Estimation of the Mean of Truncated Exponential Distribution

Transcription

1 Journal of Mathematcs and Statstcs 4 (4): 84-88, 008 ISSN Scence Publcatons Estmaton of the Mean of Truncated Exponental Dstrbuton Fars Muslm Al-Athar Department of Mathematcs, Faculty of Scence, The Hashemte Unversty, Zarqa 5, Jordan Abstract: Problem statement: In ths study, the researcher consders the problem of estmaton of the mean of the truncated exponental dstrbuton. Approach: Ths study contracted wth maxmum lkelhood and unque mnmum varance unbased estmators and gves a modfcaton for the maxmum lkelhood estmator, asymptotc varances and asymptotc confdence ntervals for the estmators. The propertes of these estmators n small, moderate and large samples were nvestgated va asymptotc theory and computer smulaton. Results: It turns out that the modfed maxmum lkelhood estmator was more effcent than the others and exsts wth probablty. Concluson: The modfed maxmum lkelhood estmator was always exst, fast and straghtforward to compute and more lkely to yeld feasble values than the unque mnmum varance unbased estmator. Its varance was well approxmated by the large sample varance of the other estmators. Key words: Truncaton modfed maxmum lkelhood estmator, fsher nformaton, smulaton, exponental dstrbuton INTRODUCTION Suppose that X be a random varable wth exponental Probablty Densty Functon (PDF) of mean ( / ), then the PDF of the random varable Y, the truncated verson of X truncated on the rght at b, s gven by: < f (y; ) 0, otherwse y b e ( e ), f 0 y b () underlyng dstrbuton s assumed to follow the exponental dstrbuton [6,9].There are dfferent approaches for samplng selecton from a subset of a larger populaton [0,]. Ths study deals wth Maxmum Lkelhood estmator, (ML) and unque mnmum varance unbased estmator, (UM), of the mean of truncated exponental dstrbuton and shows that the maxmum lkelhood estmator does not always exst, ts exstence depends upon the value of the mean of the random sample and exsts wth probablty approachng as n. A Modfed Maxmum Lkelhood estmator, (MML), s consdered and compared wth the other estmators. The results of a large scale smulatons ndcate that the modfed maxmum lkelhood where, b s a known constant. In practce, the exponental dstrbuton has been wdely used as a model n areas rangng from studes on the lfetmes of manufactured tems [,] to research estmator s more effcent and more lkely to satsfy the nvolvng survval or remsson tmes n chronc dseases [] feasblty condton, namely 0 < µ < b / for 0< <.. But n some stuatons, an estmate s desred of the mean among the elements of the Before proceedng wth the estmaton problem, t populaton belongng to a certan group. For example, can be shown that the mean, say µ ( ), of the truncated n lfe testng problems from an exponental exponental dstrbuton gven n () s: dstrbuton, separate estmate for the lfetme mean mght be requred for bulbs whose survval tmes are b µ ( ) b(e ) lmted to be less than a constant b. In ths case these () survval tmes mght follow a truncated exponental dstrbuton. The famles of truncated dstrbutons Ths functon s monotonc decreasng and provde denstes that are useful n modelng such contnuous on (0, ) wth possble range (0,b/). populatons [4-8]. The truncated exponental dstrbuton can occur n MATERIALS AND METHODS a varety of ways. It may drectly seem to be a good ft as a dstrbuton for a gven avalable data set, or t may Maxmum and modfed maxmum lkelhood result from the type of samplng used when the estmators: Assume that Y,Y, Y n be a random 84

