CONFIDENCE TUBES FOR MULTIPLE QUANTILE PLOTS VIA EMPIRICAL LIKELIHOOD

Transcription

1 The Annals of Statistics 999, Vol. 7, No. 4, CONFIDENCE TUBES FOR MULTIPLE QUANTILE PLOTS VIA EMPIRICAL LIKELIHOOD BY JOHN H. J. EINMAHL AND IAN W. MCKEAGUE Eindhoven University of Technology and EURANDOM and Florida State University The nonparametric empirical lielihood approach is used to obtain simultaneous confidence tubes for multiple quantile plots based on independent Ž possibly right-censored. samples. These tubes are asymptotically distribution free, except when both 3 and censoring is present. Pointwise versions of the confidence tubes, however, are asymptotically distribution free in all cases. The various confidence tubes are valid under minimal conditions. The proposed methods are applied in three real data examples.. Introduction. The quantile-quantile Ž Q-Q. plot is a well nown and attractive graphical method for comparing two distributions, especially when confidence limits are added. In this paper we develop Q-Q plot methods for the comparison of two or more distributions from randomly censored data. More specifically, we consider the problem of finding simultaneous confidence tubes for multiple quantile plots Ž for brevity, multi-q plots. from independent samples of possibly right-censored survival times. The multi-q plot is defined to be the -dimensional curve ŽQ Ž p.,..., Q Ž p.. parameterized by 0 p, where Q is the quantile function of the th distribution. It specializes to the ordinary Q-Q plot in the two-sample case. The comparison of quantile functions is particularly useful for the analysis of survival data in biomedical settings. Frail and strong individuals Žcorre- sponding to low and high values of p. often respond to different treatments in different ways, so treatment effects can be hard to determine from comparison of mean or median survival times alone; see, for example, Dosum Ž The approach developed here allows comparison of treatments simultaneously across all frailty levels. Our approach is based on the nonparametric empirical lielihood method. This method was originally developed by Thomas and Grunemeier Ž 975. and Owen Ž 988, 990. as a way of improving upon Wald-type confidence regions. There now exists a substantial literature on empirical lielihood indicating that it is widely viewed as a desirable and natural approach to Received May 998; revised April 999. Supported in part by a Fulbright grant and by European Union Grant ERB CHRX-CT AMS 99 subect classifications. Primary 6G5; secondary 6G0. Key words and phrases. Censoring, confidence region, distribution-free, -sample comparison, nonparametric lielihood ratio, quantile-quantile plot. 348

2 CONFIDENCE TUBES FOR QUANTILE PLOTS 349 statistical inference in a variety of settings. Moreover, there is considerable evidence that procedures based on the method outperform competing procedures. Empirical lielihood based confidence bands for individual quantile functions have recently been derived in Li, Hollander, McKeague and Yang Ž Nai-Nimbalar and Raarshi Ž 997. employed the approach to test for equality of medians; their test naturally extends to a test for equality of quantiles. We use the nonparametric empirical lielihood approach to derive asymptotic simultaneous confidence tubes for multi-q plots based on independent random samples, including confidence bands for ordinary Q-Q plots Ž.. The tubes are applicable to situations with or without random censoring. The limiting processes involved in the construction of the tubes are distribution free, except when 3 and censoring is present. In general, we are able to obtain asymptotically distribution-free pointwise confidence regions for the multi-q plot. The various confidence tubes are valid under minimal conditions, although for convenience we shall assume continuity of the underlying distribution functions. Q-Q plots are studied in detail using classical methods in Dosum Ž974, 977., Dosum and Sievers Ž 976. and Switzer Ž 976. for models without censoring; see Shorac and Wellner Ž 986., pages for a summary and discussion. For models with censoring, Wald-type simultaneous confidence bands for Q-Q plots are obtained in Aly Ž 986., but restrictive differentiability conditions on the underlying distribution functions are required. The -sample problem without censoring is studied in Nair Ž 978, 98., but essentially only pairwise comparisons are made there. A review of graphical methods in nonparametric statistics with extensive coverage of Q-Q plots can be found in Fisher Ž Some refined approximation results for normalized Q-Q plots with statistical applications have been established in Beirlant and Deheuvels Ž 990. for the uncensored case and Deheuvels and Einmahl Ž 99. in the censored case. The paper is organized as follows. The proposed confidence tubes and the main results are presented in Section. Our approach is illustrated in Section 3 using three real data examples. All the proofs are contained in Section 4.. Main results. We begin by specifying the setup precisely and introducing the basic notation. It is convenient first to recall the notation in the one-sample case. For the corresponding notation in the general -sample case, we use a further subscript to refer to the th sample. The random censorship model deals with n i.i.d. pairs Ž Z,. i i, i,..., n, obtained from two independent random samples X and Y, i,..., n, in the following way: Zi Xi Y i, i X Y 4. The distribution functions of X i i i and Yi are denoted F and G, respectively, and F is assumed to be continu- ous. We will wor with nonnegative Xi and Y i, but this restriction is in fact not needed anywhere; see the discussion at the end of this section. The right-continuous quantile function corresponding to F is denoted by Q. We i i

