Exact Non-parametric Con dence Intervals for Quantiles with Progressive Type-II Censoring

Published by Blackwell Publishers Ltd, 08 Cowley Road, Oxford OX4 JF, UK and 350 Main Street, Malden, MA 0248, USA Vol 28: 699±73, 200 Exact Non-parametric Con dence Intervals for Quantiles with Progressive Type-II Censoring OLIVIER GUILBAUD AstraZeneca, Sweden ABSTRACT. It is shown how various exact non-parametric inferences based on order statistics in one or two random samples can be generalized to situations with progressive type-ii censoring, which is a kind of evolutionary right censoring. Ordinary type-ii right censoring is a special case of such progressive censoring. These inferences include con dence intervals for a given parent quantile, prediction intervals for a given order statistic of a future sample, and related two-sample inferences based on exceedance probabilities. The proposed inferences are valid for any parent distribution with continuous distribution function. The key result is that each observable uncensored order statistic that becomes available with progressive type-ii censoring can be represented as a mixture with known weights of underlying ordinary order statistics. The importance of this mixture representation lies in that various properties of such observable order statistics can be deduced immediately from well-known properties of ordinary order statistics. Key words: con dence interval, exact inference, exceedance, mixture representation, nonparametric, order statistic, prediction interval, quantile, reliability, right censoring. Introduction Order statistics can be used for various exact non-parametric inferences, including con dence intervals for a given parent quantile, and prediction intervals for a given order statistic of a future sample. Such intervals are exact and non-parametric in that the relevant coverage probability is known exactly without any shape assumption about the parent distributionðthe only requirement is that its distribution function is continuous. It is shown in this article that these and other related exact non-parametric inferences can be generalized to situations with a certain kind of evolutionary right censoring, namely progressive type-ii censoring. A typical situation involving such censoring is as follows. A group of n experimental units is put on test at time zero in a life-testing experiment. Immediately after observing the rst death/ failure, a prespeci ed number R > 0 of the n remaining units are selected at random and removed, so that only n R units then remain under observation in the experiment. Immediately after observing the second death/failure, a prespeci ed number R 2 > 0 of the n R remaining units are selected at random and removed, so that only n R R 2 units then remain under observation in the experiment, and so on. Removed units thus become right censored at the time of death/failure of other units. The numbers R, R 2,... of units to be removed determine the censoring scheme. Successive removals of this kind may be desired and planned for various reasons; for example, to obtain deterioration or degradation data requiring the sacri ce or destruction of living or functioning units, and/or for resource reasons. Ordinary type-ii right censoring at a given order statistic is a special case of progressive type-ii censoring. Viveros & Balakrishnan (994) gave an interesting review of the background and of developments in this eld. They described various natural settings leading to this kind of experimentation, as well as recent results concerning inferences, including certain exact conditional con dence intervals and prediction intervals; see also McCool (982), Balakrishnan

700 O. Guilbaud Scand J Statist 28 & Sandu (995, 996), and Aggarwala & Balakrishnan (996, 998), with references therein. Recently Balasooriya et al. (2000) considered certain reliability sampling plans involving progressive type-ii censoring. Inferences have typically been based on parametric models, also con dence intervals for a parent quantile, and prediction intervals for a future order statistic. Models used include exponential, Weibull, log-normal, and generalized gamma models. The exact non-parametric inferences proposed in this article are not limited to certain parametric models, they are unconditional, and they are easily implemented using widely available statistical software. Of course, inferences based on the assumption that a certain parametric model holds may theoretically be more ef cient under that model, but the crucial question then is whether the model assumption is correct. The proposed non-parametric inferences have the advantage that their validity does not depend on the correctness of such an assumptionðthey are valid for any continuous parent distribution function. They may thus supplement or replace corresponding parametric inferences when model assumptions are questionable. A constructive de nition is given in section 2 of the observable uncensored order statistics that become available with a given progressive type-ii censoring scheme. This is then used in section 3 to show how each of these order statistics can be represented as a mixture of the order statistics of the underlying random sample. The mixture weights are determined by the censoring scheme and therefore known. This mixture representation is a key result for the inferences considered in this article. Its importance lies in that various properties of the observable uncensored order statistics can be deduced immediately from well-known properties of ordinary order statistics, for example certain coverage properties concerning parent quantiles. In view of its usefulness it is rather remarkable that this mixture representation appears to be new (though Stigler (977) made a very relevant observation in this context, see section 6.). Section 4 deals with its application to exact non-parametric con dence intervals for a parent quantile. A closely related mixture representation for certain exceedance probabilities is given in section 5. This representation can be applied to various exact non-parametric two-sample inferences based on exceedances, including: (a) prediction intervals for a given order statistic of a future sample (Fligner & Wolfe, 976, 979; Guilbaud, 983), (b) con dence intervals for a shift parameter (Pratt, 964, sect. 4; Hettmansperger, 984; Hettmansperger & McKean, 998, sect..2, 2.6.±2.6.2), and (c) related tests (Mathisen, 943; Westenberg, 948; Mood, 950; Gart, 963; Slivka, 970). Details are given in section 5 for prediction intervals for a future order statistic. Apart from differences in notation, the basic setup described in the next section is the same as that in Viveros & Balakrishnan (994). Results subsequently derived thus supplement that article naturally. A straightforward extension is given in section 6.2 that covers other situations, including the one considered by Balakrishnan & Sandu (996) and Aggarwala & Balakrishnan (998) that involves also type-ii left censoring. Point estimation is considered brie y in section 6.3. For convenience no notational distinction is made between random quantities and the corresponding realizations in this article. 2. Basic assumptions and de nitions Let X be a random variable with continuous distribution function F(t) ˆ Pr[X < t], and let X,, X n be the order statistics of a random sample of size n > 2 from the X - distribution. These order statistics are distinct with probability one. In the present context they are typically not all observable. Moreover, let m be a given integer satisfying 2 < m < n, and let R,..., R m be given nonnegative integers satisfying n ˆ Pm iˆ ( R i). These integers specify a progressive type-ii

