Ijetcas15 605

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International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)

ISSN (Print): 2279-0047 ISSN (Online): 2279-0055

International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS) www.iasir.net Minimax Estimation of the Scale Parameter of the Laplace Distribution under Different Loss Function Huda, A. Rasheed AL-Mustansiriyah University, College of Science, Department of Math, Baghdad, Iraq Emad, F. AL-Shareefi Southern Technical University, Basra, Iraq and Nasiriyah Technical Institute, Iraq __________________________________________________________________________________________ Abstract: In this paper, we obtained Minimax estimators of the scale parameter đ?œƒ for the Laplace distribution under different loss function by applying the theorem of Lehmann [1950]. Some Bayes estimators for the unknown scale parameter đ?œƒ of Laplace distribution have been obtained, using Non-informative prior (Jeffreys) and Informative priors (Gumbel Type II) under different loss function, represented by (Quadratic loss function, Squared – log error loss function, Entropy loss function). According to Monte-Carlo simulation study, the performance of these estimates is compared depending on the mean squared errors (MSE’s). Key words: Minimax estimator; Laplace distribution; Bayes estimator; Quadratic loss function; Entropy loss function; Squared-log error loss function; Jeffery prior; Gumbel Type II prior; Mean squared error. _________________________________________________________________________________________ I. Introduction Laplace distribution has been used for modeling data that have heavier tails than those of the normal distribution and analyze engineering, financial, industrial, environmental, and biological data (Kotz et al., 2001)[5]. It is used for modeling data that have heavier tails than those of the normal distribution. The probability density functions of a Laplace distributed random variable is given by: [9] |đ?‘Ľâˆ’đ?‘Ž| 1 đ?‘“(đ?‘Ľ|đ?‘Ž, θ) = đ?‘’đ?‘Ľđ?‘? [− ] −∞<đ?‘Ľ <∞ (1) θ

2θ

Where đ?‘Ž ∈ (−∞,∞) and θ > 0 are location and scale parameters, respectively. The cumulative distribution function is given by: 1

a−x

1 − exp [ ] for x ≼ a F(x| a, θ) = { 1 2 x−a θ exp [ ] for x < đ?‘Ž 2 θ With moment –generating function defined as: Îœđ?‘Ľ (đ?‘Ą) =

đ?‘’ đ?‘Ąđ?œ‡ 1−đ?‘? 2 đ?‘Ą 2

II. Bayes Estimators A. Jefferys Prior Information [9] Assume that, đ?œƒ has non-information prior density, defined g âˆ? √I(θ) Where I(θ) represented Fisher information which defined as follows:

(2)

đ?œ•2 đ?‘™đ?‘› đ?‘“(đ?‘Ľ;đ?‘Ž,đ?œƒ)

I(θ) = −đ?‘›đ??¸ [

] . Hence

đ?œ•đ?œƒ 2

đ?‘”(đ?œƒ) = đ?‘˜âˆšâˆ’đ?‘›đ??¸ (

đ?œ•2 đ?‘™đ?‘› đ?‘“(đ?‘Ľ;đ?‘Ž,đ?œƒ) đ?œ•đ?œƒ 2

đ?‘™đ?‘› đ?‘“(đ?‘Ľ; đ?‘Ž, đ?œƒ) = − đ?‘™đ?‘›(2đ?œƒ) − đ?œ• ln đ?‘“(đ?‘Ľ;đ?‘Ž,đ?œƒ)

=−

đ?œ•đ?œƒ đ?œ•2 ln đ?‘“(đ?‘Ľ;đ?‘Ž,đ?œƒ)

=

1 2đ?œƒ 1

đ?œ•đ?œƒ 2 2đ?œƒ 2 đ?œ•2 ln đ?‘“(đ?‘Ľ;đ?‘Ž,đ?œƒ)

− − 1

|đ?‘Ľâˆ’đ?‘Ž|

)

(3)

|đ?‘Ľâˆ’đ?‘Ž| đ?œƒ

đ?œƒ2 2|đ?‘Ľâˆ’đ?‘Ž| đ?œƒ3 2

đ??¸[ ] = 2 − 3 đ??¸[|đ?‘Ľ − đ?‘Ž|] đ?œ•đ?œƒ 2 2đ?œƒ đ?œƒ Let X~Laplace(đ?‘Ž, đ?œƒ)đ?‘Ąâ„Žđ?‘’đ?‘›, [5] |đ?‘Ľđ?‘– − đ?‘Ž| = đ?œƒW Where (W ) is standard exponential with mean and variance equal to one. Hence, |đ?‘Ľđ?‘– − đ?‘Ž|~đ??şđ?‘Žđ?‘šđ?‘šđ?‘Ž(1, đ?œƒ −1 )

IJETCAS 15-605; Š 2015, IJETCAS All Rights Reserved

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