3 350 J. H. J. EINMAHL AND I. W. MCKEAGUE write n i Ł i i i i L F F Ž Z. F Ž Z. F Ž Z. i for the lielihood, where F belongs to, the space of all distribution functions on 0,.. The ordered uncensored survival times, that is, the X i with corresponding i, are written 0 T TN, and r Ý n denotes the size of the ris set at T. The empirical lielihood i Z i T4 ratio for Ft p Ž given 0 p. is defined by 4 4 sup LŽ F. : F Ž t. p, F RŽ t.. sup LŽ F. : F Note that the sup in the denominator is attained by the KaplanMeier Žor product-limit. estimator Fn Ž t. Ł. ž r / i: Tit i It can be shown with the aid of Lagrange s method see Thomas and Grunemeier Ž 975. or Li Ž 995. that ½ 5 i i log RŽ t. Ý Ž ri. log ri log, r r i: T t i where the Lagrange multiplier D max Ž r. i: T t i satisfies the equa- i tion. p. r Ł i: Tit ž i / Now we turn to the multisample setup. The samples are assumed to be independent with sample sizes denoted n,..., n ; write n Ý n. Set Ž F,..., F. and define the multi-q plot to be 4 Q Ž p.,..., Q Ž p. :0p. Observe that this is the classical Q-Q plot when. In the sequel we consider the following more convenient version of the multi-q plot: the graph Q of the function t Q F Ž t.,..., Q F Ž t. Ž. for t 0. Denote the oint lielihood by L L F Ł and the empirical lielihood ratio at t Ž t,..., t. by ½ 5 sup LŽ.: FŽ t. FŽ t. for all,...,, RŽ t. sup LŽ.: 4.

4 CONFIDENCE TUBES FOR QUANTILE PLOTS 35 Again we find, using Lagrange s method with the constraints, F Ž t. F Ž t.,,...,, that. log RŽ t. ½ 5 i i Ý Ý Ž ri. log ri log, r r i: Tit where the,,...,, satisfy the equations i i.3 ; r r Ł Ł i: Tit i: Tit Ž here we have set Ý so Ý 0. and the should satisfy D for,...,. Later we show that this system of equations indeed has a unique solution; see Lemma 4.. In the one-sample case, it is immediately clear that the corresponding Lagrange multiplier equation. has a unique solution, but it is not obvious in the multisample case. Computation of the s can be carried out using a special-purpose root-finding procedure which exploits the monotonicity of the r.h.s. of 3. as a function of Ž see Section 3 and the proof of Lemma 4... The various confidence sets we propose are easily obtained from the main theorem below and are presented in the three subsequent theorems. These confidence sets are all of the form t: RŽ. t c 4, where c is derived using asymptotic considerations. Before stating our main theorem we introduce some more notation. We assume throughout that n n p 0asn for,..., Ž although with some care this condition can be relaxed to n.. Define s df Ž u. Ž s. H. F Ž u. F Ž u. G Ž u. 0 Ž.Ž.Ž. We will need the -matrix D DŽ. t with entries iž ti. Ž t. i, for i, pp where d i i ti pi ' i Ý li li, for i, Ł l i, l l Ž tl. pl i Ý Ł Ž t. p q l q l l l Ž the empty product is defined to be.. Also define V VŽ t. to be the random -vector with th entry W Ž Ž t.. Ž t., where the W are independent standard Wiener processes. Let be such that F Ž. 0 and let be

