Using T.O.M to Estimate Parameter of distributions that have not Single Exponential Family

Transcription

1 IOSR Journal of Mathematcs IOSR-JM) ISSN: Volume 3, Issue 3 Sep-Oct. 202), PP Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran Abdul Ameer Khuttar Al-Labban Department of Statstcs and Informatons College of Computer Scence and Mathematcs Unversty of Al- Qadsya IRAQ Abstract: We dscuss the T.O.M Term Omsson Method) to estmate the dstrbuton parameters that have not sngle exponental famly of exponental famly wth one sngle dstrbuton parameter, and compare t wth dfferent methods usng Mean Square Method MSE). Keywords - Exponental famles, Exponental form, I. INTRODUCTION: Estmators that based on T.O.M deals wth many dstrbutons wth dscrete or contnuous random varable that have not belongs to exponental famly or t s tang exponental form. Many dstrbutons not belongs to exponental famly le, Unform, Hypergeometrc, t-dstrbuton, F dstrbuton, Cauchy, shfted exponental, Pareto. [*] II. EXPONENTIAL FORM: We note that there are many defntons of such type of that representaton of exponental famles. Recall defntons of exponental famles. III. DEFINITIONS: Def. ): A famly of contnuous dscrete) random varables s called an exponental famly f the probablty densty functons probablty mass functons) can be expressed n the form f X x / ) h c )exp t, x 0,,2,... ) for x n the common doman of the f X x / ), R. Obvously h and c are non-negatve functons. The t are real-valued functons of the observatons. [2*] Def. 2): Let be an nterval on the real lne. Let {fx;): } be a famly of pdf's or pmf's). We assume that the set { x : ) 0} s ndependent of, where x x, x,..., x ). We say that the famly {fx;): 2 n } s a one-parameter exponental famly f there exst real-valued functons Q) and D) on and Borelmeasurable functons T X ) and S X ) on R n such that ) exp Q ) T D ) S 2) ) f we wrte as ) h c )exp T where exp S h, Q ) c ) exp famly n canoncal form for a natural parameter. Def. 3): Let, and DQ ) 3), then we call ths the exponental R be a -dmensonal nterval. Let {fx; ): } be a famly of pdf's or pmf's). We { x : ) 0 s ndependent of, where x x, x,..., x ). We say that the assume that the set } 2 n famly {fx; ): } s a -parameter exponental famly f there exst real-valued functons Q ),, Q ) and D ) on and Borel-measurable functons T X ),,T X ) and S X ) on R n such that: [3*] ) exp Q ) T D ) S 4) 44 Page

2 Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Def. 4): Exponental famly s a class of dstrbutons that all share the followng form: T P y / ) h exp{ T A )} 5) * s the natural parameter. For a gven dstrbuton specfes all the parameters needed for that dstrbuton. * T s the suffcent statstc of the data n many cases T = y, n whch case the dstrbuton s sad to be n canoncal form and s referred to as the canoncal parameter). * A) s the log-partton functon whch ensures that py/) remans a probablty dstrbuton. * h s the non-negatve base measure n many cases t s equal to ). Note that snce contans all the parameters needed for a partcular dstrbuton n ts orgnal form, we can express t wth respect to the mean parameter : [*] P y / ) h exp{ ) T A ))} 6) Def. 5): Regular Exponental Faml: Consder a one-parameter famly { ) : ) of probablty densty functons, where s the nterval set { : }, where and are nown constants, and where [4*] p ) s q ) e a x b ) 7) 0 o.w The form 7) s sad to be a member of the exponental class of probablty densty functons of the contnuous type, f the followng condtons satsfy: ) Nether a) nor b) depends upon. ncreasngly 2) p) s a nontrval contnuous functon of. 3) Each of 0 and s s a contnuous functon of x. and the followng condtons wth dscrete random varable X : ) The set { x : x a, a2,...} does not depend upon. 2) p) s a nontrval contnuous functon of. 3) s a nontrval functon of x. Therefore, the Exponental Form s the Exponental Class wthout satsfyng any of the condtons n above defntons or can not wrtten n Exponental Class for dscrete or contnuous random varables. IV. EXAMPLES OF DISTRIBUTIONS THAT NOT BELONGS TO EXPONENTIAL FAMILY: We wll tae here some dstrbutons as an example; to show how can wrte the p.d.f as an exponental form. Example : Let X be a contnuous random varable of Unform dstrbuton wth parameter, wth p.d.f f x; ) 0 x here we can not represent ths dstrbuton as an exponental class because x depends upon, so we can rewrte the p.d.f of ths dstrbuton represented by exponental form whch can be the same form of exponental class as follows: ) exp ln ) wth 0, p ) 0, s 0, q ) ln ) Example 2: Let X be a contnuous random varable have the shfted exponental dstrbuton wth p.d.f x - ) e 0 x wth scale parameter, and also we can here represented the p.d.f as an exponental form because random varable depends upon the parameter, x ) exp ln ) where x, p ), s, q ) ln ) 45 Page