2 sample of sze n taken from the truncated exponental dstrbuton gven n (). The lkelhood functon, say L() s: n b n L( ) ( e ) exp( n y) () where, y s the sample mean. Maxmzng ths lkelhood we get the maxmum lkelhood estmator for. It follows that: b b log L( ) / n / nbe ( e ) ny b n[ / b(e ) y] Snce the log-lkelhood functon s defned and dfferentable on an open nterval (0, ), the maxmum value of L(), f t exsts, occurs at a statonary pont * at whch log L( ) / 0 and does not occur at any boundary pont of the nterval (0, ) []. Settng log L( ) / 0 us get the equaton: Y, f Y b / < µ doesnot exst, f Y b / (5) The same argument as before shows that the MLE estmator µ exsts wth probablty approachng as n. Under the regularty condtons [5-7], ths estmator possesses the maor propertes of the maxmum lkelhood estmator, that s µ s consstent, asymptotc effcent and best asymptotcally normal estmator wth mean µ() and asymptotc varance, avar ( µ ), attans the Cramer Rao lower bound. The modfcaton to the MLE µ, gven n (5), s based on fndng an estmator whch s close as possble to the MLE µ and s more lkely to satsfy the feasblty condton 0 < µ ( ) < b / than the unque mnmum varance unbased estmator. Ths suggested estmator, say µ, can be wrtten as: b / b(e ) y 0 (4) It can be shown that the left-hand sde of Eq. 4 s monotonc decreasng n ; as tends to 0 t tends to b / y y. and as tends to nfnty t tends to ( ) Hence the soluton to (4) s unque f t exsts and t exsts f and only f 0 < y < b / and hence when 0 < y < b /, there exsts a statonary pont, say *, that satsfy Eq. 4 and 0< * <. Clearly * s the unque maxmum lkelhood estmator of when 0 < y < b /. When y b /, the Eq. 4 does not have a soluton n the doman (0, ) and hence the lkelhood functon L() does not have a maxmum. The proper defnton of the ML-estmator of s therefore: *, f Y < b / does not exstf Y b / As n, we have Y converges n probablty to the mean µ ( ) of the truncated exponental p.d.f. gven n (). Because the densty n () s monotone decreasng, a smple geometrcal argument shows that the mean µ ( ) must le n the left half of the nterval (0, b) and hence µ ( ) <b/. Then P(Y < b / ) as n, so that the MLE * exsts wth probablty approachng as n. Therefore, usng the nvarance property of the maxmum lkelhood method [,4], the maxmum lkelhood estmator, µ, of µ ( ) s gven by: 85 Y, f Y b / < µ b /, f Y b / (6) whch corresponds to the modfed maxmum lkelhood estmator, say,of,gven by: *, f Y < b / 0 f Y b / The same argument as before shows that µ µ and wth probablty approachng as n. Unque mnmum varance unbased estmator: It s obvous that the dstrbuton n () represents a regular case of the exponental class of probablty densty functons of the contnuous type and hence n y s a complete suffcent statstcs for and. Then by usng the theorem of Lehmann and Scheffe [], the unque mnmum varance unbased estmator, say µ, of s gven by: µ Y (7) The varance of µ, say var( µ ), s gven by: var( µ ) e (e ) n where, b. (8)

3 Asymptotc varances of the estmators: The asymptotc varance of µ, say avar ( µ ), s the recprocal of the Fsher nformaton: dµ E( log L( ) / )( ) d where, L( ) s n () and s n (). Thus: avar ( µ ) [ e (e ) n avar( µ ) avar( µ ) (9) Moreover, t s easy to show that avar ( µ ) b / n as 0. Havng obtaned the asymptotc varance of µ,,,,the asymptotc relatve effcency, ARE of µ relatve to µ for,,,, s: avar ( µ ) ARE avar( µ ) and the relatve effcency of,,, s defned by: MSE( µ ) RE MSE( µ ) where, MSE s the mean-squared error. µ relatve to µ for, Interval estmaton of µ ( ) : Approxmate 00(-) percent confdence nterval for µ ( ) n (5-7) can be constructed by the standard normal lmtng dstrbuton and the modfcaton of Slutsky's theorem 6 gven by [] : and / * * / b b P µ zα/ n b e (e ) ( ) * < µ / * * / b b < µ + zα / n b e (e ) * α, when Y < b / b b P( µ z α / < µ ( ) < µ z α / ) α, n n when Y b / for all,,, where z α / s the 00(-) percent pont of the standard normal dstrbuton. The smulaton technque: In order to nvestgate the propertes and the values of the estmators µ, µ and µ a large scale smulaton nvestgaton was made for the exponental p.d.f. truncated on the rght. To get the bases, varances and the mean-squared errors numercally, the smulaton technque wth the help of MATLAB, the language of techncal computng verson 6.5 s used [8]. These are computed for 50,000 samples of szes (n 0, 0, 50, 00, 00) generated from the truncated exponental dstrbuton. Pseudo-random unform numbers were obtaned from the functon RAND of the MATLAB. The transformaton to the truncated exponental dstrbuted varable s gven by: Y F (U ) log[ U ( e )] Where: F(.) The dstrbuton functon of the truncated exponental random varable U Unformly dstrbuted random varable on (0,) [0] For each combnaton of (n, ), 50,000 trals have been done to fnd 50,000 values of each estmator. These estmators are then used to estmate the means, the values of the bases, the varances and the meansquared errors for each estmator. A computer smulaton experment was run to compare three methods of estmaton of the mean of truncated exponental dstrbuton. Smulatons were performed for sample szes n 0, 0, 50, 00, 00 wth the truncaton ponts takng values 0.05, 0.5, 0.5,.0(.5)0.0. For each combnaton of values of n and, 50,000 random samples were generated from the truncated exponental dstrbuton and for each sample the mean µ ( ) was estmated by each of the three methods: (a) the method of Maxmum Lkelhood (ML), descrbed before; (b) the method of Modfed Maxmum Lkelhood (MML); and (c) the method of Unque Mnmum varance unbased estmator (UM). RESULTS The smulaton results for estmaton of the mean of truncated exponental dstrbuton are shown n Table -. 86

