Pujab Uiversity Joural of Mathematics ISSN 1016-2526) Vol. 502)2018) pp. 11-19 O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers P. Nasiri Departmet of Statistics, Uiversity of Payam Noor, 19395-4697 Tehra, I.R. Ira. Email: pasiri@pu.ac.ir, pasiri45@yahoo.com M. Jabbari Nooghabi Departmet of Statistics, Ferdowsi Uiversity of Mashhad, Mashhad-Ira. E-mail: jabbarim@um.ac.ir, jabbarim@yahoo.com *Correspodig Author Received: 21 July, 2017 / Accepted: 11 October, 2017 / Published olie: 18 Jauary 2018 Abstract. I this article, we have estimated the scale parameter of expoetial distributio with a prior iformatio. A shrikage estimator is derived for parameter of expoetial distributio cotamiated with outliers ad i the presece of LINEX loss fuctio. A admissible estimator based o the LINEX loss fuctio are compared with differet methods of estimatios. Numerical study are used to compare the estimators. AMS MOS) Subject Classificatio Codes: 62Fxx; 62F10; 62F15 Key Words: Expoetial distributio, LINEX loss fuctio, Prior iformatio, Outlier, Shrikage estimatio. 1. INTRODUCTION The followig probability desity fuctio pdf) which is expoetial distributio is ofte used i life-testig research. fx, ) = 1 e x, x > 0, > 0. Let X 1, X 2,, X ) is a radom sample of size which is derived from the expoetial distributio. I this distributio, is kow as the scale parameter ad is the mea life. is the average time to failure ad X is a ubiased estimator of it. By usig the squared error loss fuctio SELF) which is a symmetric, it is ot appropriate to estimate mea of 11
12 P. Nasiri ad M. Jabbari Nooghabi life reliability fuctio. [20] ad [18] itroduced a ew versio of loss fuctio which is asymmetric ad kow as the LINEX loss fuctio LLF). This approach was modified by [6] ad [8]. Also, a modified versio of LINEX loss fuctio is exist which is geeral etropy loss fuctio ad proposed by [7]. For ay parameter, a LLF which is ivariat is f ) = e a a 1, a 0, = ˆ 1 ), based o [6]. I this loss fuctio, the grade of asymmetry ad orietatio are deped o sig ad magitude of a, respectively. The positive value of a is usually cosidered whe overestimatio is more beefit tha the uderestimatio ad the egative value is take i the reverse situatios. Whe a is aroud zero, the loss fuctio is almost squared error ad it is a symmetric. Details are foud i [13] ad [2]. A miimum mea square error MMSE) estimator of parameter i the expoetial distributio is obtaied by [15] ad defied as X +1 uder the class of k X. [13] have used the Searls s estimator ad explaied that it was iadmissible uder the LLF. [19] derived optimal shrikage estimatios for the parameters of expoetial distributio based o record values. [14] preseted shrikage estimatio of the parameter of expoetial type-ii cesored data uder LLF. [4] have estimated P X < Y ) i the expoetial distributio with commo locatio parameter by usig shrikage method. [12] discussed several methods of shrikage estimatio of P Y < X) i the expoetial distributio mixig with expoetial distributio. [1] have used beta priors ad ew Bayes estimators of populatio proportio of respodets possessig stigmatized attribute to exted Magat Radomized Respose Techique. Also, [16] have cosidered a ew methodology for Bayesia aalysis of mixture models uder doubly cesored samples. They evaluated the Bayesia estimatio of parameters of the two-compoet mixture of Rayleigh distributio uder square root gamma, Maxwell ad half ormal priors usig two loss fuctios. Further, [3] have preseted a modified factor-type estimator uder two-phase samplig. This method is foud by icorporatig iformatio like coefficiet of variatio, kurtosis, skewess ad correlatio coefficiet. Based o [17], whe 0 idicates a cojecture of a shrikage estimator is sh = cˆ 0 ) + 0. Oe ca evaluate the shrikage factor c deped o the guessed value 0. This method of estimatio is ow used i differet subjects. If we assume source distributed which may be a small plot of plats. Whe the weather is ormal, the plats distribute the polle ad it itersperse such as a expoetial distributio far from the origi. Also, i some situatio, for example i fog or light rai, the herbage reduced their diffusio of polle, but still expoetially distributed with other scale parameter. [9] have assumed viral spores BYMDV) ad estimated the parameters of its pdf. So, i this paper we costructed the structure as: Sectio 2 is to preset desity of X 1, X 2,, X ) cotamiated with outliers. I sectios 3 ad 4, shrikage estimators of the scale parameter of a expoetial distributio cotamiated with outliers uder squared error ad LLF are discussed. I sectio 5, the miimum risk of the two loss fuctios are derived.
