The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

Transcription

1 Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, HIKARI Ltd, The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters of Beta-Biomial Distributio S. A. Salem Departmet of Statistics Faculty of Commerce Zagazig Uiversity W. S. Abu El Azm Departmet of Statistics Faculty of Commerce Zagazig Uiversity Copyright 2013 S. A. Salem ad W. S. Abu El Azm. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper samplig distributios for the maimum likelihood estimators of the Beta- Biomial model are obtaied umerically usig approimate radom umbers from the Beta-Biomial distributio. STATGRAPHICS package is used to obtai the best fitted distributios usig Chi-square ad Kolmogorov-Smirov tests. Keywords: Beta-Biomial distributio (BBD), Maimum likelihood estimatio (MLE), Samplig distributio, Chi-square ad Kolmogorov-Smirov tests

2 1264 S. A. Salem ad W. S. Abu El Azm 1. Itroductio The Beta-Biomial model is oe of the oldest discrete probability miture models ad is widely applied i the social, physical, ad health scieces, it was formally proposed by Skellam (1948), although the idea was suggested earlier by Pearso (1925) i a eperimetal ivestigatio of Bayes theorem. A familiar model i may applicatios is to assume that a observed set of couts is from a Biomial distributio with uit specific probabilities govered by a Beta distributio (the so-called Beta-Biomial model; BBM). Cosider a populatio i which for each member some evet occurs as the outcome of a Beroulli trial with fied probability P, give 0< P < 1, the umber of occurreces for i trials has the Biomial distributio with desity mass fuctio give by f ( ;, P) = P (1 P), = 0,1,..., (1.1) Suppose that P is a radom variable which follows a Beta distributio, 1 α 1 β 1 f ( P; αβ, ) = P (1 P), αβ, > 0, 0< P < 1 (1.2) B ( αβ, ) where α, β are ukow shape parameters ad B ( α, β ) is the complete Beta fuctio. Cosider the followig simple miture model 1 p ( ;, α, β) = f ( ;, P) f ( P; α, β) dp. 0 Usig equatios (1.1) ad (1.2), the Beta-Biomial model will be B( α +, β + ) p ( ;, αβ, ) =, = 0,1, 2,...,, αβ, > 0 (1.3) B ( αβ, ) The cumulative distributio fuctio c.d.f of X, ca be writte as follows: B ( α + j, β + j) F( ) =, j = 0 j B ( αβ, ) B( β+ j, α+ j) =, = 0,1,2,...,, αβ, > 0. j = 0 j ( + 1) B( j + 1, j + 1) B( αβ, ) where B ( α, β ) is the complete Beta fuctio, is o-egative iteger, ad α ad β are shape parameters. Curretly, there are little studies for estimatio of the ukow parameters of the Beta-Biomial distributio. Griffiths (1973) estimated the ukow

3 Samplig distributio of maimum likelihood estimators 1265 parameters of Beta-Biomial distributio usig maimum likelihood estimatio ad applied his results to the household distributio of the total umbers of a disease, Lee ad Sabavala (1987) developed a Bayesia procedures for the Beta- Biomial model ad, used a suitable reparameterizatio, establishes a cojugatetype property for a Beta family of priors. The trasformed parameters have iterestig iterpretatios, especially i marketig applicatios, ad are likely to be more stable proposed a Bayesia estimatio ad predictio for the Beta- Biomial Model, Tripathi et al. (1994) itroduced some alterative methods for estimatig the parameters i the Beta Biomial ad trucated Beta-Biomial models, Lee ad Lio (1999) itroduced a ote o Bayesia estimatio ad predictio for the Beta-Biomial model. The mai purpose of the paper is to eted the study of Lee ad Sabavala (1987) by umerical itegratio. This method ca be used for the geeral case of trials. Whe the umber of trials is two, the results are similar to those from Lee ad Sabavala (1987). Quitaa ad Wig-Kue (1996) itroduced a Bayesia estimatio of Beta-Biomial model by simulatig posterior desities ad Everso ad Braldlow (2002) preseted a Bayesia iferece for the Beta-Biomial distributio via polyomial epasios. I this paper we itroduced samplig distributios for the maimum likelihood estimators of the Beta-Biomial model umerically usig approimate radom umbers from the Beta-Biomial. STATGRAPHICS package is used to obtai the best fitted distributios usig Chi-square ad Kolmogorov-Smirov tests. 2. Maimum Likelihood Estimators (MLE) Lee ad Sabavala (1987), used the maimum likelihood method to estimate the ukow parameters for the Beta-Biomial distributio whe = 2 ad Lee ad Lio (1999) discussed some estimatio problem to estimate the ukow reparametrized parameters (π,θ ) whe 2. (a) Maimum Likelihood estimators for the ukow parameters of the Beta-Biomial distributio whe = 2. Lee ad Sabavala (1987) estimated the ukow parameters for Beta- Biomial distributio ( α, β ), usig the followig likelihood fuctio