4 Table : Percentage of the absolute values of the bases of the estmators ML and MML for n Method ML MML ML MML ML MML ML MML ML MML Table : Percentage values of the (n )var of the ML, MML and UM estmators and the (n ) avarun of ML and UM estmators for n Method ML MML UM ML MML UM ML MML UM ML MML UM ML MML UM avar Table : Percentage values of the relatve effcences of the ML and UM estmators relatve to the MML estmator n Method ML UM ML UM ML UM ML UM ML UM DISCUSSION Apart from the case >.5 when all the estmaton methods have comparable performance, Table shows that the MML estmator has, consstently, the lowest absolute bas of the two based estmators of, ts advantage beng partcularly marked n small samples n 0, 0 and n moderate samples n 50. Table shows that the MML estmator has slghtly larger varance than the ML estmator when <.5, but ts 87 varance s small and n most cases relatvely nsgnfcant compared to the bas n ts contrbuton to the mean-squared error. The UM estmator has the largest varance of the three estmators of when <.5. The varance of the MML estmator s well approxmated by the asymptotc varance of ML and UM estmators gven by (9) and the last lne of Table. Table gves the percentage values of the relatve effcences of ML and UM estmators defned as the rato of the means square errors, relatve to the

5 MML estmator. It s obvous from ths table that, n general, the ML and the UM estmators are less effcent than the MML especally when <.5 and ther relatve effcences ncrease wth. CONCLUSION Estmatons of the mean of truncated exponental dstrbuton have been suggested and ther propertes are studed. It turns out that the modfed maxmum lkelhood estmator has several advantages over the other estmators. It s always exst, fast and straghtforward to compute and more lkely to yeld feasble values for the estmated mean than the unque mnmum varance unbased estmator. The bas of the estmator s small and decreases rapdly as the sample sze ncreases. The varance of the MML estmator s comparable wth those of the ML and UM estmators. The varance of the MML estmator s well approxmated by the large sample varance of ML and UM estmators. REFERENCES. Davs, D.J., 95. An analyss of some falure data. J. Am. Stat. Assoc., 47: Epsten, B., 958. The exponental dstrbuton and ts role n lfe-testng. Ind. Q. Control, 5: taprefxhtml&dentferad Fegl, P. and M. Zelen, 965. Estmaton of exponental survval probabltes wth concomtant nformaton. Bometrc, : Ahmad, A.A., 00. Moments of order Statstcs from doubly truncated contnuous dstrbutons. Statstcs, 5: DOI: 0.080/ Ahmad, A.A. and M. Fawzy, 00. Recurrence relatons for sngle moments of generalzed order statstcs from doubly truncated dstrbutons. J. Stat. Plann. Inference, 7: DOI: 0.06/s (0) Ban, L.J. and G. Gaoxong, 996. Condtonal maxma and nferences for the truncated exponental dstrbuton. Can. J. Stat., 4: Josh, P.C., 979. A note on the moments of order Statstcs from doubly truncated exponental dstrbuton. Ann. Inst. Stat. Math., : -4. DOI: 0.007/BF Khan, A.H. and M.M. Al, 987. Characterzaton of probablty dstrbutons through hgher order gap. J. Commun. Stat. Theor. Math., 6: DOI: 0.080/ Deemer, W.L. and F.V. Davd, 955. Estmaton of parameters of truncated or censored exponental dstrbutons. Anal. Math. Stat., 6: DOI: 0.4/aoms/ Tryfos, P., 996. Samplng Methods for Appled Research. John Wley and Sons, New York, ISBN: , pp: Wooldrdge, J.M., 00. Econometrc Analyss of Cross Secton and Panel Data. nd Edn., MIT Press, Cambrdge, London, ISBN: : , pp: Anton, H., I. Bvens and S. Davs, 005. Calculus Sngle Varable. 8th Edn., John Wley and Sons, New York, ISBN: Hogg, R.V. and T.C. Allen, 995. Introducton to Mathematcal Statstcs. 5th. Edn., Prentce-Hall Inc., New Jersey, ISBN: 0:00557, pp: Johnson, R.A. and W.W. Dean, 998. Appled Multvarate Statstcal Analyss. 4th Edn., Prentce-Hall, Inc., New Jersey, ISBN: x, pp: Johnson, J. and J. Dnardo, 997. Econometrc Methods. McGraw-Hll, New York, ISBN: , pp: Verbeek Marno, 000. A Gude to Modern Econometrcs. John Wley and Sons, New York, ISBN: Zacks, S., 97. The Theory of Statstcal Inference. John Wley and Sons, New York, ISBN: , pp: Enander, E.P., A. Soberg, B. Meln and P. Isaksson, 996. The MATLAB, Hand Book. Addson Wesley, Longman, ISBN: