Improved Approximation Methods to the Stopped Sum Distribution

Transcription

1 Internatonal Journal of Mathematcal Analyss and Applcatons 2017; 4(6): ISSN: Improved Approxmaton Methods to the Stopped Sum Dstrbuton Aman Al Rashd *, Bashar Al Juhan, Tahan Al Saed, Hanan Al Ahmad, Mashael Al Harb, Mawada Brr, Nada Al Johan, Alya Almutar Department of Mathematcs, Faculty of Scence, Tabah Unversty, Medna, Saud Araba Emal address (A. Al Rashd) * Correspondng author Keywords Approxmatons, Stopped Sum Dstrbuton, Geometrc, Negatve Bnomal-Bernoull, Statstcs, Probabltes Receved: July 24, 2017 Accepted: October 16, 2017 Publshed: November 9, 2017 Ctaton Aman Al Rashd, Bashar Al Juhan, Tahan Al Saed, Hanan Al Ahmad, Mashael Al Harb, Mawada Brr, Nada Al Johan, Alya Almutar. Improved Approxmaton Methods to the Stopped Sum Dstrbuton. Internatonal Journal of Mathematcal Analyss and Applcatons. Vol. 4, No. 6, 2017, pp Abstract Most of the modern statstcal models used the method that requre the computaton of probabltes from the complcated dstrbuton such as (stopped sum) whch can lead to ntractable computaton. However, n ths study we wll use a modern method to solve ths problem, ths method s called "Saddlepont Approxmaton". However, In ths study, we wll derve the Saddlepont Approxmaton (CDF) for some very complcated statstcs such as stopped sum Geometrc- Geometrc dstrbuton and stopped sum Negatve bnomal-bernoull dstrbuton. 1. Introducton The method of approxmatons are very mportant n statstcs because sometmes we cannot derve the exact CDF or the PDF for some complcated dstrbutons see Ref. [1]. So, usng the approxmatons methods can save tme, effort and cost see Ref. [2]. Some bascs and fundamentals about Saddlepont Approxmaton and the stopped sum are presented below to smplfy understandng them. In most cases, the dstrbuton of the stopped sum s stll unknown; n other cases, t s already known but s too complex for the computaton of the dstrbuton functon, whch often becomes too slow for many problems see Ref. [3]. 2. Stopped Sum Dstrbuton The stopped sum dstrbuton one of the very mportant statstcs and has many applcaton n our lfe such as n branchng process and n nsurance system lke the work of [4]. Unfortunately, most of these dstrbutons are stll unknown untl now see Ref. [5]. Ths technque s based on the moment generatng functon as gven n [6]. Let X X X are d random varable and Xs and N are ndependent random varables. Ths stopped sum can be expressed mathematcally as the followng formula Ref. [7] S XX X X The mean and varance of stopped sum of the random varable S Χ are presented respectvely by:

2 Internatonal Journal of Mathematcal Analyss and Applcatons 2017; 4(6): ES = EΝEΧ VarS = EΝ varχ + EΧ varν The moment generatng functon of the stopped sum S s gven by Ref. [8]: Μ s = Μ ln Μ " s# 3. Stopped Sum of the Negatve Bnomal-Bernoull Dstrbuton The stopped sum of the Negatve Bnomal-Bernoull dstrbuton where N ~Negatve Bnomal (r, p) and X s~bernoull (p). The MGF for Negatve Bnomal dstrbuton s gven by [9] as: q / M s = ' 1 pe -. where p + q = 1 as well, the MGF for Bernoull dstrbuton s gven by: M 0 s = pe - + q. The cumulant generatng functon for N s gven by: K s = ln M s q / K s = ' 1 pe -. The cumulant generatng functon for X s gven by: K 0 s = ln M 0 s K 0 s = ln pe - + q The cumulant generatng for the stopped sum Negatve Bnomal Bernoull dstrbuton as follows: K 3 = K K 0 s q K 3 = rln 1 p 45 pq The Saddlepont equaton Κ s7 =x becomes K ŝ = rp e -7 p e -7 + pq 1 = x then s7 = ln : ;<=>? >A>; B The frst contnuty-correcton uses wd = sgns7f2 {x ln xpq 1p r x r ln q1 pq p e -7 }, see Ref. [10]. The second contnuty-correcton leads to ;<=>? LMN' u7 = K1 e >A>;. O PQ R@ RSL/ TRSL wv = sgnswf2 {ln xˉpq 1p2 r xˉxˉ r ln q1 pq p2es }, where x = x 0.5, then ;><=>? \] uw = 2snh Q >A>; U P R@ RS L /4 5W 4 5W T RS And the thrd contnuty-correcton s used uw^ = swf p pq 1re -W /p e -W + pq 1, The mean and varance are gven respectvely as EΝ = R/ S, Χ = p s = Ν Χ s = pr q p where V s s gven VΝ = pr, VΧ = pq q V s = Ν VarΧ + Χ VarΝ then V s = : R/ S B pq + p : R/ Table 1 shows the Saddlepont Approxmaton p7 / wth ts correspondng normal approxmaton ph / for the left tal, Table 2 show, the Saddlepont approxmaton p7 / wth ts correspondng normal approxmaton ph / for the center and Table 3 show, the Saddlepont Approxmaton p7 / wth ts correspondng normal approxmaton ph / for rght tal for dfferent values of Χ.