O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers 13 2. JOINT DENSITY OF X 1, X 2,, X ) WITH OUTLIERS Let a radom sample of size shows the iterval of a sampled plat from a plat from a plot of plats which is ifected by a virus. Here, most of data are comig from the airbore dispersal of the spores ad follow the expoetial distributio. A small umber of data from radom variables deote by k) are remaied ad to be trasport barley yellow mosaic dwarf virus BYMDV) have moved the virus ito the plats while the aphids cuisies o the sap. [10] have estimated the parameters of the expoetial distributio cotamiated with outliers which is comig from uiform distributio. The, cosider X 1, X 2,, X ) are distributed such that k of them are comig from pdf gx,, ) gx,, ) = e x, x > 0, > 0, > 0, 2. 1) ad the other k) are geerated from pdf fx, ) as So, the joit pdf of X 1, X 2,, X ) is fx 1, x 2,, x ) = fx, ) = 1 e x, x > 0, > 0. 2. 2) k! k)!! fx i, ) i=1 k gx Aij,, ), 2. 3) fx Aij, ) with = k+1 k+2 A 1 =1 A 2 =A 1 +1 A k =A k 1 +1. The, for gx,, ) ad fx, ) which are give i 2.1) ad 2.2), respectively, fx 1, x 2,, x ) is fx 1, x 2,, x ) = = = k! k)!! e xi k! k)!k! e xi k! k)!k! e xi k j=1 j=1 k j=1 k j=1 x Aj e x Aj e x Aj e x Aj e e 1) x A j. 2. 4) Therefore, by usig equatio 2.4), we obtai the margial desity of X as follows. fx,, ) = k k gx,, ) + fx, ) = k e x + k e x, x > 0. 3. SHRINKAGE ESTIMATION OF WITH SELF Here, the shrikage estimator of parameter whe 0 is a guess value of it, is give by sh = cˆ 0 ) + 0, c [0, 1].
14 P. Nasiri ad M. Jabbari Nooghabi Let us suppose that shsel = c X 0 ) + 0 be the estimator i the SELF. Assume that this risk deotes by R s. Hece R s = E[ shsel ] 2 = E[ c X 0 ) 0 ] 2 = E[ + c 1) 0 c X] 2 = [ + c 1) 0 ] 2 + c 2 E X 2 ) 2c[ + c 1) 0 ]E X). By usig the margial distributio of X, E X) ad V X) are obtaied respectively as ad E X) = k Hece R s = [ + c 1) 0 ] 2 + c2 2 [ k 2 Let ad the, R s will be 2c [ + c 1) 0] k) +, [ ] V X) = 2 k 2 3 + k)2. 2 2 k)2 + 2 ]. [ k + k) A = k2 k)2 + + k2 2 2 + k)2 + B = k + k), + k2 2 + k)2 + 2k k), ] 2k k) R s = [ + c 1) 0 ] 2 + c2 2 2 A 2c [ + c 1) 0]B. 3. 5) Now, we have to miimized the risk ad fid c 0 such that Therefore ad dr s dc dr s dc = 0. = 0[ + c 1) 0 ] + c2 2 A [ + c 1) 0]B c 0 B = 0, For 0 =0 ad c 0 = B A, it is give by [10]. Hece c 0 = 2 0 + 2 B + B) 0 2 0 + A 2 0B. shsel = c 0 X 0 ) + 0.
