RELIABILITY ESTIMATION IN MULTISTAGE RANKED SET SAMPLING

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1 RLIABILITY STIMATION IN MULTISTAG RANKD ST SAMPLING Authors: M. Mahdzadeh Department of Statstcs, Hakm Sabzevar Unversty, P.O. Box 397, Sabzevar, Iran hsan Zamanzade Department of Statstcs, Unversty of Isfahan, Isfahan, , Iran Abstract: A nonparametrc relablty estmator based on multstage ranked set samplng s developed. It s shown that the estmator s unbased and ts effcency relatve to the smple random samplng rval s ncreasng n the number of stages. Numercal experments are used to llustrate the theoretcal fndngs. The suggested procedure s appled on a sport data set. Key-Words: Covarate nformaton; Judgment rankng; Stress-strength model. AMS Subect Classfcaton: 62N05, 62G30.

2 2 M. Mahdzadeh and hsan Zamanzade

3 Relablty estmaton n multstage ranked set samplng 3 1. INTRODUCTION Ranked set samplng (RSS s a data collecton technque whch s advantageous n settngs where precse measurement s dffcult (.e. tme-consumng, expensve or destructve, but small sets of unts can be accurately ranked wthout actual quantfcaton. The rankng of the unts s usually done by usng expert opnon, concomtant varable, or a combnaton of them, and need not to be exact. The RSS method was ntroduced by McIntyre [7] for estmatng average yelds n agrculture. In ths setup, precse measurement entals harvestng the crops, and thus s expensve. An expert, however, can accurately rank the yelds n a small set of adacent felds by vsual nspecton. There has been a surge of research on RSS n the last two decades. The RSS has been appled n a varety of areas such as forestry, envronmental scence and medcne. For a book-length treatment of RSS and ts applcatons, see Chen et el. [4]. The RSS desgn can be elucdated as follows: 1. Draw m random samples, each of sze m, from the target populaton. 2. Apply udgement orderng, by any cheap method, on the elements of the th ( = 1,..., m sample and dentfy the th smallest unt. 3. Actually measure the m dentfed unts n step Repeat steps 1-3, p tmes (cycles, f necessary, to obtan a ranked set sample of sze M = p m. Let X k be the th udgement order statstc from the kth cycle. Then, the resultng ranked set sample s denoted by X k : = 1,..., m ; k = 1,..., p. The desgn parameter m s called set sze. A ranked set sample contans more nformaton than a smple random sample of comparable sze because t contans not only nformaton carred by quantfed observatons but also nformaton provded by the udgment rankng mechansm. Thus, statstcal procedures based on RSS tend to be superor to ther smple random samplng (SRS analogs. The success of RSS hnges on accuracy of the rankng process. To reduce possble errors, the set sze m should be kept small n the basc verson of RSS. Al-Saleh and Al-Kadr [1] suggested double RSS (DRSS that ncreases effcency of the RSS mean estmator, gven a fxed m. Al-Saleh and Al-Omar [2] generalzed DRSS to multstage RSS (MSRSS, and showed that further gan n effcency can be acheved n estmatng the populaton mean. Al-Saleh and Samuh [3] nvestgated the dstrbuton functon and the medan estmaton based on MSRSS.

