Minimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions

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1 America Joural of heoretical ad Applied Statistics 6; 5(4): -7 doi:.648/j.ajtas ISSN: (Prit); ISSN: 6-96 (Olie) Miimax Estimatio of the Parameter of Maxwell Distributio Uder Differet Loss Fuctios Lapig Li Departmet of Basic Subjects, Hua Uiversity of Fiace ad Ecoomics, Chagsha, Chia address: o cite this article: Lapig Li. Miimax Estimatio of the Parameter of Maxwell Distributio Uder Differet Loss Fuctios. America Joural of heoretical ad Applied Statistics. Vol. 5, No. 4, 6, pp. -7. doi:.648/j.ajtas Received: May, 6; Accepted: Jue 6, 6; Published: Jue, 6 Abstract: he aim of this article is to study the Bayes estimatio ad miimax estimatio of the parameter of Maxwell distributio. Bayes estimators are obtaied with o-iformative quasi-prior distributio uder differet loss fuctios, amely, weighted squared error loss, squared log error loss ad etropy loss fuctios. he the miimax estimators of the parameter are obtaied by usig Lehma s theorem. Fially, performaces of these estimators are compared i terms of risks. Keywords: Bayes Estimator, Miimax Estimator, Squared Log Error Loss, Etropy Loss, Maxwell Distributio. Itroductio he Maxwell distributio was first itroduced by Maxwell i 86, ad sice the, the study ad applicatio of Maxwell distributio have bee received great attetio. I 989, yagi ad Bhattacharya [] firstly used Maxwell distributio to model products life, ad they obtaied the miimum variace ubiased estimator ad Bayes estimator of the parameter ad reliability. Chaturvedi ad Rai [] studied Bayesia reliability estimatio of the geeralized Maxwell failure distributio. Podder ad Roy [] studied the estimatio of the parameter of this distributio uder modified liear expoetial loss (MLINEX) fuctio. Bekker ad Roux [4] discussed the maximum likelihood estimator (MLE), Bayes estimators of the trucated first momet ad hazard fuctio of the Maxwell distributio. Dey ad Maiti [5] derived Bayes estimators of Maxwell distributio by cosiderig o-iformative ad cojugate prior distributios uder three loss fuctios, amely, quadratic loss fuctio, squared-log error loss fuctio ad MLINEX fuctio. he refereces [6-8] studied the reliability estimatio of Maxwell distributio based o ype-ii cesored sample, progressively ype-ii cesored sample ad radom cesored sample, respectively. Referece [9] obtaied some Bayes estimators uder quadratic loss fuctio usig o-iformative prior, represeted by Jefferys prior ad Iformative priors as Gumbel ype II ad Cojugate (Iverted Gamma ad Iverted Levy) priors. More details about Maxwell distributio ca be foud i refereces [-]. his paper is devoted to the miimax estimatio problem of the ukow scale parameter i the maxwell distributio with the followig probability desity fuctio (pdf) (Dey ad Maiti [5]): x f ( x; ) x e, x >, > () π he miimax estimatio is a importat part i the statistical estimatio area, which was itroduced by Abraham Wald []. he miimax estimatio theory has draw great attetio by may researchers. Roy et al. [4] studied the miimax estimatio of the scale parameter of the Weibull distributio for quadratic ad MLINEX loss fuctios. Podder et al. [5] derived the miimax estimator of the parameter of the Pareto distributio uder Quaratic ad MLINEX loss fuctios. Dey [6] discussed the miimax estimator of the parameter of Rayleigh distributio. Shadrokh ad Pazira [7] studied the miimax estimator of the parameter of the miimax distributio uder several loss fuctios. he purpose of this paper is to study maximum likelihood estimatio (MLE) ad Bayes estimatio of the parameter of Maxwell distributio. Further, by usig Lehma s theorem we derive miimax estimators uder three loss fuctios, amely, weighted squared error loss, squared log error loss ad etropy loss fuctios.