3 44 Aman Al Rashd et al.: Improved Approxmaton Methods to the Stopped Sum Dstrbuton Table 1. Saddlepont Approxmaton of left tal for stopped sum of Negatve Bnomal (1, 0.8)-Bernoull (0.8) Dstrbuton. jd kp Table 2. Saddlepont Approxmaton of center for stopped sum of Negatve Bnomal (1, 0.8)-Bernoull (0.8) Dstrbuton. jd kp * * 0.5* * * * * (*) It s the value of Χ equal the value of μ. Table 3. Saddlepont Approxmaton of rght tal for stopped sum of Negatve Bnomal (1, 0.8)-Bernoull (0.8) Dstrbuton. jd kp Stopped Sum of Geometrc-Geometrc Dstrbuton The stopped sum of Geometrc- Geometrc dstrbuton where Ν~Geometrc (p), Χ s s~geometrc (p) The MGF for N s gven by: Μ s = The MGF for X s gven by: Μ " s = pe - 1 qe - pe - 1 qe - The cumulant generatng functon for Ν s gven by: Κ s = lnμ s The cumulant generatng functon for Χ s gven by: Κ " s = lnμ " s Κ s = ln ' pe- 1 qe -. Κ " s = ln ' pe- 1 qe -. The cumulant generatng functon for the stopped sum Geometrc Geometrc dstrbuton as follows: Κ s = Κ Κ " s Κ s = ln t p e - qpe- '1 1 qe- 1 qe -.v The Saddlepont equaton Κ s7 = x s becomes Κ s 1 s7 = p + 1qe -7 1 = x s7 = ln ' 1 + x xp + 1q. For the frst contnuty-correcton, we fnd Wx = sgns7y2 zx ln : LT{ B ln : R@ 4 57 SR457 {RTS LS457 1 LS4 57B, >?~; L:MN: u7 = }1 e ;<~?= BB PQ RTS457 :RTS4 57 U. The second contnuty-correcton mples W = sgnswp2 Kln : LT{ˉ B xˉ ln Q R@ 4 5W SR45W {ˉRTS LS45W :1 LS4 5WBUO, where xˉ = x 0.5, then >?~;ˉ MN: uw = 2 snh Q ;ˉ<~?= B U PQ R±S45W :RTS4 5W U. And the thrd contnuty-correcton s used

4 Internatonal Journal of Mathematcal Analyss and Applcatons 2017; 4(6): uw^ = swpq RTS45W :RTS4 5W U. The mean and the varance are gven as EΝ = R, Χ = R s = Ν Χ and VΝ = S VΧ = S V s = Ν VarΧ + Χ VarΝ V s = : R B : S + : R B : S However, Table 4, 5 and 6 show the three contnuty correcton Saddlepont p7 / wth normal approxmaton ph /. s = 1 p Table 4. Saddlepont Approxmaton of left tal for stopped sum of Geometrc (0.2)-Geometrc (0.2) Dstrbuton. jd kp Table 5. Saddlepont Approxmaton of center for stopped sum of Geometrc (0.2)-Geometrc (0.2) Dstrbuton. jd kp * (*) It s the value of Χ equal the value of μ. Table 6. Saddlepont Approxmaton of rght tal for stopped sum of Geometrc (0.2)-Geometrc (0.2) Dstrbuton. jd kp Concluson From the mathematcal computatons n the left, centre and rght, t s clear that both of the Saddlepont Approxmaton and the normal are very close to each other whch leads to the performance of the Saddlepont method. Acknowledgements The authors would lke to thank the Edtor and the referee for carefully readng the manuscrpt and for ther valuable comments and suggestons whch greatly mproved ths paper. References [1] Kuonen, Dego. "Computer-ntensve statstcal methods: saddlepont approxmatons wth applcatons n bootstrap and robust nference." PhD Thess, Swss Federal Insttute of Technology (2001). [2] Robn, Stéphane. "A compound Posson model for word occurrences n DNA sequences." Journal of the Royal Statstcal Socety: Seres C (Appled Statstcs) 51.4 (2002): [3] Johnson, NL, Kotz, S.; Kemp, AW: Unvarate dscrete dstrbutons. 2nd Ed. New York: John Wley & Sons [4] Johnson, Norman L., Adrenne W. Kemp, and Samuel Kotz. Unvarate dscrete dstrbutons. Vol John Wley & Sons, [5] Mnkova, Leda D. "The Pólya-Aeppl process and run problems." Internatonal Journal of Stochastc Analyss (1900): [6] Hogg, R. V., and A. T. Crag. "Introducton to Mathematcal Statstcs. Macmllan Publshng Company." NY, NY (1978). [7] Al Mutar Alya, O., and Heng Chn Low. "Saddlepont Approxmaton to Cumulatve Dstrbuton Functon for Posson Exponental Dstrbuton." Modern Appled Scence 7.3 (2013): 26.

5 46 Aman Al Rashd et al.: Improved Approxmaton Methods to the Stopped Sum Dstrbuton [8] Al Mutar, O., and Heng Chn Low. "Estmatons of the Central Tendency Measures of the Random-sum Posson- Webull Dstrbuton usng Saddlepont Approxmaton." Journal of Appled Scences 14 (2014): [9] Shell, Rchard. Handbook of ndustral automaton. CRC Press, [10] Butler, Ronald W. Saddlepont approxmatons wth applcatons. Vol. 22. Cambrdge Unversty Press, 2007.