4 4 M. Mahdzadeh and hsan Zamanzade The MSRSS scheme can be summarzed as follows: 1. Randomly dentfy m r1 unts from the populaton of nterest, where r s the number of stages. 2. Allot the m r1 unts randomly nto m r 1 sets of m 2 unts each. 3. For each set n step 2, apply 1-2 of RSS procedure explaned above, to get a (udgement ranked set of sze m. Ths step gves m r 1 (udgement ranked sets, each of sze m. 4. Wthout actual measurng of the ranked sets, apply step 3 on the m r 1 ranked set to gan m r 2 second stage (udgement ranked sets, of sze m each. 5. Repeat step 3, wthout any actual measurement, untl an rth stage (udgement ranked set of sze m s acqured. 6. Actually measure the m dentfed unts n step Repeat steps 1-6, p tmes (cycles, f necessary, to obtan an rth stage ranked set sample of sze M = p m. Smlar to our prevous notaton, X (r k : = 1,..., m ; k = 1,..., p denotes the rth stage ranked set sample. Clearly, the especal case of MSRSS wth r = 1 corresponds to RSS. Also, DRSS s obtaned by settng r = 2. The estmaton of system relablty has drawn much attenton n the statstcal lterature. Relablty of a component wth strength X whch s subected to stress Y s quantfed by θ = P (X > Y. Ths approach s known as the stress-strength model. The estmaton of θ has been extensvely nvestgated n the lterature when X and Y are ndependent random varables, and belong to the same famly of dstrbutons. A comprehensve account of ths topc appear n Kotz et al. [5]. In ths artcle, we study relablty estmaton n MSRSS setup. In Secton 2, a nonparametrc estmator s proposed and ts propertes are nvestgated n theory. Secton 3 s gven to a Monte Carlo analyss of the fnte sample behavor of the estmator. A sport data set s analyzed n Secton 4. The paper s concluded wth a summary n Secton STIMATION USING MSRSS Let X 1,..., X m and Y 1,..., Y n be ndependent random samples from two populatons wth densty functons f and g, respectvely. The correspondng

5 Relablty estmaton n multstage ranked set samplng 5 dstrbuton functons are denoted by F and G. The standard nonparametrc estmator of θ s ˆθ = 1 I(X > Y, mn where I(. s the ndcator functon. =1 =1 To construct an estmator under MSRSS, one needs two ranked set samples of szes m and n from f and g. It s assumed that the samples are drawn usng a sngle cycle. The results n the general setup are then easly followed. If X (r, = 1,..., m, and Y (s, = 1,..., n, are the two multstage ranked set samples, then s a natural estmator of θ. Sengupta and Mukhut [8]. Let f (r and F (r be the densty and dstrbuton functon of X (r, respec- and G (s wll be used for smlar functons assocated tvely. The notaton g (s ˆθ r,s = 1 mn =1 =1 I(X (r The especal case of r = s = 1 was treated by wth Y (s. Suppose the th order statstc of an (r 1th stage ranked set sample of sze m from f, say Z (r 1 1,..., Z m (r 1, s denoted by Z (r 1 (. Under the assumpton of no error n udgment rankng, we have X (r d = Z (r 1 (. and In our mathematcal development, the two denttes 1 m 1 n =1 =1 f (r (x = f(x g (s (y = g(y, observed by Al-Saleh and Al-Omar [2], are repeatedly used. The above denttes can be expressed n terms of dstrbuton functons, as well. It s straghtforward to see that ˆθ s unbased. The unbasedness of ˆθ r,s s verfed n the followng proposton. Proposton 1 ˆθ r,s s an unbased estmator of θ.

6 6 M. Mahdzadeh and hsan Zamanzade Proof. m =1 =1 I(X (r = = = n = n = n =1 =1 =1 =1 =1 =1 =1 = mn P (X (r P (X (r P (X (r P (X (r > Y > yg (s (y dy > yg(y dy P (x > Y f (r (x dx P (x > Y f(x dx = mnp (X > Y. We now derve varance expressons of the two estmators. Proposton 2 The varances of ˆθ and ˆθ r,s are gven by (2.1 m 2 n 2 V ar(ˆθ = m(m 1n(n 1θ 2 nm(m 1 mn(n 1 F (Y G(X 2 mnθ m 2 n 2 θ 2, 2 and (2.2 m 2 n 2 V ar(ˆθ r,s = m 2[ ] F (Y (s 2 m [ =1 m n 2[ ] 2 [ G(X mnθ m 2 n 2 θ 2. =1 =1 G (s =1 ] 2 (X ] F (r (Y (s 2 Proof. It s easy to show that (2.3 m 2 n 2 (ˆθ 2 = (A 1 A 2 A 3 A 4, where (A 1 = (2.4 m =1 =1 I(X > Y I(X > Y = m(m 1n(n 1θ 2,

