Comparing Different Estimators of Reliability Function for Proposed Probability Distribution

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1 American Journal of Mathematics and Statistics 21, (2): DOI: 192/j.ajms.212. Comparing Different Estimators of Reliability Function for Proposed Probability Distribution Dhwyia S. Hassun 1, Nathie r A Ibrahim 2,*, Suhail N. Abood 1 Department of Industrial Management, College of Administration & Economic, University of Baghdad, Iraq 2 Department of Banks Economics, College of Business Economics, Al Nahrain University, Iraq Computer Center, College of Administration & Economic, Baghdad University, Baghdad, Iraq Abstract This paper deals with building a probability distribution model, which represents some generalizations of two parameters Weibull distribution. This function is proposed using entropy like transformation, to propose uu(tt;, pp), using the cumulative distribution function { F(t) }, and the reliability function R(t), to model the time to failure of any given component when this time is different from component to one and it is not stable but have depression. After finding the proposed probability density function it is tested and the parameters of derived density is estimated by different methods including proposed method. The comparison between methods of estimator is done through simulation using mean square error as statistical tool for comparison. All the results are explained in tables. Keywords Weibull Distribution, Reliability Function, Entropy Like Transformation, Proposed Probability Density Function, MLE, Minimax Estimator, Posterior Distribution 1. Introduction Weibull two parameters distribution is considered a good model for studying time to failure of units or equipment, as well as Rayleigh and mechanical devices as drilling tools and Gamma distributions. Sometimes there is depression happen in time failure, so we work to find some probability distribution that deals with this kind of data, here this distribution is found when entropy like transformation is applied on F(t) and R(t) of two parameters Weibull to obtain a new proposed family, this function is obtained using the definition [8] ; gg(tt) = FF(tt) + RR(tt) ln{rr(tt)} (1) And then, uu(tt;, pp) = gg (tt) (2) Here we have; FF (tt;, pp) = 1 ee ttpp tt, pp, > Where, pp: Shape parameter : Scale parameter And, Using (1), we get; RR(tt) = ee ttpp * Corresponding author: nbrhms@yahoo.com (Nathier A Ibrahim) Published online at Copyright 21 Scientific & Academic Publishing. All Rights Reserved gg(tt) = 1 ee ttpp + ee ttpp ttpp () Proposed function uu(tt;, pp) obtained from gg (tt), and from this we obtain; gg (tt) = uu(tt ;, pp) = pp 2 tt2pp 1 ee ttpp tt, pp, > (4) The function in equation (4) is probability density function (pp. dd. ff) since, uu(tt ;, pp)dddd = 1 Some of statistical aspects of (4) can be obtained like (C.D.F), FF (tt), and reliability function R(t) also. The general form of r th moment is derived mm rr = EE(tt rr ). First of all the cumulative distribution function (C.D.F) is; FF (tt) = pppp( tt) = ff(zz)dddd tt tt zz pp = pp 2 zz2pp 1 ee dddd (5) Using integration by parts where; uu = zzpp, zz pp pp zzpp 1 dddd = ee dddd And applying formula { uuuuuu = uuuu vvvvvv}, we obtain the cumulative distribution function (C.D.F); FF (tt) = uu (tt) = 1 ee ttpp ttpp ee ttpp (6) From (6) we obtain; uu(tt;, pp)dddd = FF (tt) = pp 2 tt2pp 1 ee ttpp, tt, pp, > pp: Shape parameter : Scale parameter