4 1266 S. A. Salem ad W. S. Abu El Azm L ( α, β ) = [ p( ;, α, β )] = 0 ( α+ 1)( α+ 1)... α( + β 1)( + β 1)... β L( αβ, ) = = 0 ( α+ β+ 1)( α+ β+ 2)...( α+ β) (2.1) where f is the umber of the uits with occurreces has the Biomial distributio ad = f, the log likelihood is give by i = 0 = 0 l L = c+ f l[( α+ 1)( α+ 2)... α.( + β 1)( + β 2)... β] i = 0 f f l[( α+ β+ 1)( α+ β+ 2)...( α+ β)]. (2.2) The first derivatives of (2.2) with respect to α ad β are give by 1 l L = f A ( α, ) f A ( α + β, ) ad α = 1 1 l L = f A ( β, ) f A ( α + β, ) (2.3) β = 1 where A( β, ) = 1/( β) + 1/( β + 1) /( β + 1). Equatig (2.3) to zero, the maimum likelihood estimates ˆα ad ˆ β for α ad β respectively ca be obtaied as the solutio of the followig equatios ( ˆ, ) ( ˆ ˆ f A α f A α + β, ) = 0 ad = 1 = 1 f A ( ˆ β, ) f A ( αˆ + ˆ β, ) = 0 These may be solved umerically for obtaiig the estimates ˆα ad ˆ β. f (b)maimum Likelihood estimators for the ukow reparametrized parameters of the Beta-Biomial distributio whe 2. Lee ad Lio (1999) estimated the ukow reparametrized parameters ( π, θ ) of the Beta-Biomial distributio whe 2, the likelihood fuctio is give by f L ( α, β ) = p( ;, α, β ) = 0

5 Samplig distributio of maimum likelihood estimators 1267 π 1 they assumed that α = π, β = ( 1).(1 π ), Ai = (1 π)(1 θ) + iθ, θ θ B i = π (1 θ) + i θ ad Ci = (1 θ) + iθ, the reparametrized likelihood fuctio of π ad θ is f Ai f B i L( πθ, /,{ f }) = ( ( ) ) ( ((( B )( A )) /( C )) ) ( ( ) ) C (2.4) i i i i= 1 Ci i= 1 i= 0 i= 0 i= 0 i= 1 Takig the first derivatives of the atural logarithm of the likelihood fuctio (2.4) with respect to π ad θ, equatig the results to zero ad solvig these results umerically, the estimates ˆ π ad ˆ θ ca be obtaied. i f 3. Numerical Illustratio ad Samplig Distributio for the Maimum Likelihood estimators. A umerical illustratio for the maimum likelihood estimators for the Beta-Biomial distributio usig Biomial ad Poisso approimatio will be obtaied. MATHCAD program is used to evaluate the ML estimators ad to obtai Pearso Coefficiet ad Pearso family of distributio. A trail will be made to obtai the best fitted distributio to maimum likelihood estimators usig STATGRAPHICS package. (a) Biomial ad Poisso Approimatio The Beta-Biomial distributio is a combiatio of Biomial distributio with probability of Success P.Suppose P is a radom variable that follows Beta distributio with shape parameters α ad β, respectively, the Beta- Biomial distributio may be obtaied from a Biomial distributio. Teerapabolar (2008) obtaied a upper boud o Biomial approimatio to the α Beta-Biomial distributio whe P α, β = as follows: α + β 1 1 ( 1) Biα, β = (1 P( α, β) q( α, β) ) (3.1) ( + 1)(1 + α + β ) ad showed that the results gave a good Biomial approimatio to the α Beta-Biomial distributio if P α, β = or is small. α + β β