3 Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly note that the random varable n prevous two examples,2) depends on and the condton ) of def. 5) not satsfyng. Example 3: Let X be a contnuous random varable of Cauchy dstrbuton wth parameter, wth p.d.f, ) - x x ) where s the locaton parameter, and s the scale parameter, the one parameter Cauchy dstrbuton can be wrtten when =0 as follows:, ) - x x The reason of why we can not represented the Cauchy dstrbuton as an exponental famles s that we can not determne the term p) or any other terms of exponental famles, or some other reasons [5*], so we wll represent the Cauchy dstrbuton n the exponental form, ) exp ln ) ln ) ln x where p ) ln x ), s ln ), q ) ln ) V. T.O.M WITH EXPONENTIAL FORM: T.O.M can used to estmate the value of parameter n many cases, below we wll dervatve the T.O.M of dstrbutons wth one parameter) have not exponental famly or have exponental form wth two cases: Case : The dstrbuton can be wrtten as exponental class but not belongs to exponental famly because unsatsfyng one or more condtons of exponental famles, le Unform, student s t, etc. Case 2: The dstrbuton can be not wrtten as exponental class, le Shfted exponental, Cauchy, Pareto, etc.. We wll dscuss each case usng T.O.M as follows: 5. T.O.M WITH CASE : There are many dstrbutons can be wrtten as exponental class but t s not belong to exponental famly. Here we wll pont to some reasons why such dstrbutons not belongs to exponental famly p)=0, random varable depend on parameter, etc.) as shown n examples, 2), so for our case we can use T.O.M to estmate parameter of that dstrbutons. For sample wth sze n havng the p.d.f fx ;), and for any value x, =,,n x ) p ) q ) s x ) y e x by tang the natural logarthm to fx ;), snce p)=0 we have x q ) s x ) and by subtract s x ) from the fnal amount we have q ) whch s the functon of. Therefore, we can defne the q ) as follows: q ) ln y s x ) 8) where q ) represent to values that we have from prevous steps of T.O.M, =,2,,n, and from eq. 8) we from ) can educe values of q. The estmaton of can found now usng the least square error from the followng equaton: n ˆ Mn f x m, ) ym,2,..., n m where f, ) s the value of functon f on, and y x ) s the observed value. x m 5.2 T.O.M WITH CASE 2: There are many dstrbutons can not be wrtten as exponental class for some reasons. Here we wll pont to some reasons why such dstrbutons not belong to exponental famly. Often dstrbutons that gven by the locaton-scale dstrbuton famles; le Cauchy dstrbuton [*], shfted exponental densty [6*] ; such as the p.d.f n example 2. In ths case we can call T.O.M for exponental class [7*]. So n our cases we can use T.O.M to estmate parameter of those dstrbutons. m 46 Page

4 Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Example 4: In example ) of Unform dstrbuton we can obtan the followng exponental form: ) exp ln ) where 0, p ) 0, s 0, q ) ln ) snce p)=0, then we can usng eq. 8), hence q ) ln y and y Example 5: In example 2) of p.d.f we can obtan the followng exponental form: x ) exp ln ) where x, p ), s, q ) ln ) snce p)0, then we can use the same way of T.O.M [7*], thus ln y ) ln y ) p ) x x and x x ln y ) ln y ) Example 6: n example 3) the p.d.f can obtan by the followng exponental form:, ) exp ln ) ln ) ln x 9) where p ) ln x ), s ln ), q ) ln ) snce p)0, we can educe the eq. 9) usng technque of T.O.M [7*] and we can then have the followng values of parameter where 2 x x y 2 y, n-. Fnally, from the prevous examples we can fnd the estmaton of parameter as follows: n ˆ Mn f x m, ) ym,2,...,n - m 6. RESULTS: In Table below we show the comparson between T.O.M and MLE method of Scale parameter Cauchy dstrbuton. Table: Comparson between Scale Parameter for Cauchy Dstrbuton between T.O.M. and MLE [8*]. N T.O.M MLE Page

5 Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly REFERENCES: [] Andreas V. 200), Notes on exponental famly dstrbutons and generalzed lnear models. [2] Watns, Joseph C. 2009). Exponental Famles of Random Varables, Unversty of Arzona, p:. [3] Jurgen, S. 999). Mathematcal Statstcs I, Utah State Unversty, p: 20. [4] Robert V. Hogg and Allen T. Crag 978). Introducton to Mathematcal Statstcs, 4 th Edton, p: 357. [5] Anrban DasGupta A. 20). "Probablty for Statstcs and Machne Learnng", Sprnger, pp: [6] Lawrence D. Brown 986). "Fundamentals of statstcal exponental famles wth applcatons n statstcal decson theory", Vol. 9, IMS Seres. [7] Labban, J.A. 202). "Estmaton of Sngle Dstrbutons Parameter by T.O.M wth Exponental Famles", Amercan Journals Scence, Issue 55, pp [8] Ncu S. and Mchael S. Lew 2003). "Robust Computer Vson - Theory and Applcatons", Kluwer Academc Publshers, Volume 26, p Page