7 Relablty estmaton n multstage ranked set samplng 7 (2.5 (2.6 and (2.7 (A 2 = I(X > Y I(X > Y = =1 =1 =1 =1 = =1 =1 =1 =1 I(X > Y I(X > Y Y 2 2 F (Y = nm(m 1 F (Y, m (A 3 = I(X > Y I(X > Y = =1 =1 = =1 =1 I(X > Y I(X > Y X 2 2 G(X = mn(n 1 G(X, m (A 4 = I(X > Y = mnθ. =1 =1 From (2.3-(2.7 and unbasedness of ˆθ, the proof of the frst part s complete. Smlarly, (2.8 m 2 n 2 (ˆθ 2 r,s = (B 1 B 2 B 3,

8 8 M. Mahdzadeh and hsan Zamanzade where (B 1 = (2.9 = m I(X (r =1 =1 I(X (r =1 =1 = = =1 =1 =1 =1 m =1 =1 =1 =1 [ m =1 =1 =1 =1 I(X (r I(X (r I(X (r I(X (r [ (r F (Y (s [ (r F (Y (s Y (s I(X (r ][ F (r ][ F (r ] F (r (Y (s 2 m [ ][ (r F (Y (s (r F =1 =1 ] (Y (s =1 =1 (Y (s = m 2[ ] F (Y (s 2 m [ =1 ] (Y (s ] I(X (r Y (s [ ] (r F (Y (s 2 ] F (r (Y (s 2, Y (s (2.10 and (2.11 (B 2 = = m = m = m m I(X (r =1 =1 =1 =1 =1 I(X (r I(X I(X I(X I(X [ ][ G (s (X = m n 2[ ] 2 [ G(X (B 3 = m =1 =1 I(X (r =1 G (s G (s ] (X ] 2 (X, = mnθ. X

9 Relablty estmaton n multstage ranked set samplng 9 Now the second part follows from (2.8-(2.11 and unbasedness of ˆθ r,s. The varances of ˆθ and ˆθ r,s are compared n the next proposton. Proposton 3 For any m, n 2 and r, s 1, V ar(ˆθ r,s V ar(ˆθ. Proof. Usng equatons (2.1 and (2.2, t can be shown m 2 n 2[ ] V ar(ˆθ V ar(ˆθ r,s = C 1 C 2 C 3, where m [ C 1 = =1 =1 m ( = =1 =1 ] F (r (Y (s 2 [ ] m F (Y (s 2 =1 ] 2 [ F (r (Y (s F (Y (s, 2 C 2 = mn(n 1 G(X m n 2[ ] 2 [ G(X [ = m =1 [ = m =1 G (s G (s ] 2 [ ] 2 (X n G(X ] 2 (X G(X, =1 G (s ] 2 (X and C 3 = m(m 1n(n 1θ 2 nm(m 1 m(m 1 = m(m 1 =1 [ [ =1 ] F (Y (s 2 (1 1 n ( =1 F (Y (s [ = m(m 1 2 = m(m 1 =1 2 =1 F (Y (s F (Y (s F (Y (s ] F (Y (s F (Y 2 F (Y ( 1 n =1. 2 F (Y (s 2 ] Clearly, C 0, = 1, 2, 3, as was asserted.

10 10 M. Mahdzadeh and hsan Zamanzade As mentoned earler, ncreasng the number of stages leads to mprovement n the context of mean and dstrbuton functon estmaton based on MSRSS. So, t s natural to observe smlar trend n the case of relablty estmaton. The next result attends to ths problem. Proposton 4 For fxed m and n, V ar(ˆθ r,s s decreasng n r and s. Proof. It suffces to show that V ar(ˆθ r,s V ar(ˆθ r 1,s and V ar(ˆθ r,s V ar(ˆθ r,s 1. From the begnnng of proof for the second part of Proposton 2, one can wrte (2.12 m 2 n 2 (ˆθ 2 r,s = m I(X (r =1 =1 I(X (r =1 =1 I(X (r =1 =1 =1 =1 I(X (r I(X (r I(X (r I(X (r. We now establsh some equaltes and nequaltes regardng the four expectaton terms on the rght-hand sde of the above equaton. Let W (r 1 ( be the th order statstc of an (r 1th stage ranked set sample of sze m from f. As to the frst term, we have I(X (r I(X (r = I(X (r I(X (r Y (s, Y (s [ =, Y (s (2.13 = [ = I(X (r I(X (r I(W (r 1 ( Y (s Y (s I(W (r 1 ( I(W (r 1 ( I(W (r 1 (, Y (s Y (s Y (s ], Y (s, Y (s ] I(W (r 1 ( I(W (r 1 ( where the nequalty holds owng to the postve covarance between any par of order statstcs n a sample (see Lehmann [6]. Y (s,, Y (s