2 Lapig Li: Miimax Estimatio of the Parameter of Maxwell Distributio Uder Differet Loss Fuctios. Prelimiary Kowledge.. Maximum Likelihood Estimatio LetX ( X, X,, X ) be a sequece of idepedet ad idetically distributed (i. i. d.) radom variables of Maxwell distributio with pdf (), ad x x, x,, x ) is the ( observatio of X. he likelihood fuctio of for the give sample observatio is x i i l( ; x) ( x ) e () π Which is balaced ad L(, δ ) as δ or. he risk fuctio has miimum at the followig Bayes estimator: ˆ δ exp[ E(l X )]. (7) I may practical situatios, it appears to be more realistic to express the loss i terms of the ratio δ /. I this case, Dey et al. [] poited out that a useful asymmetric loss fuctio is etropy loss fuctio: ( ˆ δ δ L, ) l (8) hat is ( ) xi t / l( ; x) e e () ˆ he Bayes estimator uder the etropy loss is deoted by, give by ˆ δ [ ( )] E X. (9) where t x is the observatio of i Xi. By solvig log likelihood equatio, the maximum likelihood estimator of is easily derived as follows: ˆ M Ad by Eq. (), we ca easily show that is distributed the Gamma distributio Γ (, ), which has the followig probability desity fuctio ([5], p. 8): ( ; ) t f t t e, t >, > Γ( ) x t Here, Γ ( x) t e dt is the Gamma fuctio. +.. Loss Fuctio Loss fuctio plays a importat role i Bayesia aalysis ad the most commo loss are symmetric loss fuctio, especially squared error loss fuctio are cosidered most. Uder squared loss fuctio, it is to be thought the overestimatio ad uderestimatio have the same estimated risks. However, i may practical practical problems, overestimatio ad uderestimatio will have differet cosequeces. o overcome this difficulty, Zeller [8] proposed a asymmetric loss fuctio kow as the LINEX loss fuctio, Brow proposed a ew asymmetric loss fuctio for scale parameter estimatio i 968, which is called squared log error loss (Kiapoura ad Nematollahib [9]): (4) (5) L (, δ ) (lδ l ) (6). Bayesia Estimatio I this sectio, we estimate by cosiderig weighted square error loss, squared log error loss ad etropy loss fuctios. We further assume that some prior kowledge about the parameter is available to the ivestigatio from past experiece with the Maxwell model. he prior kowledge ca ofte be summarized i terms of the so-called prior desities o parameter space of. I the followig discussio, we assume the followig Jeffrey s oiformative quasi-prior desity defied as, π ( ), > () d Hece, d leads to a diffuse prior ad d to a oiformative prior. Combig the likelihood fuctio () ad the prior desity (), we obtai the posterior desity of is t h( x) Γ( d + ) d + ( d + ) t / e () his is a Gamma distributio Γ( d +, t ). heorem. Let X ( X, X,, X ) be a sample of Maxwell distributio with pdf (), ad x ( x, x,, x ) is the observatio of X. t i i x is the observatio of X. he (i) Uder the weighted square error loss fuctio ( δ ) L (, δ ) ()

3 America Joural of heoretical ad Applied Statistics 6; 5(4): -7 4 the Bayes estimator is give by ˆ δ E E X [ X ] d [ ] () (ii) he Bayes estimator uder the squared log error loss fuctio is come out to be Where Ψ( d + ) ˆ e δ exp[ E(l X )] / d l + y y e y Ψ ( ) l Γ ( ) dy d Γ( ) (4) is a Digamma fuctio. (iii) he Bayes estimator uder the etropy loss fuctio is obtaied as ˆ d δ [ ( )] E X XU ( ) Proof. (i) By formula (), we kow that he E X ~ Γ( d +, ), E[ X ], d ( d)( d ) [ X ] (5) hus, the Bayes estimator uder the weighted square error loss fuctio is give by ˆ δ For the case (ii) By usig (), d E X d E [ ] [ X ] d + ( d + ) / (l ) l E X e d Γ( d + ) d l Γ( d + ) l( / ) Ψ( d + ) l( / ) d he the Bayes estimator uder the squared log error loss fuctio is come out to be ˆ e δ exp[ E(l X )] Ψ( d + ) / (iii) By Eqs. () ad (7), the Bayes estimator uder the etropy loss fuctio is give by d d ˆ δ [ ( )] E X 4. Mimimax Estimatio he most importat elemets i the miimax approach are the specificatio of the prior distributio ad the loss fuctios by usig a Bayesia method. he derivatio of miimax estimators