Nonparametric Demand Forecasting with Right Censored Observations

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1 J. Software Engneerng & Applcatons, 2009, do /jsea Publshed Onlne Noveber 2009 (http// 259 Nonparaetrc Deand Forecastng wth Rght Censored Observatons Bn ZHANG 1,2 ; Zhongsheng HUA 2 1 Insttute for Econocs and Lngnan College, Sun Yat-sen Unversty, Guangzhou, Chna; 2 School of Manageent, Unversty of Scence and Technology of Chna, Hefe, Chna. Eal 1 bzhang3@al.ustc.edu.cn Receved July 17 th, 2009; revsed August 10 th, 2009; accepted August 18 th, ABSTRACT In a newsvendor nventory syste, deand observatons often get rght censored when there are lost sales and no backorderng. Deands for newsvendor-type products are often forecasted fro censored observatons. The Kaplan-Meer product lt estator s the well-known nonparaetrc ethod to deal wth censored data, but t s undefned beyond the largest observaton f t s censored. To address ths shortfall, soe copleton ethods are suggested n the lterature. In ths paper, we propose two hypotheses to nvestgate estaton bas of the product lt estator, and provde three odfed copleton ethods based on the proposed hypotheses. The proposed hypotheses are verfed and the proposed copleton ethods are copared wth current nonparaetrc copleton ethods by sulaton studes. Sulaton results show that bases of the proposed copleton ethods are sgnfcantly saller than that of those n the lterature. Keywords Forecastng, Deand, Censored, Nonparaetrc, Product Lt Estator 1. Introducton In a newsvendor nventory syste, a decson aker places an order before the sellng season wth stochastc deand. If too uch s ordered, stock s left over at the end of the perod, whereas f too lttle s ordered, sales are lost. The optal order quantty s often set based on the well-known crtcal rato [1], therefore deand observatons often get rght censored when there are lost sales and no backorderng. Because lost sales cannot be observed, the avalable sales data actually reflect the stock avalable for sale, rather than the true deand. Deands for newsvendor-type products are often forecasted fro censored observatons. The proble of deand forecastng n the presence of stockouts s a well-known proble of handlng censored observatons, whch was recognzed by [2]. Approaches of handlng censored observatons can be dvded nto two classes (1) paraetrc ethod, whch often assues that the observatons coe fro specfc theoretcal dstrbuton and then estate paraeters of the assued dstrbuton by applyng axu lkelhood estaton or soe updatng procedures [3]; Ths ethod s often used n densty forecastng [4]; (2) nonparaetrc ethod, whch s often establshed based on the product lt estator [5], and attepts to address the proble of the undefned regon beyond the largest observaton when t s censored [6]. Paraetrc ethods for deand forecastng fro censored observatons have been nvestgated n [7 14]. These works have been brefly revewed n [15], and t has been ndcated n [15] that t s dffcult to deterne the shape or faly of deand dstrbuton n advance when deand observatons are censored. The product lt (PL) estator s a nonparaetrc axu lkelhood estator of a dstrbuton functon based on censored data. If the largest observaton s censored, the PL estator s developed to estate the left-hand sde of deand dstrbuton, but t s undefned for the rght-hand sde of dstrbuton functon. Under the assupton that there are ore nforaton besdes the censored observatons, Lau and Lau [3] and Zhang et al. [15] have nvestgated the probles of estatng the rght-hand sde of deand dstrbutons. Wthout addtonal nforaton besdes the censored observatons, truncaton technques or copleton ethods are usually eployed to defne the whole dstrbuton functon. Truncaton technques are based on the data-drvng rules, whch nclude two coon truncaton rules (1) truncatng at the largest observaton f t s censored, and (2) truncatng at (n l)th order statstcs [6]. These truncaton rules ay ntutvely appear to have good propertes by avodng probles n tal, but they wll ncur large bas because the locaton of the gnored Copyrght 2009 ScRes

2 260 Nonparaetrc Deand Forecastng wth Rght Censored Observatons regon s a rando event. Copleton ethods a to redefne the PL estator beyond the largest observaton f t s censored. We wll brefly revew nonparaetrc copleton ethods n the next secton. In ths paper, we propose two hypotheses to nvestgate estaton bas of the PL estator, and provde three odfed copleton ethods based on the proposed hypotheses. The proposed hypotheses are verfed and the proposed copleton ethods are copared wth current nonparaetrc copleton ethods n the lterature by sulaton studes. The reander of ths paper s structured as follows. We brefly ntroduce the PL estator and revew current nonparaetrc copleton ethods suggested n the lterature. Then we propose two hypotheses to nvestgate estaton bas of the PL estator, and provde three odfed copleton ethods. We further verfy the two hypotheses and copare the proposed copleton ethods wth current nonparaetrc copleton ethods by sulaton studes. The paper ends wth soe concludng rearks. 2. Nonparaetrc Copleton Methods In ths secton, we frst ntroduce the PL estator n the context of an nventory syste, and then we revew current nonparaetrc copleton ethods for the PL estator suggested n the lterature. 2.1 Product Lt Estator Let X, 1, 2,, n, be d (ndependent dentcally-dstrbuted) deand fro dstrbuton F, and nventory level Y, 1, 2,, n be d fro dstrbuton G. It s often to assue that both F and G are contnuous and defned on the nterval 0,. In an nventory syste, deand X s censored on the rght by the avalable nventory level Y, and we observe Z n X, Y and I X Y, 1, 2,, n, where I stands for the ndcator functon, and ndcates whether deand observaton Z s censored ( 0 ) or not ( 1). Kaplan and Meer [5] ntroduced the PL estator for the survval functon S t 1F t, whch s estated as follows ˆ n n S t 1 1 n1 I Z where Z n denotes the th ordered observaton aong all Z, and n corresponds to Z n. Fro the above defnton, t s observed that the PL estator s undefned beyond the largest observaton,.e., for n t (1) t Z and Revew of Current Copleton Methods To overcoe the shortfall of the PL estator that t s undefned beyond the largest observaton, soe copleton ethods are suggested n the lterature. Efron [16] ntroduced the noton of self-consstency,.e., SˆE t 0, for t Z nn. (2) Gll [17] defned the survval functon by I Z ˆ 1 n 1 n SG t n n 1 n1 Chen and Phada [18] odfed t as n I Z t for t Z nn (3) t n ˆ n1 n SC t c1 n n1 1 n1 for t Z (4) where c 0,1 s deterned by nzng the ean squared error loss E 2 ˆ F t 2 d ˆ F t F t E S t S t ds t (5) 0 0 Clearly, the extree values of scalar c yeld Efron s and Gll s versons, respectvely. Besdes the above three constant copleton ethods, there are two curve copleton ethods suggested n the lterature. Brown et al [19] suggested an exponental copleton ethod as follows ˆ B S e t B t, for t Z nn. (6) The paraeter where Z l Z Z s set by solvng B. Let () 0, 0 S ˆ Znn e B nn, Z, 1,, denote the ordered uncensored deand observatons, the reanng n observatons are censored ones. Moeschberger and Klen [20] attepted to coplete Ŝt by a two-paraeter Webull functon as follows SˆM t k e M t, for t Z (7) The two paraeters M and k n Equaton (7) are deterned by solvng Sˆ k M t Z e and k M t 1 e Sˆ Z. 1 When a copleton ethod s used, the bas of B t E Sˆ t S t, s entrely deterned by the Ŝt, copleton ethod [21]. For a copleton ethod, t s clear that the undefned regon has the ost contrbuton to the bas of the PL estator. One ght thnk that ths regon could be n soe sense gnored, as t s sug- Copyrght 2009 ScRes

3 Nonparaetrc Deand Forecastng wth Rght Censored Observatons 261 gested n truncaton technques. Because the locaton of ths regon s a rando event, sply gnorng the undefned regon wll result n a large bas [6]. The bas of SˆE t s negatve and asyptotcally zero as t, whereas the bas of SˆG t s postve and ncreasng as t. The bas usng any other copleton ethod wll be bounded by the bases of SˆE t and SˆG t [6]. The bas of SˆC t changes fro negatve to postve and t s ncreasng as t. If an estator s asyptotcally zero as t, we say that t has copleteness, whch s necessary for estatng oents of dstrbuton. SˆE t, S ˆB t, and SˆM t have copleteness snce they are asyptotcally zero as t, whereas SˆG t and SˆC t do not have the copleteness. The curve copleton ethods, SˆB t, and SˆM t satsfy the downward slopng onotoncty of survval functon, but the constant copleton ethods, S t, S t and S t do not. ˆE ˆG 3. New Copleton Methods In ths secton, we frst propose two hypotheses to nvestgate estaton bas of the PL estator at two specal ponts, and provde three odfed copleton ethods based on the proposed hypotheses. Then we splfy show the nonparaetrc copleton ethods by an exaple. ˆC 3.1 Estaton Bas of the PL Estator If deand observatons X, 1, 2,, n, are observable, then ts eprcal survval functon expressed as follows n 1 S t 1 I X t (8) n 1 S t can be Snce X Z, the value of S t at pont Z be rewrtten as nn can n 1 1 S Z 1 1 (11) n1 Fro Equatons (9 11), S ˆ Z can be vewed as a odfcaton of S Z by replacng I Xn Z by 1 (fro Equaton (9 11)), and then replacng 1 by n (fro Equaton (11,10)). By ntroducng these two replaceents, t s clear that S Znn S Znn and ˆ S Znn SZnn. Ths ndcates that S Z wll underestate S Z, S ˆ Z wll overestate S Z Sˆ Z wll underestate or overest-, but ate S Z. The sgn of bas ˆ SZnn terned by I Xn Z Snce I X Z n and n S Z 1 s copletely de- and n, 1, 2,, n1. are rando varables deterned by X and Y, 1, 2,, n, the bas s also a rando varable and ts sgn also depends on X and Y, 1, 2,, n. In the case when X n 1 n Z n n s satsfed and there s at least one censored observaton aong Z n, 1, 2,, n1, Sˆ Z ust overestate S Z. We argue that n has ore portant nfluence on the estaton bas than IXn Z does,.e., the PL estator wll statstcally overestate at pont Z. Based on ths percepton, we present the followng hypothess Hypothess 1 Denote by ˆ B Znn S Znn S Z, then BZ s statstcally larger than zero. Snce the PL estator s a pecewse rght contnuous functon, and the largest uncensored observaton Z s n 1 S Z 1 IX Z a rght contnuous pecewse pont, so the relatve estaton bas of the PL estator at pont Z n 1 (9) should be I Xn Znn n1 1 statstcally saller than that at pont Z. That s, the 1 1 n1 PL estator statstcally provde ore accurate estaton at pont Z Accordng to Equaton (1), the estaton value of the PL than at Z. Therefore, we have the estator at pont Z s followng hypothess 1 Hypothess 2 ˆ n1 n SZ 1 1 n1 (10) Denote by Rt Sˆ t S t S t, then RZ s To copare S Z statstcally saller than and Sˆ Z, we ntroduce RZ nn. Copyrght 2009 ScRes

4 262 Nonparaetrc Deand Forecastng wth Rght Censored Observatons 3.2 Modfed Copleton Methods In the sprt of the exponental curve copleton ethod suggested by [19], we provde three odfed copleton ethods based on the proposed hypotheses. Hypothess 1 ples that the PL estator wll statstcally overestate at pont Z. Therefore bas can be reduced f paraeter of the exponental curve s set by solvng ˆ D Z d S Z e nn nstead of ˆ B Z S Z e nn, where d 0,1 s an adjusted factor for overcong the overestaton of the PL estator at pont Z. Chen and Phada [18] proposed an optal constant copleton by settng Sˆ Z csˆ Z. Slarly, we set d n 2 c,1. Ths paraeter settng s presented because scalar d should not be larger than one n solvng D. Hypothess 2 ndcates that the relatve estaton bas of the PL estator at pont Z s statstcally saller than that at pont Z. Therefore bas can be reduced f paraeter of the exponental curve s set by solvng ˆ L Z S Z e nstead of ˆ B Z S Z e nn. Snce the exponental curve ay approxately pass the two ponts ˆ Z, S and Z Z, d S ˆ Z nn, the paraeter of the exponental curve can also be set as. 2 A L D 3.3 An Illustratve Exaple In a case study of a newsvendor nventory syste, Lau and Lau [3] presented 20 ordered daly sales observatons 34, 34, 37 *, 38, 44 *, 45 *, 47 *, 50, 50, 50, 60, 60 *, 65 * (8 tes), where astersk ndcates a censored observaton, e.g., the thrd entry 37 * eans that Z3 37 was observed on a day when Y3 37, plyng that X For ths exaple, the varous aforeentoned copleton ethods are plotted n Fgure 1. Fro Fgure 1 t can be observed that, the fve curve copleton ethods satsfy the downward slopng onotoncty of survval functon, and that the fve curve copleton ethods and Efron s self-consstent copleton have the copleteness. 4. Sulaton Studes In ths secton, we verfy the two proposed hypotheses and copare the aforeentoned copleton ethods by sulaton studes. 4.1 Sulaton Experents In our sulaton studes, we desgn two experents under an nventory syste wth soe specfc dstrbutons as follows Experent 1 Followng [22 23], we take deand dstrbuton F to be a Webull dstrbuton, a F x 1exp x for x 0 wth a=1 and 2. To reflect a varety of censorng dstrbuton patterns, we also follow [23] to take Webull dstrbuton, G y b 1exp uy for x 0 wth b a 2, b a, and b 2a, as our censorng nventory dstrbuton. Ths b b gves the hazard rate ht ut a, whch s decreasng for ba 1, constant for ba 1, and ncreasng ba 1. The scale factor u n Gy s adjusted n such a way so that the expected stockout probablty (ESP) Fgure 1. Coparson of the eght copleton ethods for the PL estator Copyrght 2009 ScRes

5 Nonparaetrc Deand Forecastng wth Rght Censored Observatons 263 Mean of Std. Dev. of 95% C.I. of Table 1. Statstcal results of B Z n Experent 1 b/a =0.5 b/a =1 b/a =2 a=1 a=2 a=1 a=2 a=1 a=2 B Z B Z Lower B Z Upper Mean of Std. Dev. of 95% C.I. of Table 2. Statstcal results of B Z n Experent 2 ESP =1/3 ESP =1/2 ESP =2/ B Z B Z Lower B Z Upper s 1/3, 1/2, or 2/3. These values thus copletely specfy the hazard rate. The reader s referred to [23] for further detals. Ths experent s appled for nvestgatng the case when hazard rate s decreasng, constant or ncreasng. Experent 2 Analogous to [8], we express the relaton between deand X and sales Z by wrtng sales as a rando proporton of deand,.e., Z WX, where W s a rando varable takng values on the nterval [0.5,1]. For perods wth no stockout, W 1, and therefore sales and deand are equal; for perods n whch a stockout has occurred, sales wll be less than deand wth W 1. We assue that stockouts occur n each perod (ndependently) wth probablty ESP and when a stockout occurs, sales Z s a rando (unforly dstrbuted) proporton of deand X. In our case studes, we take F to be a lognoral dstrbuton wth locaton paraeter 4, and shape paraeter 1 and 2, and we also set ESP=1/3, 1/2, and 2/3. Ths experent s desgned for nvestgatng the case when hazard rate changes fro ncreasng to decreasng. In the above two experents, we have four dfferent cases n ters of hazard rate decreasng, constant, ncreasng, and changng fro ncreasng to decreasng. In coparson wth Experent 1, Experent 2 akes an addtonal assupton on the relaton between deand and sales,.e., sales s a rando (unforly dstrbuted) proporton of deand. Consderng the cobnaton of the paraeters n the above two experents, under each of four cases of hazard rate, we have 6 dfferent cobnatons of the paraeters. Under each paraeters cobnaton, we set the nuber of observatons n=20, and randoly generate 1000 sulaton runs. To ensure the applcablty of the copleton ethod suggested by Moeschberger and Klen [20], the nuber of uncensored observatons n each sulaton run s restrcted to be larger than 3. For the convenence of coparson, the largest observaton n each sulaton run s restrcted to be a censored one. 4.2 Hypotheses Verfcaton Under each of four cases of hazard rate, we calculate BZ under 1000 sulaton runs for verfyng Hypothess 1. Statstcal results of BZ are reported n Tables 1 and 2. In these tables, 95% C.I. s short for 95% confdence level. Results shown n Tables 1 and 2 verfy Hypothess 1,.e., the PL estator statstcally overestates at pont Z. Table 1 also llustrates that B Z ncreases wth the ncrease of b/a, ths ples that the estaton bas of the case wth ncreasng hazard rate s larger than that of the case wth decreasng hazard case. Table 2 also llustrates that BZ ncreases as the expected stockout probablty ncreases. To verfy the correctness of Hypothess 2, we calculate RZ and R Z under each of four cases of hazard Copyrght 2009 ScRes

6 264 Nonparaetrc Deand Forecastng wth Rght Censored Observatons rate. Statstcal results of R Z and R Z n the two experents are reported n Tables 3 and 4, respectvely. The last two rows of these two tables present results of pared 2-taled t-tests on R Z R Z. Fro, Tables 3 and 4, t s observed that the relatve estaton bas of the PL estator of pont Z s statstcally saller than that of pont Z at the 0.01 sgnfcance level. Table 4 also ples that the relatve estaton bases of the PL estator at ponts Z and Z ncrease wth the ncrease of the expected stockout probablty. 4.3 Coparson wth Current Copleton Methods In ths subsecton, we assess perforance of the proposed copleton ethods n ters of ntegral absolute bas, IAB ˆ ES t S t ds t. In our sulaton 0 results, Efron and Gll denote Efron s and Gll s copleton ethods respectvely; CP, BHK and MK stand for the copleton ethods of Chen and Phada [18], Brown et al [19], and Moeschberger and Klen [20], respectvely; Left, Down and Ave represent the proposed exponental Table 3. Statstcal results of Mean of Std. Dev. of curve copleton ethods wth paraeter L, D and A, respectvely. Results of pared 2-taled t-tests on IAB aong the copared eght copleton ethods under each of four cases of hazard rate are shown n Tables 5 8, respectvely. These tables report t-statstcs on IAB between the row ethod and colun ethod. One negatve t-statstc n these tables eans that the row ethod s better than the correspondng colun ethod n ters of IAB, whereas postve t-statstc ples that the colun ethod s better than the correspondng row ethod. t-statstc n parentheses represents that the coparson s at the 0.05 sgnfcance level; t-statstc n square brackets ples that there s no sgnfcant dfference between the row and colun ethods; t-statstc wthout parentheses or square brackets expresses that the coparson s at the 0.01 sgnfcance level. Accordng to the followng results shown n Tables 5 8, we coe to the followng conclusons n ters of IAB (1) Ave s the leadng copleton ethod; (2) Left perfors better than the optal constant copleton ethod CP; (3) CP s always better than the current curve copleton ethods (.e., BHK and MK); (4) Efron and Gll are the two worst copleton ethods. R Z and R Z n Experent 1 b/a =0.5 b/a =1 b/a =2 a=1 a=2 a=1 a=2 a=1 a=2 R Z R Z Mean of R Z Std. Dev. of R Z t-statstcs P-value Table 4. Statstcal results of Mean of Std. Dev. of R Z RZ Std. Dev. of R Z and R Z n Experent 2 ESP =1/3 ESP =1/2 ESP =2/ R Z Mean of R Z t-statstcs P-value Copyrght 2009 ScRes

7 Nonparaetrc Deand Forecastng wth Rght Censored Observatons 265 Table 5. Results of pared 2-tal t-test on IAB n Experent 1 wth b/a=0.5 Gll CP BHK MK Left Down Ave Efron Gll CP ( ) BHK MK Left Down (2.4231) Table 6. Results of pared 2-tal t-test on IAB n Experent 1 wth b/a=1 Gll CP BHK MK Left Down Ave Efron Gll CP (2.5103) BHK MK Left Down Table 7. Results of pared 2-tal t-test on IAB n Experent 1 wth b/a=2 Gll CP BHK MK Left Down Ave Efron Gll CP BHK MK Left Down [1.6798] Table 8. Results of pared 2-tal t-test on IAB n Experent 2 Gll CP BHK MK Left Down Ave Efron (2.3348) ( ) Gll CP BHK MK [ ] Left ( ) Down Conclusons Deands for newsvendor-type products are often forecasted fro censored observatons. The Kaplan-Meer product lt estator s the well-known nonparaetrc ethod to deal wth censored data, but t s undefned beyond the largest observaton. In ths paper, we propose two hypotheses to nvestgate estaton bas of the PL estator, and provde three odfed copleton ethods based on the proposed hypotheses. Sulaton results show that bases of the proposed copleton ethods are sgnfcantly saller than that of the copleton ethods n the lterature. Accordng to these results, we know that the proposed copleton ethods can prove deand forecastng wth rght censored observatons. We also show that sulaton s a useful way to verfy probablty result whch s dffcult to be proved by usng classcal statstcal theory and ethods. The developed ethods are easy to pleent n software packages. Many forecastng technques have been ntegrated nto enterprse software packages such as anageent nforaton systes, enterprse resources plannng systes, decson support systes. The proposed forecastng technques n ths paper are sple and easly pleented n enterprse software packages. Copyrght 2009 ScRes