11 Relablty estmaton n multstage ranked set samplng 11 Smlarly, t follows that I(X (r (2.14 In addton, I(X (r I(X (r I(X (r (2.15 = = = [ [ = = = = I(X (r I(X (r I(X (r I(X (r I(W (r 1 ( Y (s Y (s I(W (r 1 ( I(W (r 1 ( I(W (r 1 ( I(X (r I(W (r 1 ( I(W (r 1 ( ] Y (s Y (s ] Y (s I(W (r 1 ( I(W (r 1 ( I(X (r Y (s I(W (r 1 ( I(W (r 1 ( Y (s., Y (s Y (s,, Y (s and (2.16 I(X (r = = = I(X (r I(W (r 1 ( I(W (r 1 ( Y (s Y (s.

12 12 M. Mahdzadeh and hsan Zamanzade Puttng (2.12-(2.16 together, we get m m 2 n 2 (ˆθ r,s 2 =1 =1 I(W (r 1 ( =1 =1 I(W (r 1 ( =1 =1 =1 =1 I(W (r 1 ( I(W (r 1 ( I(W (r 1 ( I(W (r 1 ( I(W (r 1 ( = m 2 n 2 (ˆθ 2 r 1,s. Ths mples that V ar(ˆθ r,s V ar(ˆθ r 1,s because ˆθ r,s s unbased for any r, s 1. A smlar argument proves the second part. The above theoretcal development assumes perfect rankngs. It s possble to obtan some results n the mperfect rankng stuaton. Suppose the rankng mechansm s such that and f (r 1 m 1 n =1 =1 f (r (x = f(x, g (s (y = g(y, where and g (s are the densty functons of the multstage udgment order statstcs drawn from the two populatons. Then one can smply verfy that Propostons 1 and 3 stll hold. However, t may not be an easy ob to prove Proposton 4 n ths setup. In the next secton, effect of the rankng errors s assessed usng Monte Carlo smulatons. 3. NUMRICAL RSULTS Ths secton reports results of smulaton studes carred out to compare the performances of ˆθ and ˆθ r,s. It s assumed that both populatons follow normal, exponental or unform dstrbuton. Suppose X and Y µ are standard normal random varables. Then, t s smply shown that ( µ θ = Φ, 2 where Φ(. s the dstrbuton functon of X. Smlarly, for standard exponental random varables X and Y/α, we have θ = 1 1 α.

