An inductive proof for a closed form formula in truncated inverse sampling

Transcription

1 Journal of Proagatons n Probablty and Statstcs Vol. No. August Internatonal ed.. 7- An nductve roof for a closed for forula n truncated nverse salng Kuang-Chao Chang Fu Jen Catholc Unversty Abstract Inverse salng s a seuental salng rocedure such that salng s contnued untl a redeterned nuber of unts ossessng certan attrbute are ncluded n the sale. Ths salng rocedure does not have control over the total sale sze n artcular when the attrbute under consderaton revals wth rare freuency. A reedy for ths shortcong s to truncate the salng rocedure when the total sale sze reaches a secfed axu nuber. We roose a closed for forula to coute the exected total sale sze n ths truncated verson of nverse salng and we rove the forula by atheatcal nducton. Keywords: Inverse salng Truncated nverse salng atheatcal nducton. Kuang-Chao Chang s Assocate Professor Deartent of Statstcs and Inforaton Scence Fu Jen Catholc Unversty Tae Tawan ROC. -al: stat6@als.fu.edu.tw

2 JPPS Volue Nuber August. 7-. Introducton In sale survey theory and ethodology nverse salng s often used to estate the roorton of a certan rare te n the oulaton. In so-called standard nverse salng sales are taen randoly and seuentally untl a secfed nuber of the rare te has been observed see Haldane 945 or Cochran 977. A drawbac n such salng rocedure s the lac of control on the fnal rando sale sze whch ay be very large f the value of the roorton to be estated s very sall. To overcoe ths drawbac we ay consder truncatng the nverse salng rocedure when the rando total sale sze reaches a redeterned ostve nteger. Ths odfed verson of nverse salng wll be called the truncated nverse salng TIS. In ths aer we roose a closed for forula to coute the exected fnal rando total sale sze n TIS under the assuton of nfnte oulaton and we rove the forula by the ethod of atheatcal nducton. The contents of ths artcle ay be used as suleentary teachng ateral for teachers of robablty statstcs atheatcs and other related felds.. A closed for forula n TIS We begn ths secton wth the followng lea. Lea. Let X be dstrbuted as Bnoaln then n Pr X n where. Proof. Let Y be dstrbuted as Bnoaln then n Pr X n Pr Y Y n snce Y s nonnegatve and nteger-valued see Karr Corollary 4.4. Q..D. Next n the followng theore we gve the closed for forula to coute the exected fnal rando total sale sze n TIS assung the oulaton s nfnte. Theore. Let Z have negatve bnoal dstrbuton wth..f. z z f z z L and let Z be the rando varable defned by Z n{z } where s a ostve nteger and. Then the exected value of Z denoted by s - 8 -

3 A closed for forula n truncated nverse salng Kuang-Chao Chang f > f. Proof. The case that s trval. If > we rove by nducton on. Let U be the rando varable defned by otherwse. results n a success the frst tral f U Then when we have U PrU U PrU [ ].. [ ] L L /. Next we assue that the theore s true f the araeter value of the reured nuber of successes for Z s. Then when the araeter value s we have U PrU U PrU [ ]. [ ]. { } { } - 9 -

4 JPPS Volue Nuber August. 7-. Now and see Feller 968 Vol. I.64 euaton.8. Thus where. Let XBnoal then by Lea.. Pr X Cobnng all the above results we obtan { } - -

5 A closed for forula n truncated nverse salng Kuang-Chao Chang. Q..D. The dstrbuton of Z n Theore. ay be consdered as a rght truncated negatve bnoal dstrbuton dfferng fro those left truncated ones largely dscussed n lterature see Saford 955 Rder 955 Cacoullos and Charalabdes 975 Johnson Kotz and Ke etc.. 3. Concluson In ths aer we roosed a closed for forula to coute the exected fnal rando total sale sze n TIS and we roved the forula by atheatcal nducton. The ethod of atheatcal nducton ay be used to rove any forulas n salng theory. We conclude ths aer by ntroducng the followng Lea 3. n whch a well-nown forula n cature-recature salng s gven see Sngh and Chaudhary or Chaan 95. The nductve roof of the forula s left as an exercse for readers. Lea 3. Let N be the sze of a fnte oulaton consstng of two strata and let th N h be the h stratu sze h. Sales are taen seuentally wthout relaceent untl observatons are obtaned fro the frst stratu. Then the exected value of the fnal rando sale sze denoted by N N N s where N. N N N N N The roof wll be gven n the next ssue of ths ournal. Acnowledgeents The author would le to than Professor Chen-Pa Han and Professor Shaw-Hwa Lo for ther careful readng and helful coents. References Cacoullos T. and Charalabdes C. A On VU for truncated bnoal and negatve bnoal dstrbutons Annals of the Insttute of Statstcal atheatcs

6 JPPS Volue Nuber August. 7- Chaan D. G. 95. Inverse ultle and seuental sale censuses Boetrcs Cochran W. G Salng Technues 3 rd ed. Wley. Feller W An Introducton to Probablty Theory and Its Alcatons Vol. I 3 rd ed. Wley. Haldane J. B. S On a ethod of estatng freuences Boetra Johnson N. L. Kotz S. and Ke A. W. 99. Unvarate Dscrete Dstrbutons nd ed. Wley. Karr A. F Probablty. Srnger-Verlag. Rder Paul R.955. Truncated bnoal and negatve bnoal dstrbutons Journal of the Aercan Statstcal Assocaton Saford. R The truncated negatve bnoal dstrbuton Boetra Sngh D. and Chaudhary F. S Theory and Analyss of Sale Survey Desgns Wley. - -