6 1268 S. A. Salem ad W. S. Abu El Azm Teerapabolar (2009) preseted a Poisso approimatio to the Betaα Biomial distributio. He cocluded that if is small, the the Biomial α + β distributio with distributio parameters ad α P α, β = ca be approimated by the Poisso distributio with mea α + β α λ( α, β, ) = Pα, β =. α + β He obtaied a differet upper boud o Poisso approimatio to the Beta- Biomial distributio as follows. α If α ( 1) ad λ =, the upper boud o Poisso approimatio α + β to the Beta-Biomial distributio will be (,, ) ( α + 1)( α + β) β Pois, (1 e λ α β α β = ). (3.2) ( α + β )(1 + α + β ) α If α = ( 1) ad λ =, the upper boud o Poisso approimatio α + β to the Beta-Biomial distributio will be (,, ) α Pois, (1 e λ α β α β = ). ( α + β )(1 + α + β ) A result of this approimatio, without the coditio α ( 1), ca be obtaied by the followig fact Pois α, β α β 1 α = α + β α + β (1 + α + β ) ( α + β ) (,, ) (1 ( ) ( ) ) (1 e λ α β ), 1 α which gives a good approimatio if ad are small. (1 + α + β) ( α + β ) A MATHCAD program (2) is writte to evaluate (3.1) ad (3.2) for = 5,10, α = ad β = , for Poisso approimate α must be greater tha 1. The followig relative values betwee the upper boud o Biomial approimatio ad the upper boud o Poisso approimatio Biα, β Rαβ, = Pois α, β

7 Samplig distributio of maimum likelihood estimators 1269 are obtaied, to show which oe is the best approimatio, all results are listed i program (2), from these results we coclude that If = 5, most relative values are less tha oe, the the Biomial approimatio gave more accurate results tha Poisso approimatio. If = 10, most relative values are more tha oe, the the Poisso approimatio gave more accurate results tha the Biomial approimatio. (b) Maimum Likelihood Illustratio ad suggested Pearso Family of distributios A umerical illustratio for the maimum likelihood estimators for the Beta-Biomial distributio usig Biomial ad Poisso approimatio will be obtaied. MATHCAD program 3 ad 4 are used to evaluate the ML estimators usig equatios (1.3) ad (2.1) respectively usig the followig steps Step 1: For α = 30, 35 ad β = 65, 70, 100, radom samples of size = 5, 10 ad 15 form approimatio Beta-Biomial distributio are geerated, that is, for α Biomial approimatio with parameters = 100 ad p = ad Poisso α + β α approimatio with parameters = 100 ad λ =. α + β Step 2: A Chi-square goodess-of-fit test for every geerated sample (Biomial, Poisso) approimatio, is carried out usig the followig hypotheses H 0: Data follows the Beta-Biomial Distributio. H 1: Data does ot follow the Beta-Biomial Distributio. 2 Step 3: Usigα = 0.05, if P-values for χ tests > 0.05, do ot reject H 0 ad go further for obtaiig maimum likelihood estimates. If P-values for χ 2 tests < 0.05, stop ad geerate aother set of radom umbers. Step 4: Repeated (1), (2) ad (3) 100 times ad compute square root of mea square error (MSE), Skewess β 1, Kurtosis β 2, Pearso coefficiet K p ad Pearso type for maimum likelihood estimators. For differet, α ad β all results are displayed i tables 1 ad 2, ad we coclude the followig results

8 1270 S. A. Salem ad W. S. Abu El Azm 1. For α = 30,35 ad β = 65,70 maimum likelihood estimators have the miimum MSE for most sample sizes. 2. As the sample size icreases, MSE of the estimated parameters decreases. 3. Geerally, we observe that the samplig distributio of the maimum likelihood estimators ˆα ad ˆ β have Pearso type I distributio with the followig form where m m1 1 2 f ( ) = G( a ) ( a ) = a+ a, m = a+ a ( ) ( ) c2 a2 a1 c2 a2 a1 m 2 for both ( a1 ) ad ( a2 ) to be positive it must to be a1 < < a2. Where a 1 ada 2 is the root of equatio (2-14) witha 1 < 0 < a 2, a1 = c1from equatio (2.15) ad G is a costat, for differet samples size. (c) Best Fitted Distributio usig Statgraphics package. Usig STATGRAPHICS package ad geerated radom umbers i sectio 3, the fitted distributios for the maimum likelihood estimators ˆα ad ˆ β are obtaied, the fitted distributios will be tested usig Kolmogrov Smirov ad Chi-square goodess-of-fit tests to determie the best fit. Goodess-of-fit test for the maimum likelihood estimators for the Beta-Biomial distributio are obtaied usig Biomial ad Poisso approimatio for differet, α ad β. We coclude that the maimum likelihood estimators of Beta Biomial distributio usig Poisso ad Biomial approimatio have ormally desities whe = 5, 10 ad 15.

9 Samplig distributio of maimum likelihood estimators 1271 STATGRAPHICS package is used to obtai samplig distributio for the maimum likelihood estimators, all results are listed i table 3 ad 4. From these tables we coclude that the samplig distributios for the maimum likelihood estimators of the Beta-Biomial distributio have ormal desities. Table (1): Maimum likelihood estimates, stadard deviatio Square root of mea square error (MSE), Skewess β 1, Kurtosis β 2, Pearso coefficiet ad Pearso type, for Beta-Biomial distributio by usig Biomial approimatio with α = 30,35, β = 65,70 ad differet samples size = 5,10 ad 15. Number of repetitios N = 100. Biomial Parameters ML estimates Square root of MSE α β p ˆα ˆβ ˆα ˆβ

10 1272 S. A. Salem ad W. S. Abu El Azm Table (1) (cotiued): Maimum likelihood estimates, stadard deviatio Square root of mea square error (MSE), Skewess β 1, Kurtosis β 2, Pearso coefficiet ad Pearso type, for Beta-Biomial distributio by usig Biomial approimatio with α = 30,35, β = 65,70 ad differet samples size = 5,10 ad 15. Number of repetitios N = 100. Skewess β 1 β2 Kurtosis Pearso coefficiet Pearso type ˆα ˆβ ˆα ˆβ ˆα ˆβ ˆα ˆβ I I I I IV I IV I IV I I I I I IV IV I I I I IV IV I I

11 Samplig distributio of maimum likelihood estimators 1273 Table (2): Maimum likelihood estimates, stadard deviatio, Square root of mea square error (MSE), Skewess β 1, Kurtosis β 2, Pearso coefficiet ad Pearso type, for Beta-Biomial distributio by usig Biomial approimatio with α = 30,35, β = 65,70 ad differet samples size =5,10 ad 15. Number of repetitios N = 100. Poisso Parameters ML estimates Square root of MSE α β λ ˆα ˆβ ˆα ˆβ

12 1274 S. A. Salem ad W. S. Abu El Azm Table (2) (cotiued): Maimum likelihood estimates, stadard deviatio, Square root of mea square error (MSE), Skewess β 1, Kurtosis β 2, Pearso coefficiet ad Pearso type, for Beta-Biomial distributio by usig Biomial approimatio with α = 30,35, β = 65,70 ad differet samples size =5,10 ad 15. Number of repetitios N = 100. Skewess β 1 β2 Kurtosis Pearso coefficiet Pearso type ˆα ˆβ ˆα ˆβ ˆα ˆβ ˆα ˆβ I I I I I I I IV I I IV IV I I VI I IV VI IV I I IV I I

13 Samplig distributio of maimum likelihood estimators 1275 Table (3): Samplig distributio for the maimum likelihood estimators for Beta- Biomial distributio usig Biomial approimatio with α = 30, 35, β = 65, 70 ad differet samples size = 5, 10 ad 15. Biomial Parameters ˆα Biomial approimatio ˆβ Distributio uder ull hypothesis P- value Distributio uder ull hypothesis P- value Weibull Weibull Normal Normal α β p Normal Normal Normal Normal Normal Normal Cauchy Loglogistic Weibull Loglogistic Normal Normal Logormal Cauchy Logormal Logormal Normal Normal Cauchy Logormal

14 1276 S. A. Salem ad W. S. Abu El Azm Table (4): Samplig distributio for the maimum likelihood estimators for Beta- Biomial distributio usig Poisso approimatio with α = 30, 35, β = 65, 70 ad differet samples size = 5, 10 ad 15. Poisso Parameters ˆα Poisso approimatio ˆβ α β λ Distributio uder ull hypothesis Distributio uder ull hypothesis Distributio P- value uder ull hypothesis Distributio uder ull hypothesis P- value Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Logormal Loglogistic Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal

15 Samplig distributio of maimum likelihood estimators 1277 Refereces 1- E. S. Pearso, Bayes' Theorem i the Light of Eperimetal Samplig, Biometrika, 17 (1925), J. C. Lee ad D. J. Sabavala, Bayesia Estimatio ad Predictio for the Beta Biomial Model, Joural of the America Statistical Associatio, 5(3) (1987), J. C. Lee ad Y. L. Lio, A Note o Bayesia Estimatio ad Predictio for the Beta-Biomial Model. Joural of Statistical Computatio ad Simulatio, 9 (1999), J. G. Skellam, A Probability Distributio Derived From the Biomial Distributio by Regardig the Probability of Success as Variable Betwee the Sets of Trials. Joural of the Royal Statistical Society, Ser. B, 10 (1948), K. Teerapabolar, A Boud o the Biomial Approimatio to the Beta-Biomial Distributio, Iteratioal Mathematical Forum, 3, o. 28 (2008), K. Teerapabolar, Poisso Approimatio to the Beta-Biomial Distributio. Iteratioal Mathematical Forum, 4, o. 25(2009), Received: April 7, 2013