Scand J Statist 28 Non-parametric inferences with censoring 70 censoring scheme. The uncensored order statistics Y,, Y m that become available with this scheme are de ned in m steps as follows in terms of the order statistics X,, X n. These Y i are all observable, and inferences considered in this article are based on them. In the following steps, n and X,, X n are denoted n 0 and X (0),, X (0) n 0, ``draw a SRS'' means ``draw a simple random sample without replacement'', and ð i is a certain nite population of integers of size N i from which a SRS of size n i is drawn. Brie y, ð i is an index set corresponding to N i remaining values. Y i of which n i are retained and R i are removed in step i; see ()±(4). Step. (a) Set Y ˆ X (0) ; (b) From ð ˆf2, 3,..., n 0 g of size N ˆ n 0, draw a SRS s ð of size n ˆ N R ; and (c) Let X () (),, X n denote the ordered X (0) j with j 2 s.... Step i, m. (a) Set Y i ˆ X (i )... Step m. ; (b) From ð i ˆf2, 3,..., n i g of size N i ˆ n i, draw a denote the ordered SRS s i ð i of size n i ˆ N i R i ; and (c) Let X (i),, X n (i) i X (i ) j with j 2 s i. Set Y m ˆ X (m ), N m ˆ n m and n m ˆ 0. It is assumed that the samples s i ð i are based on additional random experiments that are independent of each other and of X,, X n. Note that the nite populations ð i, their sizes N i, and the sample sizes n i, are completely determined by the integers n, m, and R,..., R m, and that n i ˆ N i R i equals n ( R ) ( R i ). 0 for < i, m, whereas n m ˆ 0 equals N m R m. The fact that the sample size in step i, m is of the form n i ˆ N i R i means that R i can be interpreted as the number of values. Y i that are removed in this step. That is, for each step i, m, the two ordered sequences, Y,, (Y i X (i ) ), X (i ) 2,, X (i ) n i, () Y,, Y i, X (i), X (i) (i) 2,, X n i, (2) can be interpreted as the ordered data that are available in this step before/after the removal of R i of the N i ˆ n i values X (i ) j. Y i, see the simpli ed notation in (3) and (4). Because N m ˆ n m ˆ R m, this interpretation of () and (2) can be made also for step i ˆ m. That is, it is understood that all n m values X (m ) j. Y m are removed, if there are any such values. The ordered data that are available after the last step m then consist of the uncensored order statistics Y,, Y m. Note that the situation with no censoring corresponds to the special case with m ˆ n and R ˆˆR m ˆ 0, whereas the situation with ordinary type-ii censoring corresponds to the special case with m, n, R ˆ ˆ R m ˆ 0 and R m ˆ n m. 3. Mixture representation for order statistics The key result for the inferences considered in this article is that under the assumptions made, each of the uncensored order statistics Y,, Y m can be represented as a mixture with known weights of the underlying order statistics X,, X n. This mixture representation is important in that various properties of the Y i s can be deduced immediately from well-known properties of the X j s. This representation is stated in theorem in terms of a certain matrix W of mixture weights de ned through a sequence of auxiliary mixtures.

702 O. Guilbaud Scand J Statist 28 3.. Auxiliary mixtures Consider an arbitrary step i < m in the de nition of Y,, Y m in section 2, and let Y,, Y i, Z,, Z N i, (3) Y,, Y i, Z,, Z ni, (4) denote the ordered data () and (2) that are available before/after the removal in that step, with the natural interpretation for i ˆ m described previously. Of course, if R i ˆ 0, then (3) and (4) are identical. Now, let i, m and recall from section 2 how Z,, Z ni in (4) are de ned through a SRS of size n i ˆ N i R i from a set of N i ˆ n i indices corresponding to Z,, Z N i in (3). A well-known result concerning the probability that the rth order statistic in a SRS equals the sth ordered value in a nite population (Wilks, 962, p. 243; David, 98, p. 25) then implies that Pr[Z r ˆ Z s 8 ] equals >< s Ni s Ni h (i) r,s ˆ, if r < s < r R r n i r n i, i (5) >: 0, otherwise, for < r < n i and < s < N i. Let H (i) be the n i 3 N i matrix with elements (5). This matrix H (i) ˆ (h (i) r,s ) has unit row sums, with h(i) r,s. 0ifr < s < r R i, and h (i) r,s ˆ 0 otherwise. In particular, if R i ˆ 0, then H (i) is an identity matrix. The rst i values in (3) are identical to the rst i values in (4). It follows that for each step i, m, the uth value among the i n i ordered values available after the removal is equal to the íth value among the i N i ordered values available before the removal with probability equal to the (u, í) element of the (i n i ) 3 (i N i ) matrix K (i) ˆ I (i) 0 (i)!, (6) 0 (i) 2 H (i) where I (i) is an i 3 i identity matrix, whereas 0 (i) and 0 (i) 2 are conformable zero matrices. This actually holds also for i ˆ m, with K (m) equal to the m 3 (m N m ) matrix (I (m) j0 (m) )ifn m. 0 and equal to I (m) if N m ˆ 0, because N m ˆ R m. This natural interpretation of (6) for i ˆ m is understood in the sequel. Thus for each step i < m, the uth value among the i n i ordered values (4) is equal to a mixture of the i N i ordered values (3), with mixture weights given by the uth row of K (i). The matrices K (i) are completely determined by the integers n, m, and R..., R m through (5) and (6), so these matrices are known. 3.2. The mixture representation Let W ˆ (w ij ) be the m 3 n matrix de ned as a product of matrices (6) by W ˆ K (m) K (m ) K () : (7) It follows immediately from (7) that W has non-negative elements and unit row sums, because each factor K (i) has non-negative elements and unit row sums (postmultiplication by a column vector of s results in a column vector of s). Further results concerning W are given in the next section, including a recursive relation that is convenient for numerical evaluations. The key result for the inferences considered in this article is now stated as theorem in terms of: (a) the m 3 n matrix W ˆ (w ij ) just de ned, (b) the uncensored order statistics