13 Relablty estmaton n multstage ranked set samplng 13 Table 1: Parameter values correspondng to case A, B and C. Parameter A B C µ α 3 1 1/3 β 2 1 1/2 Fnally, let X and Y/β be unformly dstrbuted on the unt nterval. Then, t follows that 1 β/2 0 < β < 1 θ =. 1/(2β β 1 Under each parent dstrbuton, three values were assgned to the assocated parameter so as to produce θ = 0.25, 0.5, 0.75 whch are referred to as case A, B and C, respectvely. The approprate parameter values are gven n Table 1. Also, sample szes (m, n (3, 3, (4, 4, (5, 5 and stage numbers (r, s (1, 1, (2, 2, (2, 4, (3, 3, (4, 4, (4, 6, (5, 5 were selected. We assume that the rankng the varables of nterest X and Y are done based on concomtant varables X and Y whch are related accordng to equatons ( X µx X = ρ 1 1 ρ 2 1 σ Z 1, x and ( Y µy Y = ρ 2 1 ρ 2 2 σ Z 2, y where ρ [0, 1] ( = 1, 2, and Z 1 (Z 2 s a standard normal random varable ndependent from X (Y. Moreover, Z 1 and Z 2 are ndependent. The qualty of rankngs are controlled by the parameter ρ s. It s easy to see that Corr(X, X = ρ 1 and Corr(Y, Y = ρ 2. The chosen values of (ρ 1, ρ 2 are (1, 1 for perfect rankngs of X and Y, (1, 0.8 for perfect rankng of X and farly accurate rankng of Y, and (0.8, 0.8 for farly accurate rankngs of X and Y. For each combnaton of dstrbuton, sample szes and correlatons, 5,000 pars of samples were generated n SRS and MSRSS (wth the aforesad stage numbers. The two estmators were computed from each par of samples, and ther varances were determned. The relatve effcency (R s defned as the rato of V ar(ˆθ to V ar(ˆθ r,s. The R values larger than one ndcate that ˆθ r,s s more effcent than ˆθ. Tables 2-4 dsplay the results. It s observed that that MSRSS based estmator outperforms ts SRS contender n all stuatons consdered. Moreover, for any (m, n, the R s ncreasng n both r and s, when the other factors are fxed. For example, compare entres for m = n = 3. In general, no comparson can be made between Rs n two setups that one stage number s ncreased, and the other one s decreased. The effcency gan could be substantal f the set szes and stage numbers are large, e.g. when m = n = r = s = 5, the parent dstrbuton s unform, and the

14 14 M. Mahdzadeh and hsan Zamanzade Table 2: stmated Rs for dfferent sample szes and stage numbers under normal dstrbuton. (ρ 1, ρ 2 = (1, 1 (ρ 1, ρ 2 = (1, 0.8 (ρ 1, ρ 2 = (0.8, 0.8 (m, n (r, s A B C A B C A B C (3,3 (1, (2, (2, (3, (4, (4, (5, (4,4 (1, (2, (2, (3, (4, (4, (5, (5,5 (1, (2, (2, (3, (4, (4, (5, rankngs are perfect. It s to be mentoned that when (ρ 1, ρ 2 = (1, 1, the Rs for cases A and C are n good agreement (and smaller than that of case B for all dstrbutons and sample szes, partcularly when r = s. As expected, the Rs dmnsh n the presence of rankng errors. The smallest values are obtaned for (ρ 1, ρ 2 = (0.8, APPLICATION TO RAL DATA The MSRSS can be very effcent f the varable of nterest s hghly correlated to a concomtant varable. In ths case, f the second varable can be measured wth neglgble cost, then we may use t n udgment rankng process (see Stokes [9] for more detals. In dong so, n step 2 of the RSS procedure, the elements of the th sample are ordered accordng to the concomtant varable, and then study varable s actually measured for unt ranked th smallest. The MSRSS case s treated smlarly.

15 Relablty estmaton n multstage ranked set samplng 15 Table 3: stmated Rs for dfferent sample szes and stage numbers under exponental dstrbuton. (ρ 1, ρ 2 = (1, 1 (ρ 1, ρ 2 = (1, 0.8 (ρ 1, ρ 2 = (0.8, 0.8 (m, n (r, s A B C A B C A B C (3,3 (1, (2, (2, (3, (4, (4, (5, (4,4 (1, (2, (2, (3, (4, (4, (5, (5,5 (1, (2, (2, (3, (4, (4, (5, In ths secton, we llustrate the proposed procedure usng a data set collected at the Australan Insttute of Sport. It s made up of thrteen measured varables on 102 male and 100 female athletes 1. We wll consder lean body mass (LBM and body mass ndex (BMI for each athlete. The LBM s a component of body composton, calculated by subtractng body fat weght from total body weght. xact measurement of the LBM s done usng varous technologes such as dual energy X-ray absorptometry (DXA whch s costly. On the other hand, the BMI s a well-accepted measure of obesty whch s easy to calculate and readly accessble. A BMI value s s smply weght (n kg dvded by square of heght (n m. The correlaton coeffcent between the two varables s So, the BMI can serve as a concomtant varable. Let X and Y be the LBM varable for the male and female populatons, respectvely. It s of nterest to estmate θ = P (X > Y. For m = n = 4, 50,000 samples were drawn from the two hypothetcal populatons based on SRS and MSRSS (wth r = s = 1, 2 desgns. The samplng s done wth replacement 1 The data set can be found at