5 35 J. H. J. EINMAHL AND I. W. MCKEAGUE such that F Ž., G Ž. and G ŽQ ŽF Ž... for,...,. We assume throughout that the F are continuous. THEOREM.. When R, D and V are evaluated at t Žt, Q ŽF Ž t..,..., Q ŽF Ž t... for t, we have.4 log R DV D on D,, with the -dimensional Euclidian norm. Write the restriction of Q to t, as Q,. In the next theorem we consider the important case, in which the multi-q plot reduces to the usual Q-Q plot. Define c s, s for 0 by ž / ss, s P sup W Ž s. s c s, s. Set ˆc c ˆ Ž., ˆ Ž., where ˆ 5 ½ ˆ t Qn F n Ž t. 5. ˆ Ž t. n, n n with ˆ.6 s n Ý i: T s i r r and with Q the Ž right-continuous. n quantile function corresponding to F n. Now we define the confidence band for Q, to be i i B t, 0,. : log RŽ t. ˆc 4. THEOREM.. In the censored case, for and 0, lim P Ž Q, B.. n 7. REMARK.. In the uncensored case Ž and. we have that Ž. Ž t. Q FŽ t. Ž t. p p ž p p / FŽ t. p p F Ž t. t.

6 CONFIDENCE TUBES FOR QUANTILE PLOTS 353 ˆ Therefore for this case we can replace the Ž t. defined in 5. by the simpler but almost equivalent estimator Fn Ž t. ˆ Ž t. n. ž n n F Ž t. / For use in ˆc, we can replace ˆ Ž t. by n F n Ž t.. F Ž t. n Observe that this last expression is not an estimator of Ž t. but of Ž t.. This however maes no difference because of 7. and the fact that for c 0 sup W Ž s. s sup W Ž s. s. D ss, s scs, cs Of course, here the KaplanMeier estimator Fn is ust the empirical distribution function of the first sample. Next we return to general, but assume that there is no censoring. Note that in this case the assumptions on reduce to F Ž.. Define C s, s for 0 by Ý ž s / s s, s 8. P sup W Ž s. C s, s. ˆ ˆ ˆ Set C C Ž., Ž., where F n Ž t. ˆ Ž t.. F Ž t. Define the confidence tube for Q, by n. ˆ 4 T t, 0, : log R t C. THEOREM.3. In the absence of censoring, for all and 0, lim P Ž Q, T.. n Now we allow censoring and but tae. Set Q Q,. Define the confidence region for Q by 4. 4 R t 0, : log R t, where is the upper -quantile of the chi-square distribution with degrees of freedom. In the case note that R amounts to a confidence interval for the F Ž.-quantile of F.