Scand J Statist 28 Non-parametric inferences with censoring 703 Y,, Y m de ned in section 2, and (c) the order statistics X,, X n of the underlying random sample. Theorem For each given i ˆ,..., m, it holds with probability one that Y i ˆ Xn jˆ ^w ij X j, (8) where ( ^w i,..., ^w in ) is a random vector independent of (X,..., X n ) that has multinomial distribution with parameters and (w i,..., w in ), and, as a consequence, it holds that Pr[Y i 2 B] ˆ Xn w ij Pr[X j 2 B] (9) jˆ for any given Borel set B. Proof. See appendix A. Theorem provides a strong and a weak mixture representation for each Y i marginally. Relation (8) is stronger than (9) in that the latter only asserts that the left and right members of (8) have the same distribution. A strong mixture representation for all Y i s simultaneously, as well as the stochastic behaviour of the m 3 n matrix ^W ˆ ( ^w ij ), is given as part of the proof of theorem in appendix A. Clearly, if the n probabilities in the right member of (9) are known, then the left member is known for each < i < m, because all weights w ij are known. This is used in section 4 with B equal to the interval [î p, ) or(î p, ) to construct exact non-parametric con dence intervals for a p-quantile î p. 3.3. The matrix W of mixture weights The multiplicative structure (7) of W is not related to stochastic properties of the order statistics X,, X n. It re ects the fact that the samples s i ð i underlying the auxiliary mixtures described in section 3. are independent of each other, cf. appendix A. As mentioned previously, it follows immediately from this structure that W has non-negative elements and unit row sums. Some additional results concerning W are given in this section. It is clear from the steps de ning the Y i s in section 2 that with probability one: (a) Y equals X, and (b) for each i > 2, Y i equals one of the order statistics X j with j > i. In view of (8), this implies that the rst row of W has only one positive element, namely w ˆ, and that in row i > 2, the elements w ij with < j, i are equal to zero. To evaluate W it is therefore only necessary to evaluate the elements w ij with 2 < i < j < n. It is shown in appendix B that this can be done conveniently as follows through a recursive relation. Recall that H (i) ˆ (h (i) r,s ) is the n i 3 N i matrix given by (5) for < i, m, with N i ˆ n i, n i ˆ N i R i and n 0 ˆ n. Set T (0) equal to the (n ) 3 (n ) identity matrix, and evaluate the elements w i,i, w i,i 2,..., w i,n successively for i ˆ,..., m through! w i,i w i,i 2,..., w i,n ˆ H (i) T (i ) : (0) 0 (i) T (i) The order of the right member is n i 3 (n i), and it is understood that the part of the left member below the horizontal partition line should be deleted if the right member has only one

704 O. Guilbaud Scand J Statist 28 row, which occurs for i ˆ m ifr m ˆ 0 but not otherwise. Only the matrices T (i) with i < m 2 are actually required for the evaluation of W through (0), and these matrices are well de ned through the partition of the left member. The positive elements of W have a certain structure. As mentioned previously, the rst row has the positive element w ˆ, and it can be shown (Guilbaud, 998, app. B.2) that row i > 2 has R R i consecutive positive elements, with w ii being the rst of these positive elements. All other elements are equal to zero. The number of positive elements in the successive rows of W is thus non-decreasing. Finally, recall that in the special case with no censoring it holds that m ˆ n and R ˆ ˆ R m ˆ 0, so in this case each factor K (i) in (7), as well as their product W, equals the n 3 n identity matrix. 4. Con dence intervals for a quantile 4.. Theoretical development Suppose that p 2 (0, ) is speci ed and that the p-quantile î p of the X-distribution is to be estimated. It is assumed that î p is uniquely de ned through î p ˆ ö(f) where ö() is a functional [given by some convention] that satis es G(ö(G)) ˆ p for all continuous distribution functions G. For example, î p may be de ned as the midpoint or as the left endpoint of the possibly degenerate interval fx; F(x) ˆ pg of all p-quantiles. Let r and s be given integers satisfying < r, s < m, and consider the interval estimators (, Y s ], [Y r, ) and [Y r, Y s ]. It is now shown that each of these interval estimators has a probability of covering î p that depends on the lower tail binomial probabilities b j ˆ Xj n p k ( p) n k, () k kˆ0 < j < n, and on W ˆ (w ij ), but not on F. The key idea is to use (9) with B ˆ [î p, ) and B ˆ (î p, ), and the fact that Pr[î p < X j ] ˆ Pr[î p, X j ] ˆ b j, to show that Pr[Y s, î p ] ˆ Xn w sj Pr[î p < X j ] ˆ Xn w sj b j, (2) Pr[î p, Y r ] ˆ Xn jˆ jˆ w rj Pr[î p, X j ] ˆ Xn jˆ jˆ w rj b j : (3) The probability of the coverage events [î p < Y s ], [Y r < î p ] and [Y r < î p < Y s ] can then easily be expressed in terms of (2) and (3) by considering complementary events. It is convenient to introduce the n 3 matrix b ˆ (b j ) with elements (), and the m 3 matrix a ˆ (a i ) de ned in terms of W ˆ (w ij )by a ˆ Wb: (4) Note that the probability (2) and (3) equals a s and a r, respectively, and that it follows from the latter relation that a < < a m, because Y,, Y m with probability one. In the special case with no censoring, W is the n 3 n identity matrix, so a equals b. The coverage probability of the interval estimators of î p can be expressed as follows in terms of the elements of a, Pr[î p < Y s ] ˆ a s, (5)