16 16 M. Mahdzadeh and hsan Zamanzade Table 4: stmated Rs for dfferent sample szes and stage numbers under unform dstrbuton. (ρ 1, ρ 2 = (1, 1 (ρ 1, ρ 2 = (1, 0.8 (ρ 1, ρ 2 = (0.8, 0.8 (m, n (r, s A B C A B C A B C (3,3 (1, (2, (2, (3, (4, (4, (5, (4,4 (1, (2, (2, (3, (4, (4, (5, (5,5 (1, (2, (2, (3, (4, (4, (5, to ensure that the measured unts are ndependent of each other. From each sample, the correspondng estmator was computed, and ts varance was fnally determned. The effcences of ˆθ 1,1 and ˆθ 2,2 relatve to ˆθ are estmated as and 1.275, respectvely. As expected, the SRS estmator s outperformed by ts RSS and DRSS versons. It s to be noted that the R values are not much bgger than unty. Ths may root n the relatvely low correlaton of 0.71 between the varable of nterest and the concomtant varable. 5. CONCLUSION The RSS desgn s known to be a vable alternate to the usual SRS n stuatons that cost-effcency s of hgh mportance. It employs auxlary nformaton to drect attenton toward the actual measurement of more representatve unts n the populaton under study. The success of RSS largely depends on the qualty of rankng process. Snce udgment rankng on large sets of unts s prone to

17 Relablty estmaton n multstage ranked set samplng 17 errors, the set sze s chosen small n practce. The MSRSS allows to construct more effcent procedures by ncreasng the number of stages rather that the set sze. Ths artcle deals wth relablty estmaton for the stress-strength model usng MSRSS. A nonparametrc estmator s presented, and shown to be unbased wth smaller varance as compared wth the usual estmator n SRS. It s further proved that the estmator becomes more effcent by ncreasng the number of stages for ranked set samples drawn from the two populatons. Results of smulaton studes support the mathematcal fndngs. An applcaton to a real data set clarfes how udgment rankng can be mplemented usng a concomtant varable. ACKNOWLDGMNTS The authors are grateful to the revewer and the Assocate dtor for ther comments that have largely contrbuted to mprove the orgnal manuscrpt. RFRNCS [1] Al-Saleh, M.F., and Al-Kadr, M. (2000. Double-ranked set samplng. Statstcs and Probablty Letters 48, [2] Al-Saleh, M.F., and Al-Omar, A.I. (2002. Multstage ranked set samplng. Journal of Statstcal Plannng and Inference 102, [3] Al-Saleh, M.F., and Samuh, M.J. (2008. On multstage ranked set samplng for dstrbuton and medan estmaton. Computatonal Statstcs and Data Analyss 52, [4] Chen, Z., Ba, Z., and Snha, B.K. (2004. Ranked set samplng: Theory and Applcatons. Sprnger, New York. [5] Kotz, S., Lumelsk, Y., and Pensky, M. (2003. The stress-strength model and ts generalzatons. Theory and applcatons. World Scentfc, Sngapore. [6] Lehmann,.L. (1966. Some concepts of dependence. The Annals of Mathematcal Statstcs 37, [7] McIntyre, G.A. (1952. A method of unbased selectve samplng usng ranked sets. Australan Journal of Agrcultural Research 3, [8] Sengupta, S., and Mukhut, S. (2008. Unbased estmaton of P (X > Y usng ranked set sample data. Statstcs 42, [9] Stokes, S.L. (1977. Ranked set samplng wth concomtant varables. Communcatons n Statstcs: Theory and Methods 6,