O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers 15 Substitute c 0 i 3.5), imply that MiR s = [ + c 0 1) 0 ] 2 + c2 0 2 2 A 2c 0 [ + c 0 1) 0 ]B. 4. SHRINKAGE ESTIMATOR WITH LLF I this sectio, a estimator of is derived uder LLF based o shrikage method. The followig fuctio is LLF ) f ) = e a a 1, a 0, = where shll = cˆ 0 ) + 0. Let us suppose that shll = c X 0 ) + 0, shll 1, be the estimator uder LLF ad the risk is deoted by LR s. Hece LR s = EL )) = E e a a 1 ), where ad So where = shll 1 = 1 [cˆ c 0 + 0 ] 1 = cˆ + 1 c) 0 1 = cˆ + s, s = 1 c) 0 1. LR s = EL )) = E e a a 1 ) ) = E e a cˆ +s cˆ a + s 1 = e as E e ac Eˆ) = E X) = k ) ˆ ) ac Eˆ) as 1, k) +,
16 P. Nasiri ad M. Jabbari Nooghabi Therefore ) E e ac ˆ = E LR s = e as k ac) e ac k = 0 e acx = k 0 = k ac) ) X = E e ac ) Xi = E e ac X) x k) e dx + ) x k dx + e 1 ac + k) 1 1 ac). 0 0 e acx x e dx 1 e 1 ac) x dx ) ) k) k k) + ac + as 1. 1 ac) Now, we have to miimize the risk. Hece, c 0 is foud such that dlr s dc hc) = a 0 a eas k ac) k k) + ) k) + + e as 1 ac) ka ac) 2 + = 0, ie. ) k)a 1 ac) 2 ) + a 0 = 0. 4. 6) Therefore, hc) = 0 is solved by usig Newto-Raphso method as c j = c j 1 hc j 1) h, j = 1, 2,..., c j 1 ) where h c) = dhc) dc. After selectio the proper value of c, LR s will be miimum for c = c 0, where c 0 is the solutio of the equatio 4.6). 5. NUMERICAL EXPERIMENTS AND DISCUSSIONS Here, to see the performace of the estimators ie. R s ad LR s, samplig experimets by usig R statistical software are used. The results are give i Tables 1-6 for k=1,2, =0.5, 1.5, =2,3, 0 =0.75 ad a=-0.2. Bias of ˆ ie. Bias )) is defied as Eˆ). Table 1. Bias of the estimators, R s ad LR s for k=1, =1.5, =2, 0 =0.75 ad a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-7.370582 6461.77-0.867558 0.002586 20-7.622106 5706.11-0.874153 0.002691 30-6.558582 1661.77-0.888976 0.002655 40-6.317610 862.48-0.889309 0.002671 50-6.094307 702.66-0.896411 0.002644 60-6.102083 677.75-0.889351 0.002701
O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers 17 Table 2. Bias of the estimators, R s ad LR s for k=2, =1.5, =2, 0 =0.75 ad a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-6.671847 1735.64-0.885200 0.002698 20-6.462133 1665.02-0.879905 0.002660 30-6.379222 1380.26-0.873978 0.002758 40-6.145077 724.14-0.885146 0.002711 50-6.115715 683.22-0.881738 0.002747 60-5.970897 611.50-0.890469 0.002696 Table 3. Bias of the estimators, R s ad LR s for k=1, =1.5, =3, 0 =0.75, a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-19.151860 63021.03-1.384436 0.004786 20-16.937740 19807.44-1.389674 0.004950 30-15.755640 13068.62-1.413402 0.004930 40-15.565560 11551.14-1.407027 0.004982 50-15.369630 10245.82-1.406485 0.005003 60-14.905750 8715.08-1.421279 0.004972 Table 4. Bias of the estimators, R s ad LR s for k=2, =1.5, =3, 0 =0.75 ad a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-17.621820 75474.21-1.340366 0.004911 20-15.478530 14127.74-1.410833 0.004901 30-15.574820 12204.55-1.393489 0.005014 40-15.120370 10115.33-1.406576 0.005007 50-14.813120 8989.30-1.417287 0.004981 60-15.012150 8839.64-1.403421 0.005041 Table 5. Bias of the estimators, R s ad LR s for k=1, =0.5, =3, 0 =0.75 ad a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-31.177520 133815.00-1.512323 0.004325 20-20.504490 52664.87-1.474874 0.004652 30-18.030170 23910.56-1.457887 0.004785 40-17.160260 18089.33-1.442318 0.004867 50-16.324460 13794.63-1.446929 0.004867 60-16.018730 12267.45-1.440376 0.004900 Table 6. Bias of the estimators, R s ad LR s for k=2, =0.5, =3, 0 =0.75 ad a=-0.2. Bias shsel ) R s Bias shll ) LR s 10-31.341310 122380.00-1.624322 0.003857 20-24.419250 105833.60-1.529319 0.004465 30-19.762300 39089.02-1.507678 0.004602 40-18.353870 24394.75-1.477247 0.004758 50-17.301090 18342.36-1.472677 0.004784 60-17.016030 15612.78-1.451695 0.004880 From Tables 1 to 6 LR s is less tha R s. Also, comparig the bias shows that the absolute value of bias of the estimators of is decreasig respect to. I additio, the bias