Scand J Statist 28 Non-parametric inferences with censoring 705 Pr[Y r < î p ] ˆ a r, (6) Pr[Y r < î p < Y s ] ˆ a s a r : (7) The restriction r, s is of course relevant only for (7). That is, (5) holds for < s < m, and (6) holds for < r < m. Provided a has been evaluated through (4), the integers r and/or s can be determined so that the con dence level of an interval is equal to or larger than a speci ed value á. It is understood here that a is such that this is possible. The restriction a s a r > á leaves some exibility with respect to the choice of r, s in (7). A simple and reasonable approach is to try to choose r, s so that a r a s. Another approach is the following. In the situation without any censoring it is reasonable (David, 98, p. 6) to try to minimize the index difference s r corresponding to the relevant interval [X r, X s ], that is, to minimize the expectation of the probability mass F(X s ) F(X r ) of the X- distribution within the interval. In the present situation it follows from (8) that the expectation of F(Y i ) ˆ Pn jˆ ^w ij F(X j ) equals e i ˆ Xn w ij j=(n ), (8) jˆ so the expectation of the probability mass within the interval [Y r, Y s ] is equal to e s e r.it therefore seems reasonable to try to choose r, s so that the difference e s e r is minimized. Computationally it is convenient to note that the m 3 matrix e ˆ (e i ) with elements (8) is equal to (n ) WJ with J ˆ (, 2,..., n) T. 4.2. Implementation and illustration The implementation is simple. Suppose the following quantities are given: (a) the m integers R,..., R m specifying the censoring scheme, and (b) the value of 0, p, specifying the quantile î p to be estimated. Then, the weight matrix W can be evaluated through (5) and (0) as described in section 3.3, the elements () of b can be evaluated through some appropriate algorithm or function for the binomial distribution function, and the vector a can be evaluated through (4). Moreover, the vector e can be evaluated as just described in the previous paragraph. Widely available software can be used for these calculations. A SAS program (SAS Institute Inc., 989) is given in Guilbaud (998). For illustration purposes, Viveros & Balakrishnan (994) used a progressively type-ii censored sample generated from certain times to breakdown of an insulating uid tested at 34 kv given by Nelson (982, tab. 6.), with: n ˆ 9, m ˆ 8, R,..., R 8 equal to 0, 0, 3, 0, 3, 0, 0, 5, and observed uncensored times Y,, Y 8 in original scale equal to 0.9, 0.78, 0.96,.3, 2.78, 4.85, 6.50, 7.35. They considered certain exact conditional inferences about the X-distribution based on these data under: (a) a two-parameter Weibull model (through an equivalent two-parameter extremevalue model for log(x )); and (b) an ordinary exponential model with Pr[X < x] ˆ exp( x=ì), x. 0. Under the Weibull model, they obtained an exact conditional 90% con dence interval for the shape parameter that covers the value corresponding to the simpler exponential model, so in this particular sense the data are consistent with the simpler model. They also showed that under this simpler model, the inferences are actually unconditionalðthe key result being that with ^ì ˆ Pm iˆ (R i )Y i =m, 2m^ì=ì has a chi-square distribution with 2m degrees of freedom. Inferences about a given function of ì such as the p-quantile î p ˆ ìlog( p) can be based on this result. The observed value of ^ì with these data is 9.09, so for example, with p ˆ 0:20, the intervals (, 3:48], [:38, ) and [:23, 4:70] for î 0:20 have con dence level equal to 0.90 under this model.