depeds primarily o a theorem due to Lehma which ca be stated as follows: Lemma (Lehma s heorem) If τ { F ; Θ} be a family of distributio fuctios ad D a class of estimators of. Suppose that δ D is a Bayes estimator agaist a prior δ R δ, distributio ( ) o Θ, ad the risk fuctio ( ) equals costat o Θ ; the δ is a miimax estimator of. heorem Let X ( X, X,, X ) be a sample of Maxwell distributio with probability desity fuctio (), the d (i) δˆ is the miimax estimator of the parameter for the weighted square error loss fuctio. Ψ( d + ) (ii) ˆ e δ is the miimax estimator of the / parameter for the squared log error loss fuctio. d (iii) δˆ is the miimax estimator of the parameter for the etropy loss fuctio. Proof. o prove the theorem we shall use Lehma s theorem, which has bee stated before. he we have to fid the Bayes estimator δ of. hus if we ca show that the risk of d is costat, the the theorem will be proved. (i) he risk fuctio of the estimator ˆ δ is d R( ) E L, ( ) ( ) ( ) ( ) + d E d E From the coclusio ~ Γ(, ), we have

4 5 Lapig Li: Miimax Estimatio of the Parameter of Maxwell Distributio Uder Differet Loss Fuctios he E E ( ) ( ), ( )( 4) R( ) ( ) d ( )( 4) ( d ) d + 4d ( )( 4) he R ( ) is a costat. So, accordig to the Lehma s theorem it follows that, ˆ δ is the miimax estimator for the parameter of the Maxwell distributio uder the weighted square error loss fuctio. Now we are goig to prove the case (ii) he risk fuctio of the estimator ˆ δ is R ( ) E L (, ˆ δ ) ( l ˆ δ l ) E E[l ˆ δ ] l E l[ ˆ δ ] + (l ) From the coclusio ~ Γ(, ), we have hus E(l ) Ψ( ) l + l ˆ E[l δ ] E( Ψ( d + ) l + l ) Ψ( d + ) [ Ψ( ) l + l ] + l Ψ( d + ) Ψ ( ) + l E[l ˆ δ ] E[ Ψ( d + ) l( / )] Ψ ( d + ) Ψ( d + ) E[l( / )] + E[(l / ) ] Usig the fact (l y) y (l ) Ψ ( ) e y y y Ψ( ) Γ( ) e y dy dy Γ( ) E Y Ψ [(l ) ] ( ) where Y ~ Γ(,). ad we ca show that U ~ Γ(,) hus Ψ ( ) + Ψ ( ) E[(l U) ] E[(l( / ) + l ) ] E[(l( / )) ] + l E(l( / )) + (l ) E[(l( / )) ] + l ( Ψ( ) l ) + (l ) he we get the fact E[(l( / )) ] Ψ ( ) +Ψ ( ) l Ψ ( ) + (l ) herefore ˆ E δ Ψ d + Ψ d + E + E [l ] ( ) ( ) (l( / )) [(l( / )) ] ( ) ( )[ ( ) l ] + Ψ ( ) + Ψ ( ) l Ψ ( ) + (l ) Ψ d + Ψ d + Ψ ( ) R E[l ˆ δ ] l E l[ ˆ δ ] + (l ) ( ) ( )[ ( ) l ] + Ψ ( ) + Ψ ( ) l Ψ ( ) + (l ) l [ Ψ( d + ) Ψ ( ) + l ] + (l ) Ψ d + Ψ d + Ψ ( d ) ( ) ( d ) ( ) ( ) Ψ + Ψ Ψ + + Ψ + Ψ he R ( ) is a costat. So, accordig to the Lehma s theorem it follows that, ˆ δ is the miimax estimator for the parameter of the Maxwell distributio uder the squared log error loss fuctio. Fially we are goig to prove the case (iii) he risk fuctio of the estimator ˆ δ is, d d R E L E l + ( ) ( ˆ δ ) d E l( d ) + l + E (l ) + d l( d) + l + Ψ( ) l + l + d l( d) + Ψ ( ) + l + he ( ) R is a costat. So, accordig to the Lehma s theorem it follows that, ˆ δ is the miimax estimator for the parameter of the Maxwell distributio uder the etropy loss fuctio.

5 America Joural of heoretical ad Applied Statistics 6; 5(4): Risk Fuctio he risk fuctios of the estimators ˆ δ, ˆ δ ad BL ˆ δ relative to squared error loss L ( ˆ, ) ( δ ) are deoted by R( ˆ δ ), ( ˆ R δbl) ad R ( ˆ δ ), respectively, are ca be easily show as R( ˆ δ ) E[ L ( ˆ δ, )] ˆ d E δ [( ) ] E[( ) ] risk fuctio L L L ( d ) ( d ) ( ) + f tdt t t, d d ( )( 4) ( ) ( ) + Ψ( d+ ) e R( ˆ δ ) [( ˆ E δ ) ] E[( ) ] / ( )( 4) Ψ( d+ ) Ψ( d+ ) 4e 4e + R( ˆ δ ) E[ L ( ˆ δ, )] ˆ d E δ [( ) ] E[( ) ] ( ) ( ) d d + ( )( 4) he Figs. -4 have plotted the ratio of the risk fuctios to, i.e. R( ˆ δ ˆ ) R( δ ) R( ˆ δ ) L, L, ad L risk fuctio Fig.. Ratio of the risk fuctios with Fig.. Ratio of the risk fuctios with. L L L L L L.5. L L L risk fuctio.5. risk fuctio Fig. 4. Ratio of the risk fuctios with Fig.. Ratio of the risk fuctios with. From Figs. -4, it is clear that o of the estimators uiformly domiates ay other. We therefore recommed choose the estimators accordig to the value of d whe the quasi-prior desity is used as the prior distributio.