2 American Journal of Mathematics and Statistics 21, (2): The cumulative D.F in (6) uu (tt) represents the C.D.F for proposed distribution uu(tt;, pp) defined in (4). According to (4) and (6) the reliability function for uu(tt;, pp) is denoted by; RR uu (tt) = 1 uu (tt) (7) RR uu (tt) = ee ttpp + ttpp tt pp ee = (1 + ttpp ) ee ttpp (8) Now the r th moment of the random variable (T) having density uu(tt;, pp) (equation 4) is given by; Let; mm rr = EE( rr ) = tt rr uu(tt;, pp)dddd = pp yy = ttpp yy = ttpp pp = 1 +2pp 1 ttrr 2 ee ttpp dddd dddd = ppttpp 1 dddd tt pp = yy tt = yy pp 1 ttrr+pp 2 ttrr+pp rr+pp = 1 yy pp 1 tt pp 1 ee ttpp dddd ee ttpp pp ttpp 1 dddd ee yy dddd = 1 rr +pp yy pp 1 rr +pp pp ee yy dddd = 1 rr pp +2 yyrr pp +1 ee yy dddd rr mm rr = 1 pp +2 ΓΓ rr + 2 (1) pp The r th moment for the proposed (pp. dd. ff) can be used in obtaining moment estimators, since expression (1) can be used easily to obtain the mean, variance and coefficients of skewness and kurtosis of distribution in (4). 2. Estimation of Parameter We using the method of maximum likelihood to estimate the parameters (, pp) of the density uu(tt;, pp). If ( 1, 2,, ) are times to failure fo r a given samp le of size (), taken fro m uu(tt;, pp), then the likelihood function LL(;, pp) is; Fro m; LL = pp tt 2pp 1 2 ii exp ( tt ii ) (11) ln LL = lnpp 2 ln +(2pp 1) lntt ii tt pp ii=1 ii (12) ln LL = pp + 2 lntt ii tt ii ln LL pp pp pp (1) lntt ii (1) 2 + tt ii (14) 2 Solving (1 &14) for (, pp), gives; pp tt ii 2 = 2 tt ii pp MMMMMM = (15) lntt pp ii = 1 pp (ln tt ii )(tt ii ) MMMMMM pp ) = (2) (lntt ii )(tt ii pp (tt ii ) pp MMMMMM = (2 ) pp ii=1 (ln tt ii )(tt ) ii pp ii=1 (tt ) ii 2 ln tt ii (16) Equation (16) is non linear equation can be solved by Newton Raphson or fixed point to find MLE for (pp) and then this estimated value can be used in (15) to find ( MMMMMM ). This can be explained through simulation. The second method of estimation is method of moment, since the first moment is; mm 1 = 1 1 pp +2 Γ 1 pp + 2 And second moment ii mm 2 = 1 2 pp +2 Γ 2 pp + 2 Therefore equating sample moments since; mm 1, mm 2 with first and second mm rr = tt ii rr mm 1 = μμ 1 tt ii 1 = 1 pp +2 Γ pp (17) Fro m mm 2 = μμ 2 tt ii 2 2 = 1 pp +2 Γ pp (18) tt = 1 1 pp +2 Γ 1 pp + 2 If pp = 1 as (special case), this gives; But when pp 1 tt 2 = 1 tt = 1 Γ() mmmmmm = 2 tt Γ pp tt = 1 pp +2 mmmmmm = mmmmmm 1 pp +2 Γ 1 pp + 2 = Γ 1 tt pp +2 tt = 2 tt pp 1+2pp (19) Which is an implicit function of ( pp ) can be solved numerically.

3 86 Dhwyia S. Hassun et al.: Comparing Different Estimators of Reliability Function for Proposed Probability Distribution The estimator ( mmmmmm ) from (19) can be put in (18) to find (pp mmmmmm ), also by using numerical method.. Bayes Estimator Here we try to find another estimator for scale parameter ( ), for the ( pp. dd. ff) defined in (4), which is the Bayes estimator, where ( ) is considered as a random variable having prior distribution; ππ() = kk Γ(αα)ββ αα (αα +1) ee 1 ββββ, αα,ββ > (2) In order to find the Bayes estimator (dd 1 ) for () and the minimax estimator ( dd 2 ), first of all we try to find the posterior distribution [ππ( xx)]; Let; ππ( tt) = LL tt,,pp ππ () ff (tt ) LL tt,, pp ππ() = pp 2 tt 2pp 1 ii ff tt = pp tt ii 2pp ee tt pp ii = pp 2pp tt 1 ii kk 1 2 +αα+1 Γ(αα)ββ αα = kk αα+1 ee 1 dddd ee tt pp ii ππ()dddd ee 1 1 ββ + tt pp ii dddd yy = 1 = yy = yy, dddd = yy 2 dddd ff tt = kk 1 yy +1 2+αα ee yy yy 2 dddd Where; ππ( tt) = = kk 1 yy 2 +αα+1 ee yy yy 2 dddd = kk 1 2+αα yy2 +αα 1 ee yy dddd = kk 1 Γ(2 + αα) 2+αα kk 1 1 ππ( tt) 2+αα+1 ee = kk 1 2 +αα Γ(2 + αα) 2 +αα +αα+1 Γ (2+αα ) 1 2 ee (21),, αα, ββ > (22) = 1 ββ + ttiipp We can check ππ( tt)dddd = 1 ππ( tt) iiii aa pp. dd. ff under square error loss function. LL(, dd) = ( dd ) 2 RRRRRRRR = EE[LL(, dd)] = ( dd) 2 ff( tt)dddd = 2 ff( tt)dddd 2 ( tt)dddd + dd 2 ff( tt)dddd = 2EE ( tt) + 2dd = dd 1 = BBBBBBBBBB eeeeeeeeeeeeeeeeee Let; yy = = EE( tt) which is the posterior mean, i.e; EE( tt) = ππ( tt)dddd 2+αα 2 +αα+1 = Γ(2 + αα) 1 ee dddd 2 +αα 2 +αα = Γ(2 + αα) 1 ee dddd 1 = yy = yy, dddd = yy 2 dddd 2+αα EE( tt) = Γ(2 + αα) yy +αα 2 ee yy ( yy 2)dddd 2+αα +1 1 EE( tt) = Γ(2 + αα) 2 +αα yy2 +αα 2 ee yy dddd = Γ(2 + αα 1) Γ(2 + αα) (2 + αα 2)! = (2 + αα 1)! dd 1 = BBBBBBBBBB = 2+αα 1 Where = 1 ββ + tt ii pp (2) The second Bayes estimator is the minimax estimator (dd 2 ) which obtained under the following loss function; LL = dd 2 Since risk = expected loss; = dd 2 2 ππ( tt)dddd = 2 2 2dd 2 + dd 2 ππ( tt)dddd 2 = ππ( tt)dddd 2dd ππ ( tt)dddd + dd ππ ( tt)dd = 2EE 1 tt + 2dd 2EE 1 2 tt = EE 1 tt dd 2 = EE 1 2 tt Which is the minimax estimator, where; EE αα tt = EE 1 2 tt = ( 2 + αα)(2 + αα + 1) 2 Therefore; dd 2 = 2 + αα Application The application has been done through simulation procedure to compare between four estimators, for scale parameter of proposed probability distribution and then to estimate reliability. The comparison for scale parameter was