706 O. Guilbaud Scand J Statist 28 Table. Components a i of vector a in (4) for selected values of p i p 2 3 4 5 6 7 8 0.0 0.35 0.4203 0.7054 0.9006 0.9746 0.9962 0.9996 0.5 0.0456 0.985 0.443 0.786 0.890 0.9736 0.9953 0.9994 0.20 0.044 0.0829 0.2369 0.5003 0.7377 0.94 0.9775 0.9957 0.25 0.0042 0.030 0.3 0.3064 0.5482 0.8000 0.9329 0.983 0.50 0.0000 0.0000 0.0004 0.0043 0.028 0.20 0.2960 0.5399 The censoring scheme is given by: n ˆ 9, m ˆ 8, and R,..., R 8 equal to 0, 0, 3, 0, 3, 0, 0, 5. The same data are used here to illustrate the exact non-parametric con dence intervals for a p-quantile. Table gives the value of the components a i of the vector a for some selected values of p. Con dence levels of intervals of the form (, Y s ], [Y r, ) and [Y r, Y s ] can then easily be determined through (5)±(7). For example, from the row corresponding to p ˆ 0:20 in Table it follows that the intervals (, Y 6 ] ˆ (, 4:85], [Y 2, ) ˆ [0:78, ), and [Y, Y 6 ] ˆ [0:9, 4:85] for î 0:20 have exact con dence level equal to 0.94, 0:0829 ˆ 0:97 and 0:94 0:044 ˆ 0:8970, respectively. Exact con dence levels of intervals for other quantiles î p can be obtained similarly from the other rows of Table. A well-known practical problem in the situation without any censoring is that the available con dence levels may be rather few and separated in the neighbourhood of the desired level, particularly if the sample size is small. Of course, this may occur also with progressive type-ii censoring, and Table illustrates this. For example, the largest possible con dence level (6) of an interval [Y r, ) for the 0th percentile î 0:0 is 0:35 ˆ 0:8649, and the largest possible con dence level (5) of an interval (, Y s ] for the median î 0:50 is 0.5399. This is not surprising with these data. One should therefore ensure at the planning stage of an experiment that the n and the censoring scheme to be used lead to an acceptable con dence level for the interval and the quantile of interest. Moreover, the development of interpolation methods like those available for the situation without any censoring (Guilbaud, 979; Hettmansperger & Sheather, 986; Nyblom, 992) should be of considerable interest. 5. Two-sample inferences based on exceedances Inferences involving two independent samples are considered in this section. The key result is a mixture representation for certain exceedance probabilities that is closely related to theorem. This representation is stated in theorem 2. Various exact non-parametric inferences based on two independent random samples can be generalized through this representation to situations with progressive type-ii censoring, see section. Details are given here only for prediction intervals for a future order statistic. 5.. Assumptions and de nitions Let X ;,, X n ; and X ;2,, X n2 ;2 be the order statistics of two independent random samples of sizes n and n 2 from a common parent X-distribution with continuous distribution function. These n n 2 order statistics are distinct with probability one. In the present context they are typically not all observable. Moreover, for k ˆ, 2: (a) let m k be a given integer satisfying 2 < m k < n k ; (b) let R ;k,..., R mk ;k be given non-negative integers satisfying n k ˆ Pm k iˆ ( R i;k); (c) let Y ;k

Scand J Statist 28 Non-parametric inferences with censoring 707,, Y mk ;k be de ned in m k steps in terms of these integers and X ;k,, X nk ;k as described in section 2; and (d) let the m k 3 n k matrix W k ˆ (w ij;k ) of mixture weights (that relates each Y i;k to the underlying X j;k s through theorem ) be de ned as in section 3, with n k, m k and R i;k substituted for n, m and R i everywhere. Random quantities involved in the m steps de ning the Y i; s are assumed to be independent of random quantities involved in the m 2 steps de ning the Y i;2 s, so Y ;,, Y m ; are independent of Y ;2,, Y m2 ;2. These m m 2 uncensored order statistics are all observable. 5.2. Mixture representation for exceedance probabilities A convenient result in the theory of exceedances is that for any given < j < n and < j 2 < n 2, the probability Pr[X j ; < X j2 ;2] equals q j, j 2 ˆ X n 2 n n2 n, (9) j j j 2 j j j 2 j< j 2 which is a lower tail probability of the hypergeometric distribution with parameters j j 2, n 2 and n 2 n ; see Gastwirth (968, pp. 699±700) for a proof and some historical notes. Such exceedance probabilities are basic for the exact non-parametric two-sample inferences mentioned previously. It is then of interest to determine the analogous exceedance probabilities Pr[Y i ; < Y i2 ;2] through which these exact inferences can be generalized to situations with progressive type-ii censoring. Theorem 2 provides a convenient solution to this problem. The result is stated in terms of: (a) the m 3 n matrix W ˆ (w ij; ) and the m 2 3 n 2 matrix W 2 ˆ (w ij;2 ) of mixture weights de ned in section 5., and (b) the n 3 n 2 matrix Q ˆ (q j, j 2 ) with elements (9). Theorem 2 For each given i ˆ,..., m and i 2 ˆ,..., m 2, Pr[Y i ; < Y i2 ;2] equals the (i, i 2 )-element g i,i 2 of the m 3 m 2 matrix G ˆ W QW T 2 : (20) Proof. See appendix C. Recall from section 3.3 that if m k ˆ n k and R ;k ˆ ˆ R mk ;k ˆ 0, then the matrix W k equals the n k 3 n k identity matrix. Theorem 2 thus covers situations where both or only one of the samples are subject to progressive type-ii censoring, as well as the situation without any censoring. The numerical evaluation of (20) is simple. Each W k can be evaluated as described in section 3, and the elements (9) of Q can be evaluated through an appropriate algorithm or function for the hypergeometric distribution function. 5.3. Exact non-parametric prediction intervals for a future order statistic It is understood here that quantities indexed by k ˆ correspond to the present sample, whereas quantities indexed by k ˆ 2 correspond to the future sample. Let r, s and t be any given integers satisfying < r, s < m and < t < m 2, and consider prediction intervals of the form (, Y s; ], [Y r;, ) and [Y r;, Y s; ] for the order statistic Y t;2 in the future sample. The problem is to determine the probability that such an interval will contain Y t;2. It is emphasized