18 P. Nasiri ad M. Jabbari Nooghabi of shrikage estimator of uder LLF is less tha the bias of the correspodig estimator uder SELF. Further more, the values of LR s ad R s are decreasig whe icreased. Overall coclusio is that we ca say that to estimate uder g ad f the LLF has better performace tha the SELF. 6. ACTUAL EXAMPLE To show the performace of the estimators, we have selected a actual example which is based o a data set discussed by [11] ad [5]. Whe the size of the crushed rock is larger tha the rocks crushed, a rock crushig machie is workig. Else, it has bee reset. The followig data shows the sizes of the crushed rocks up to the third reset of the machie. 9.30, 0.60, 24.40, 18.10, 6.60, 9.00, 14.30, 6.60, 13.00, 2.40, 5.60, 33.80 By usig Kolmogorov-Smirov Statistic=0.20685, critical value at the level of 5%=0.37543 ad p=0.61239) ad Aderso-Darlig Statistic=0.33653, critical value at the level of 5%=2.5018) tests, data follows expoetial distributio with parameter ˆ=11.97461. Here, we assume that =1.5, 0 is the media ad a=-0.2. So, the shrikage estimators of, R s, LR s ad the correspodig likelihood respect to k=1, 2 ad 3 are show i Table 7. Table 7. Shrikage estimators of, R s, LR s ad the correspodig likelihood for k=1,2,3, 0 =0.75 ad a=-0.2. k shsel R s L shsel ) shll LR s L shsel ) 1 8.919453 0.1000459 3.401296e-19 9.332078 7.137392e-05 4.115446e-19 2 8.864743 0.1514201 2.779157e-19 9.372426 8.634380e-05 3.613990e-19 3 8.800492 0.2294922 2.204463e-19 9.419562 1.044919e-04 3.149668e-19 Table 7 shows that the likelihood is maximized whe k=1 uder both loss fuctios. Therefore, i the example the umber of outliers k) is equal to 1 ad the value of the shrikage estimators of ukow parameter, R s ad LR s could be take from the first row of Table 7. 7. ACKNOWLEDGMENTS We would like to thak the respected editors. We would also like to thak all the respected referees for valuable commets. REFERENCES [1] A. O. Adepetu ad A. A. Adewara, Extesio of Magat Radomized Respose Techique Usig Alterative Beta Priors, Pujab Uiv. J. Math. Vol. 48, No. 1 2016) 29-45. [2] J. Ahmadi, M. Doostparast ad A. Parsia, Estimatio ad predictio i a two-parameter expoetial distributio based o k-record value uder LINEX loss fuctio, Commuicatio i Statistics-Theory ad Methods 34, 2005) 795-805. [3] A. Audu ad A. A. Adewara, A Modified Factor-type Estimator uder Two-phase Samplig, Pujab Uiv. J. Math. Vol. 49, No. 2 2017) 59-73. [4] A. Baklizi ad A. E. Q. El-Masri, Shrikage estimatio of P X < Y ) i the expoetial case with commo locatio parameter, Metrika 59, 2004) 163-171. [5] N. Balakrisha ad P. S. Cha, Record values from Rayleigh ad Weibull distributios ad associated iferece, Natioal Istitute of Stadards ad Techology Joural of Research, Special Publicatio, Proceedigs of the Coferece o Extreme Value Theory ad Applicatios 866, 1994) 4151. [6] A. P. Basu ad N. Ebrahimi, Bayesia approach to life testig ad reliability estimatio usig asymmetric loss fuctio, Joural of Statisical Plaig ad Iferece 19, 1991) 21-31.
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