8 266 Nonparaetrc Deand Forecastng wth Rght Censored Observatons 6. Acknowledgeents Ths work s supported by natonal Natural Scence Foundaton of Chna (No ). REFERENCES [1] B. Zhang, X. Xu, and Z. Hua, A bnary soluton ethod for the ult-product newsboy proble wth budget constrant, Internatonal Journal of Producton Econocs, Vol. 117, No. 1, pp , January [2] G. Hadley and T. M. Whtn, Analyss of nventory systes, Prentce-Hall, Englewood Clffs, [3] H. S. Lau and A. H. L. Lau, Estatng the deand dstrbutons of sngle-perod tes havng frequent stockouts, European Journal of Operatonal Research, Vol. 92, No. 2, pp , July [4] Z. Hua and B. Zhang, Iprovng densty forecast by odelng asyetrc features An applcaton to S&P500 returns, European Journal of Operatonal Research, Vol. 185, No. 2, pp , March [5] E. L. Kaplan and P. Meer, Nonparaetrc estaton fro ncoplete observatons, Journal of the Aercan Statstcal Assocaton, Vol. 53, No. 282, pp , June [6] B. Gllespe, J. Gllespe, and B. Iglewcz, A coparson of the bas n four versons of the product-lt estator, Boetrka, Vol. 79, No. 1, pp , March [7] S. A. Conrad, Sales data and the estaton of deand, Operatonal Research Quarterly, Vol. 27, No. 1, pp , [8] W. Wecker, Predctng deand fro sales data n the presence of stockouts, Manageent Scence, Vol. 24, No. 10, pp , June [9] D. B. Braden and M. Freer, Inforatonal dynacs of censored observatons, Manageent Scence, Vol. 37, No. 11, pp , Noveber [10] N. S. Agrawal and A. Sth, Estatng negatve bnoal deand for retal nventory anageent wth unobservable lost sales, Naval Research Logstcs, Vol. 43, No. 6, pp , Septeber [11] P. C. Bell, Adaptve sales forecastng wth any stockouts, Journal of the Operatonal Research Socety, Vol. 32, No. 10, pp , October [12] P. C. Bell, A new procedure for the dstrbuton of perodcals, Journal of the Operatonal Research Socety, Vol. 29, No. 5, pp , May [13] P. C. Bell, Forecastng deand varaton when there are stockouts, Journal of the Operatonal Research Socety, Vol. 51, No. 3, pp , March [14] X. Dng, Estaton and optzaton n dscrete nventory odels, Ph.D. thess, The Unversty of Brtsh Coluba, Vancouver, Canada, [15] W. Zhang, B. Zhang, and Z. Hua, Quas-bootstrap procedure for forecastng deand fro sales data wth stockouts, n Proceedngs of the 38th Internatonal Conference on Coputers and Industral Engneerng, Vol. 1 3, pp , October [16] B. Efron, The two saple proble wth censored data, n Proceedngs of the 5th Berkeley Syposu on Matheatcal Statstcs and Probablty, Vol. 4, pp , [17] R. D. Gll, Censorng and stochastc ntegrals, Matheatcal Centre Tract No. 124, Matheatsch Centru, Asterda, [18] Z. Chen and E. Phada, An optal copleton of the product lt estator, Statstcs & Probablty Letters, Vol. 76, No. 9, pp , May [19] B. W. Brown, M. Jr. Hollander, and R. M. Korwar, Nonparaetrc tests of ndependence for censored data, wth applcatons to heart transplant studes, n F. Proschan and R. J. Serflng, (Eds.), Relablty and boetry statstcal analyss of lfelength, Phladelpha Socety for Industral and Appled Matheatcs, pp , [20] M. L. Moeschberger and J. P. Klen, A coparson of several ethods of estatng the survval functon when there s extree rght censorng, Boetrcs, Vol. 41, No. 1, pp , March [21] P. Meer, Estaton of a dstrbuton functon fro ncoplete observatons, n J. Gan (Eds.), Perspectves n probablty and statstcs, Appled Probablty Turst, Sheffeld, England, pp , [22] J. H. J. Geurts, Soe sall-saplel nonproportonal hazards results for the Kaplan-Meer estator, Statstca Neerlandca, Vol. 39, No. 1, pp. 1 13, March [23] J. H. J. Geurts, On the sall-saple perforance of Efron s and of Gll s verson of the product lt estator under nonproportonal hazards, Boetrcs, Vol. 43, No. 3, pp , Septeber Copyrght 2009 ScRes