708 O. Guilbaud Scand J Statist 28 that the general development here covers the situations where the present sample and/or the future sample are subject to progressive type-ii censoring, as well as the situation without any censoring. Now, by considering relevant complementary events as in section 4., it can be veri ed using theorem 2 that Pr[Y t;2 < Y s; ] ˆ g s,t, (2) Pr[Y r; < Y t;2 ] ˆ g r,t, (22) Pr[Y r; < Y t;2 < Y s; ] ˆ g r,t g s,t : (23) The restriction r, s is of course relevant only for (23). That is, (2) holds for < s < m, and (22) holds for < r < m. Provided t is speci ed and G in (20) has been evaluated, the integers r and/or s can be determined so that the prediction probability is equal to or larger than a speci ed value á. It is understood here that G is such that this is possible. The restriction g r,t g s:t > á leaves some exibility with respect to the choice of r, s in (23). A simple and reasonable approach is to try to choose r, s so that g s,t g r,t. Another approach that seems reasonable is to try to minimize the expected proportion of future X j;2 s within [Y r;, Y s; ]. It can be shown using (8) and e.g. Fligner & Wolfe (976, th. 4.3) that the expected proportion of future X j;2 s less than or equal to Y i; is e i; ˆ Xn w ij; j=(n ), (24) jˆ so it seems reasonable to try to choose r, s so that e s; e r; is minimized. Note the similarity between (24) and (8), and that the m 3 matrix e ˆ (e i; ) with elements (24) is equal to (n ) W J with J ˆ (, 2,..., n ) T. Particular situations of interest are when a prediction interval is desired that will contain: (a) the order statistic Y t;2 in the future progressively type-ii censored sample for some given < t < m 2 ; or (b) the order statistic X t;2 in the future (underlying complete) sample for some given < t < n 2, for example t ˆ n 2 [which may be of interest even if one plans to censor this sample]. Situation (a) is covered by (2)±(23) in the previous general development; whereas situation (b) only requires the modi cation that W 2 is set equal to the n 2 3 n 2 identity matrix in (20). Other situations of interest are when a prediction interval is desired that will contain: (c) at least a speci ed number m9 2 < m 2 of the Y i;2 s in the future progressively type-ii censored sample; or (d) at least a speci ed number n9 2 < n 2 of the X j;2 s in the future (underlying complete) sample [which may be of interest even if one plans to censor this sample]. With a prediction interval of the form [Y r;, ) (or of the form (, Y s; ]), situation (c) corresponds to the choice t ˆ m 2 m9 2 (or the choice t ˆ m9 2 ); whereas situation (d) corresponds to the choice t ˆ n 2 n9 2 (or the choice t ˆ n9 2 ), with W 2 set equal to the n 2 3 n 2 identity matrix in (20). See Hahn & Meeker (99, sec. 5.4±5.5) for a discussion of such prediction intervals in situations without any censoring. 6. Concluding comments and additional results 6.. Comments It has been shown how exact non-parametric con dence intervals for quantiles and prediction intervals for future order statistics can be constructed in situations with progressive type-ii

Scand J Statist 28 Non-parametric inferences with censoring 709 censoring. These intervals are natural generalizations of corresponding intervals without censoring, they are unconditional, and they are valid for any continuous parent distribution function. Moreover, they are easily implemented. It has also been indicated how various other related exact non-parametric inferences based on exceedances can be generalized to situations with progressive type-ii censoring. The key results for the inferences considered in this article are: (a) the marginal mixture representation for each observable uncensored order statistic Y i given by theorem, and (b) the related mixture representation for an exceedance probability involving two such independent order statistics given by theorem 2. The importance of these representations lies in that various results concerning such observable uncensored ordered statistics can be deduced immediately from well-known results concerning ordinary order statistics. These mixture representations and applications appear to be new. It should, however, be mentioned that during the revision of this article the author discovered that Stigler (977, p. 549) made a very relevant observation in this context. Brie y, although Stigler did not give any explicit representation, he pointed out that the set of ordered uncensored data that become available with progressive type-ii censoring could be viewed as being selected from the set of underlying ordered data with probabilities that are determined by the censoring scheme. (The simultaneous mixture representation (25) derived in appendix A provides an explicit formulation of this fact.) Stigler did not give any potential application of this observation like those in the present article. He mentioned mixtures of Dirichlet processes in Bayesian non-parametric problems (Antoniak, 974), and an interpretation of the censored order statistic process in terms of a single Dirichlet process subjected to random monotone deformations of the time scale. 6.2. An extension A straightforward extension of the results derived in sections 2±4 is described in this section. It covers the particular situation considered by Balakrishnan & Sandhu (996) and Aggarwala & Balakrishnan (998) that also involves type-ii left censoring. In this particular situation, the R 0 smallest order statistics of the random sample from the X-distribution are not observed (with R 0 given), and the progressive type-ii censoring scheme is applied to the other part of the data. Suppose that in the de nition of the Y i s in section 2, X,, X n consist of a given subset (e.g. the n largest) of the order statistics X 9,, X9 n9 of a random sample of size n9.n from the X-distribution. It can then be veri ed that: (a) the mixture representations in theorem and (25) still hold as stated; (b) the coverage probabilities (5)±(7) hold with the only modi cation in (4) and () that n and j in the right member of () should be replaced by n9 and j9, where j9 is the appropriate index of X j in the random sample (given by the relation X9 j9 X j ); and (c) the expectation of F(Y i ) is given by (8) with n and j in the right member similarly replaced by n9 and j9. Exact non-parametric con dence intervals for î p can thus also be easily constructed in this case. In the particular situation just mentioned with type-ii left censoring of the R 0 smallest order statistics in a random sample of size n9, this extension is very simple, because it then holds that n9 ˆ R 0 n and j9 ˆ R 0 j. The results derived in section 5 can be extended similarly. 6.3. Point estimation In addition to the con dence intervals derived in section 4, one may wish to have a point estimator of the quantile î p of interest. Two estimators are described in this section. The second