6 7 Lapig Li: Miimax Estimatio of the Parameter of Maxwell Distributio Uder Differet Loss Fuctios 6. Coclusio I this paper, we have derived Bayes estimators of the parameter of Maxwell distributio uder weighted squared error loss, squared log error loss ad etropy loss fuctios. Simulatio results show that the risk fuctios of the estimators ˆ δ, ˆ δbl ad ˆ δ relative to squared error loss decrease as sample size icreases. Whe the sample size is large, such as >5, the risks are almost the same. Ackowledgemet his study is partially supported by Natural Sciece Foudatio of Hua Provice (No. 5JJ) ad Foudatio of Hua Educatioal Committee (No. 5C8). he author also gratefully ackowledge the helpful commets ad suggestios of the reviewers, which have improved the presetatio. Refereces [] yagi R. K. ad Bhattacharya S. K., 989. Bayes estimatio of the Maxwell s velocity distributio fuctio, Statistica, 9 (4): [] Chaturvedi A. ad Rai U., 998. Classical ad Bayesia reliability estimatio of the geeralized Maxwell failure distributio, Joural of Statistical Research, : -. [] Podder C. K. ad Roy M. K.,. Bayesia estimatio of the parameter of Maxwell distributio uder MLINEX loss fuctio. Joural of Statistical Studies, : -6. [4] Bekker A. ad Roux J. J., 5. Reliability characteristics of the Maxwell distributio: a Bayes estimatio study, Comm. Stat. heory & Meth., 4 (): [5] Dey S. ad Sudhasu S. M.,. Bayesia estimatio of the parameter of maxwell distributio uder differet loss fuctios. Joural of Statistical heory & Practice, 4 (): [6] Krisha H. ad Malik M., 9. Reliability estimatio i Maxwell distributio with ype-ii cesored data. Iteratioal Joural of Quality & Reliability Maagemet, 6 (): [7] Krisha H. ad Malik M.,. Reliability estimatio i Maxwell distributio with progressively ype-ii cesored data. Joural of Statistical Computatio & Simulatio, 8 (4): -9. [8] Krisha H. ad Vivekaad K. K., 4. Estimatio i Maxwell distributio with radomly cesored data. Joural of Statistical Computatio & Simulatio, 85 (7): -9. [9] Rasheed H. A., Khalifa Z. N., 6. Bayes estimators for the Maxwell distributio uder quadratic loss fuctio usig differet priors. Australia Joural of Basic ad Applied Scieces, (6): 97-. [] Gui W., 4. Double acceptace samplig pla for time trucated life tests based o Maxwell distributio. America Joural of Mathematical & Maagemet Scieces, (): [] Hossai A. M. ad Huerta G., 6. Bayesia Estimatio ad Predictio for the Maxwell Failure Distributio Based o ype II Cesored Data. Ope Joural of Statistics, 6 (): [] Modi K., Gill V., 5. Legth-biased Weighted Maxwell Distributio. Pakista Joural of Statistics & Operatio Research, (4): [] Wald A., 95. Statistical Decisio heory, Mc Graw-Hill, New York. [4] Roy M. K., Podder C. K. ad Bhuiya K. J.,. Miimax estimatio of the scale parameter of the Weibull distributio for quadratic ad MLINEX loss fuctios, Jahagiragar Uiversity Joural of Sciece, 5: [5] Podder C. K., Roy M. K., Bhuiya K. J. ad Karim A., 4. Miimax estimatio of the parameter of the Pareto distributio for quadratic ad MLINEX loss fuctios, Pak. J. Statist., (): [6] Dey S., 8. Miimax estimatio of the parameter of the Rayleigh distributio uder quadratic loss fuctio, Data Sciece Joural, 7: - [7] Shadrokh A. ad Pazira H.,. Miimax estimatio o the Miimax distributio, Iteratioal Joural of Statistics ad Systems, 5 (): [8] Zeller A., 986. Bayesia estimatio ad predictio usig asymmetric loss fuctio. Joural of America statistical Associatio, 8: [9] Kiapoura A. ad Nematollahib N.,. Robust Bayesia predictio ad estimatio uder a squared log error loss fuctio. Statistics & Probability Letters, 8 (): [] Dey D. K., Ghosh M. ad Sriivasa C., 987. Simultaeous estimatio of parameters uder etropy loss, J. Statist. Pla. ad Ifer., 5: 47-6.