4 American Journal of Mathematics and Statistics 21, (2): done by MSE, and for reliability, the integrated mean square error used. Fro m table (1), for ( = 2), the estimator which have smallest MSE is proposed one (d2), and the MLE, and third one is moment estimator are finally (dd 1 ) which is Bayes estimator, so (dd 2 ), (Minimax estimator) is the best one among four different estimators ( = 2 ), same results reflect for the estimator of reliability, where ( dd 2 ) have smallest IMSE, and then MLE, Moment, finally (d1). Fro m table (4), for ( = 2), for all chosen set of parameters, the minimax estimator (d2) is the best, and then maximum likelihood and then (d1) and the fourth one is MOM. Also the estimator of reliability function have the order of performance (d2, MLE, d1 ) for ( = 5). Ta ble ( 1). Estimator of () by MLE, MOM and dd 1, dd 2 with MSE for each ( = 2) Methods of Estimations MLE MSE MOM MSE d 1 MSE d 2 MSE

5 88 Dhwyia S. Hassun et al.: Comparing Different Estimators of Reliability Function for Proposed Probability Distribution Ta ble ( 2). Estimation of Reliability function ( = 2) Reliability MLE MOM d 1 d

6 American Journal of Mathematics and Statistics 21, (2): Ta ble ( ). Integrated MSE for Reliability Estimator ( = 2) IMSE of Reliability MLE MOM d 1 d

7 9 Dhwyia S. Hassun et al.: Comparing Different Estimators of Reliability Function for Proposed Probability Distribution Ta ble ( 4). Estimator of () by MLE, MOM and dd 1, dd 2 with MSE for each ( = 5) Methods of Estimations MLE MSE MOM MSE d 1 MSE d 2 MSE

8 American Journal of Mathematics and Statistics 21, (2): Ta ble ( 5). Estimation of Reliability function ( = 5) Reliability.1.5 MLE MOM d 1 d

9 92 Dhwyia S. Hassun et al.: Comparing Different Estimators of Reliability Function for Proposed Probability Distribution Ta ble ( 6). Integrated MSE for Reliability Estimator ( = 5) IMSE of Reliability MLE MOM d 1 d

10 American Journal of Mathematics and Statistics 21, (2): Conclusions 1). The proposed distribution can be used for time to failure experiment when the data have depression. 2). The proposed estimator (d2) which is the minimax estimator for scale parameter (), have smallest MSE for all sample size and all chosen set of parameter. ). The maximum likelihood estimator for () is also good estimator for ( = 5), this from the fact that MLE g ives sufficient and efficient estimators for large sample size. 6. Suggestion 1). Generalize this new generated distribution, to another form which include location parameter for the system which have failure after certain time. 2). Generalize the concept of experiments to include ( = 1), where this size is necessary in application. ). Extend the research to three parameters Weibull family to obtain a new proposed one using entropy like transformation. REFRENCES [1] Al-athari, Faris M. Hassan, Dhwyia S. Ibrahim Nathier A. (212), "Using Decision Theory Approach to build a model for Bayesian Sampling Plans" American Journal of Mathematics & Statistics, Vol.2, No [2] Bather, J. (2). "Decision Theory". Chichester: John Wiley & Sons, Inc. [] Choudhury, A. (25). "A Simple Derivation of Moments of the Exponentiated Weibull Distribution". Metrika 62 (1):

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