑟𝑒𝑓𝑒𝑟 to the first order statistic which is represents the smallest values of t. If (α ̂ , β̂) is unbiased estimator to (α , β) respectively then the equationrealized


IHSCICONF 2017

Special Issue


https://doi.org/ 10.30526/2017.IHSCICONF.1814

𝛼 𝛽 𝑛 Thus, we have = 𝑡(1) −

(15)

𝛼 + 𝛽 (𝑙𝑛2) = 𝑡𝑚𝑒 → 𝛼 = 𝑡𝑚𝑒 − 𝛽 (𝑙𝑛2) (16) By equating equations (17) and (18), we get 𝑡𝑚𝑒 − 𝛽(𝑙𝑛2) = 𝑡(1) −

𝛽 𝑛

→ 𝑡𝑚𝑒 − 𝑡(1) = 𝛽(𝑙𝑛2) −

𝛽 𝑛

1 𝑡𝑚𝑒 − 𝑡(1) = 𝛽 ((ln2) − ) 𝑛 𝛽̂𝑀𝐷 (𝑡𝑚𝑒 −𝑡(1) ) = 1 ((ln2) − 𝑛)

(17)

Then the approximate estimation of Reliability function for the two-parameter Exponential distribution using Median-First Order Statistics Method (MD) will be 𝑅̂𝑀𝐷 = 𝑒𝑥𝑝 (−

(𝑡𝑖 − 𝛼) ) 𝛽̂𝑀𝑑

(18)

Ridge Regression method (RR) "The ridge regression (RR) estimates of A and B can be obtained by minimizing the error sum of squares for the model 𝑌𝑖 = a+b𝑋𝑖 Subject to the single constraint that a2+b2= ρ where ρ is a finite positive constant "; [1]. i.e.; 𝐿 = ∑𝑛𝑖=1(𝑌𝑖 − 𝑎 − 𝑏𝑋𝑖 )2 + λ(𝑎2 + 𝑏 2 − 𝜌) w.r.t a and b. when these derivatives are equated to zero, we obtain the following two equations 𝑛

𝑛

∑ 𝑌𝑖 = (𝑛 + 𝜆)𝑎 + 𝑏 ∑ 𝑋𝑖 𝑖=1 𝑛

𝑖=1 𝑛

𝑛

∑ 𝑋𝑖 𝑌𝑖 = 𝑎 ∑ 𝑋𝑖 + 𝑏(𝜆 + ∑ 𝑋𝑖2 ) 𝑖=1

𝑖=1

𝑖=1

Solving above two equations for a and b we get 𝑎̂ =

(∑ni=1 Xi ) (∑ni=1 Xi Yi ) − ∑ni=1 Yi (λ + ∑ni=1 X i2 )

2 (∑ni=1 Xi ) − (n + λ) (λ + ∑ni=1 Xi2 ) For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/ www.ihsciconf.org

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𝑏̂ =


(∑𝑛𝑖=1 𝑋𝑖 )(∑𝑛𝑖=1 𝑌𝑖 ) − (𝑛 + 𝜆) ∑𝑛𝑖=1 𝑋𝑖 𝑌𝑖 2

(∑𝑛𝑖=1 𝑋𝑖 ) − (𝑛 + 𝜆)(𝜆 + ∑𝑛𝑖=1 𝑋𝑖2 )

For two parameter Exponential distribution when α is known we recognize that 𝑌𝑖 = 𝑡𝑖 , b = 𝛽 𝑋𝑖 = [-ln (1-F (𝑡𝑖 ))]

𝑖 = 1, 2,…, n .

𝛽̂ =

∑ 𝑡𝑖 ∑(− ln(1 − 𝐹(𝑡𝑖 ))) − (𝑛 + 𝜆) ∑ 𝑡𝑖 (− ln(1 − 𝐹(𝑡𝑖 ))) 2

[∑(− ln(1 − 𝐹(𝑡𝑖 )))]2 − (𝑛 + 𝜆)(𝜆 + ∑[− ln(1 − 𝐹(𝑡𝑖 ))] )

(19)

Then the approximate estimation of Reliability function for the two-parameter Exponential distribution using ridge regression technique (RR) will be 𝑅̂𝑅𝑅 = 𝑒𝑥𝑝 (−

(𝑡𝑖 − 𝛼) ) 𝛽̂𝑅𝑅

(20)

Where 0