70 O. Guilbaud Scand J Statist 28 is computationally somewhat more demanding than the rst, but it has an appealing unbiasedness property for certain values of p. A simple possibility is to interpolate linearly between the m points (Y i, e i ), i ˆ,..., m, with e i equal to the expected value of F(Y i ) given by (8). For any given p satisfying e < p < e m, the resulting point estimator of î p then equals ëy i ( ë)y i with ë ˆ (e i p)=(e i e i ) and i satisfying e i < p < e i, whereas the point estimator is left unde ned for p outside the interval [e, e m ]. This is analogous to an old proposal in the situation without any censoring where one interpolates linearly between the n points (X j, j=(n )), see def. 6 in Hyndman & Fan (996, p. 363) with references. When p equals e i for some i, this estimator of î p : (a) equals Y i ; (b) is invariant in an obvious sense under continuous and strictly monotone transformations of the data; but (c) is not unbiased, because the expectation E(F(Y i )) typically differs from F(E(Y i )). A variant is to use the median M i of the F(Y i )-distribution instead of its mean e i in the linear interpolation just described. This estimator of î p is thus de ned for any M < p < M m, and left unde ned for p outside the interval [M, M m ]. Recall that Pr[F(Y i ) < t] equals P n jˆ w ij Pr[F(X j ) < t] where F(X j ) has a Beta( j, n j ) distribution, so each M i can be determined numerically through a suitable search algorithm. This estimator is analogous to the one proposed by Hyndman & Fan (996) in the situation without any censoring, see their def. 8. When p equals M i for some i, this estimator of î p : (a) equals Y i ; (b) is invariant in an obvious sense under continuous and strictly monotone transformations of the data; and (c) is median unbiased. More elaborate point estimators are certainly possible to develop, and further research is required in this context. Acknowledgements The research reported in this article was partially carried out at Stockholm University, and partially supported by the National Network in Applied Mathematics in Sweden. Parts of the results were presented at the International Biometrics Conference of the International Biometric Society that took place in Cape Town, South Africa, December 998. The author thanks two referees for valuable comments and suggestions. References Aggarwala, R. & Balakrishnan, N. (996). Recurrence relations for single and product moments of progressive type-ii right censored order statistics from exponential and truncated exponential distributions. Ann. Inst. Statist. Math. 48, 757±77. Aggarwala, R. & Balakrishnan, N. (998). Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inferences and simulation. J. Statist. Plann. Inference 70, 35±49. Antoniak, C. E. (974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 52±74. Balakrishnan, N. & Sandhu, R. A. (995). A simple simulation algorithm for generating progressive type-ii censored samples. Amer. Statist. 49, 229±230. Balakrishnan, N. & Sandhu, R. A. (996). Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type-ii censored samples. Sankhya Ser. B 58, ±9. Balasooriya, U., Saw, S. L. C. & Gadag, V. (2000). Progressively censored reliability sampling plans for the Weibull distribution. Technometrics 42, 60±67. David, H. A. (98). Order statistics, 2nd edn. Wiley, New York. Fligner, M. A. & Wolfe, D. A. (976). Some applications of sample analogues to the probability integral transformation and a coverage property. Amer. Statist. 30, 78±85.

Scand J Statist 28 Non-parametric inferences with censoring 7 Fligner, M. A. & Wolfe, D. A. (979). Nonparametric prediction intervals for a future sample median. J. Amer. Statist. Assoc. 74, 453±456. Gart, J. J. (963). A median test with sequential application. Biometrika 50, 55±62. Gastwirth, J. L. (968). The rst-median test: a two-sided version of the control median test. J. Amer. Statist. Assoc. 63, 692±706. Guilbaud, O. (979). Interval estimation of the median of a general distribution. Scand. J. Statist. 6, 29±36. Guilbaud, O. (983). Nonparametric prediction intervals for sample medians in the general case. J. Amer. Statist. Assoc. 78, 937±94. Guilbaud, O. (998). Exact nonparametric con dence intervals for quantiles with progressive type-ii censoring. Research Report 208, Institute of Actuarial Mathematics and Mathematical Statistics, Stockholm University. Hahn, G. J. & Meeker, W. Q. (99). Statistical intervals, a guide for practitioners. Wiley, New York. Hettmansperger, T. P. (984). Two-sample inference based on one-sample sign statistics. J. Roy. Statist. Soc. Ser. C 33, 45±5. Hettmansperger, T. P. & Mc Kean, J. W. (998). Robust non-parametric statistical methods. Arnold, London. Hettmansperger, T. P. & Sheather, S. J. (986). Con dence intervals based on interpolated order statistics. Statist. Probab. Lett. 4, 75±79. Hyndman, R. J. & Fan, Y. (996). Sample quantiles in statistical packages. Amer. Statist. 50, 36±364. Mathisen, H. C. (943). A method of testing the hypothesis that two samples are from the same population. Ann. Math. Statist. 4, 88±94. McCool, J. I. (982). Censored data. In Encyclopedia of statistical sciences (eds. S. Kotz & N. L. Johnson). Wiley, New York. Mood, A. M. (950). Introduction to the theory of statistics. McGraw-Hill, New York. Nelson, W. (982). Applied life data analysis. Wiley, New York. Nyblom, J. (992). Note on interpolated order statistics. Statist. Probab. Lett. 4, 29±3. Pratt, J. W. (964). Robustness of some procedures for the two-sample location problem. J. Amer. Statist. Assoc. 59, 665±680. SAS Institute, Inc. (989). SAS/IML software: usage and references, Ver. 6, st edn. Author, Cary. Slivka, J. (970). A one-sided nonparametric multiple comparison control percentile test: treatments versus control. Biometrika 57, 43±438. Stigler, S. M. (977). Fractional order statistics, with applications. J. Amer. Statist. Assoc. 72, 544±550. Viveros, R. & Balakrishnan, N. (994). Interval estimation of parameters of life from progressively censored data. Technometrics 36, 84±9. Westenberg, J. (948). Signi cance test for median and interquartile range in samples from continuous populations of any form. Proc. Konink. Nederl. Akad. Wetensch. 5, 252±26. Wilks, S. S. (962). Mathematical statistics. Wiley, New York. Received January 2000, in nal form November 2000 Olivier Guilbaud, Biostatistics, AstraZeneca, S-585 SoÈdertaÈlje, Sweden. Appendix A: proof of theorem Let X ˆ (X,..., X n ) T and Y ˆ (Y,..., Y m ) T, and for each i < m, let B (i) and A (i) equal the (i N i ) 3 and (i n i ) 3 matrices with elements consisting of the ordered data (3) and (4) that are available before/after the removal in that step. Note that X ˆ B (), Y ˆ A (m), and A (i) ˆ B (i ) for i, m. For each i, m, let ^K (i) equal the right member of (6) with H (i) replaced by ^H (i) ˆ (^h (i) (i) where ^h r,s, is de ned in terms of (3) and (4) as the indicator of [Z r ˆ Z s ] that equals or 0 as this event occurs or not. Then, with probability one, each row of ^H (i) has one element equal to and the others equal to 0 in such a way that (Z,..., Z ni ) T ˆ ^H (i) (Z,..., Z N i ) T, so it holds that A (i) ˆ ^K (i) B (i). Recall also from section 3. that A (m) ˆ K (m) B (m) where K (m) equals (I (m) j0 (m) )ifn m. 0 and I (m) if N m ˆ 0. Combined together, these results show that with probability one that r,s ), ^W ˆ K (m) ^K (m ) ^K (), it holds with

72 O. Guilbaud Scand J Statist 28 Y ˆ ^WX: Each row of the m 3 n random matrix ^W has one element equal to and the others equal to 0, and each column has at most one element equal to. Provided ^W and X are independent (which will be shown), this means that (25) is a simultaneous mixture representation of the elements of Y in terms of the elements of X. The stochastic behaviour of ^W can be determined as follows. Recall from (3), (4) and section 2 how for each i, m, the indicators of the events [Z r ˆ Z s ] are de ned through a SRS s i ð i. This implies that: (a) the simultaneous distribution of the rows of ^H (i) is given by the probability that a SRS of size n i from a nite population of size N i equals a given subset of size n i of that population, that is, Pr[^h (i) r,s r ˆ, for r ˆ,..., n i ] ˆ Ni n i for each set of n i integers s r satisfying < s,, s ni < N i ; and (b) marginally, ^h (i) r,s (25) (26) ^h (i) r,s is a Bernoulli variable with probability parameter and thus expected value (5), which means that ^H (i) has expected value H (i) and thus that ^K (i) has expected value K (i). Another consequence is that the matrices ^K (i),< i, m, are independent of each other and of X, as the samples s i ð i. This determines the stochastic behaviour of ^W ˆ K (m) ^K (m ) ^K () in (25). In particular, it is clear that ^W is independent of X, and that ^W has expectation W given by (7). Now to the proof of theorem. The representation (25) implies that for each given < i < m, (8) holds with probability one, with ( ^w i..., ^w in ) behaving as described in theorem. The representation (9) then easily follows from (8) by intersecting [Y i 2 B] with the event [ n jˆ [ ^w ij ˆ ] which has probability one. Appendix B: proof of recursive relation (0) The fact that K (m) equals (I (m) j0 (m) )ori (m) in (7) means that W is given by the rst m rows of the product K (m ) K (2) K (), which is of the form I (m ) 0 (m )! I (2) 0 (2)! I () 0 ()! : (27) 0 (m ) 2 H (m ) 0 (2) 2 H (2) 0 () 2 H () Recall from (6) that the identity matrices I (), I (2),..., I (m ) in the successive factors from the right are of order 3, 2 3 2,...,(m ) 3 (m ). An immediate consequence of (27) is that the rst two rows of K () remain unaffected when K () is successively multiplied from the left by the other factors in (27). The rst two rows of the m 3 n matrix W ˆ (w ij ) are thus as follows. In the rst row, w ˆ, whereas w j ˆ 0 for 2 < j < n. In the second row, w 2 ˆ 0, whereas the subsequent elements w 2 j,2< j < n, are equal to the elements in the rst row of H (). Thus, the elements w 2 j,2< j < n, in the second row of W are as given by (0) with i ˆ. Similarly, the rst three rows of the product K (2) K () remain unaffected when this product is successively multiplied from the left by the other factors in (27), and, therefore, the third row of W is as follows. Its rst two elements, w 3 and w 32, are equal to 0, whereas the subsequent elements w 3 j,3< j < n, are equal to the elements in the rst row of the product H (2) T (), with T () equal to the submatrix of H () obtained by deleting the rst row and column of H (). Thus, the elements w 3 j,3< j < n, in the third row of W are as given by (0) with i ˆ 2. More generally, it can be veri ed that the rst i rows of the product K (i) K (2) K () remain unaffected when this product is successively multiplied from the left by the other factors in (27), and that the (i )th row of W therefore has its rst i elements, w i, j,< j < i, equal to 0, whereas the subsequent elements, w i, j, i < j < n, are equal to the elements in the

Scand J Statist 28 Non-parametric inferences with censoring 73 rst row of the product H (i) T (i ), as given by (0). This shows that W can be determined as described in section 3.3. Appendix C: proof of theorem 2 Apply (8) to each of the two independent order statistics in Pr[Y i ; < Y i2 ;2]. That this probability equals X n X n 2 j ˆ j 2ˆ w i, j ;w i2, j 2 ;2 Pr[X j ; < X j2 ;2] (28) then easily follows from the fact that the two involved multinomial vectors are independent of each other and of all X j;k s: intersect [Y i ; < Y i2 ;2] with the two events [ n j [ ^w ˆ i, j ; ˆ ] and [ n 2 j [ ^w 2ˆ i 2, j 2 ;2 ˆ ], which have probability one. Finally, substitute q j, j 2 in (9) for the probability in the general term in (28).