7 354 J. H. J. EINMAHL AND I. W. MCKEAGUE THEOREM.4. In the censored case, for all and 0, lim P Ž Q R.. n The asymptotic null distribution in the test for equality of medians developed by Nai-Nimbalar and Raarshi Ž 997. can be essentially derived from the proof of Theorem.4 by taing as their estimator * of the common median. Finally we establish an interval property for the confidence tube T Žwhich also applies to B and R.: one-dimensional cross-sections parallel to a given axis are intervals. This is useful for computing the various confidence sets because points belonging to them can then be found by a simple search strategy that sweeps along each axis. Žl. Ž Žl. THEOREM.5. Suppose that t t,..., t,..., t. T for l,, Ž. Ž. Ž. Ž. where t t. Then t* t,..., t,..., t T for any t t, t. In the two-sample case Ž. we have a somewhat stronger result. THEOREM.6. Let Žt Žl., t Žl.., l,, belong to the confidence band B and Ž. Ž. Ž. Ž. suppose t t and t t. Then Žt, t. also belongs to B whenever t Ž. t t Ž. and t Ž. t t Ž.. Ž. Ž. This theorem as well as Theorem.5 implies, by taing t t or t Ž. t Ž., that the intersection of the band B with a vertical or horizontal line is an interval. In addition, it shows that the bands are nondecreasing in the sense that their lower or upper boundaries are nondecreasing. Discussion. We wish to emphasize that our approach, including the definition of the multi-q plot, is new even in the uncensored case. We also remind that nonnegativity of the observations is not needed anywhere in the proofs. This is especially useful in the uncensored case, where often the samples do not represent life or failure times or when a transformation is applied to the data Ž see the third example in Section 3.. Another desirable feature of our approach is that the confidence bands and tubes are essentially invariant under permutations of the order of the samples involved. ŽOnly at the two ends of the tube does the first sample play a somewhat special role.. We did not formulate a version of our confidence tubes in the censored case for 3 since then DV in 4. is not distribution free, even when only one of the samples is subect to censoring. Our approach can, however, be generalized to this situation by estimating all the unnowns appearing in D and V and then using simulation. This means that we replace D by C Žgiven. Ž. in the proof of Theorem., t by Qn F n t for,...,, and by. The process to be simulated has the form CV, where Vˆ ˆ is the estimated version of V. Hence, approximate confidence tubes can be

8 CONFIDENCE TUBES FOR QUANTILE PLOTS 355 constructed for the censored case as well, but we do not pursue this in further detail here. The one-sample Q-Q plot, t QF Ž Ž t.. 0 with F0 nown, is essentially treated in Li, Hollander, McKeague and Yang Ž 996., since their confidence bands for QŽ p. can be transformed to bands for QF Ž Ž t.. 0 by the time change p F Ž t. 0. The present paper can be seen as a generalization of their approach to the -sample case. For uncensored data, in the two-sample Q-Q plot case, our confidence bands perform well in the tails due to the weighting which naturally arises when using the empirical lielihood method. Our bands share this property with the weighted bands Ž W bands. introduced in Dosum and Sievers Ž 976., which are based on the standardized two-sample empirical process. The bands in Switzer Ž 976. and Aly Ž 986. for the censored case are much wider in the tails, since they are based on the unweighted empirical process. All these procedures as well as our procedures are essentially based on the inversion of a distance between empirical distribution functions Žor KaplanMeier estimators.. In fact, the W bands are asymptotically equivalent to our bands in the uncensored case. 3. Applications to real data. In this section we illustrate our approach in three real data examples. First we consider a biomedical example for the two-sample case with censored survival data. The data come from a Mayo Clinic trial involving a treatment for primary biliary cirrhosis of the liver; see Fleming and Harrington Ž 99. for discussion. A total of n 3 patients participated in the randomized clinical trial, 58 receiving the treatment Ž D-penicillamine. and 54 receiving a placebo. Censoring is heavy Ž87 of the 3 observations are censored.. Figure displays the 90% confidence band Žand pointwise confidence intervals. for the Q-Q plot of treatment versus placebo for survival time in days. The standard empirical Q-Q plot based on quantiles of the KaplanMeier estimator is also displayed. Note that although the diagonal departs from the pointwise confidence region at some points, it remains within the simultaneous band, so there is no overall evidence of a difference between treatment and placebo. The second example also illustrates the two-sample case. Hollander, McKeague and Yang Ž 997. analyzed data on 43 manuscripts submitted to the Theory and Methods Section of JASA during 994. Each observation consists of the number of days between a manuscript s submission and its first review or the end of the year, along with a censoring indicator Ž if a paper received its first review by the end of the year; 0 otherwise.. Similar data Ž on 444 manuscripts. are available for 995. The censoring is light Ž330 of the 876 observations are censored. compared with the previous example. It is of interest to loo for differences in the pattern of review times for the two years. Figure displays the 95% confidence band Žand pointwise confidence intervals. for the Q-Q plot. The lower endpoints of the pointwise confidence intervals touch the diagonal between 0 and 5 days, which might suggest

9 356 J. H. J. EINMAHL AND I. W. MCKEAGUE FIG.. 90% confidence band Ž solid line. for the treatment versus placebo Q-Q plot in the Mayo Clinic trial, for 86 t 976 days; pointwise confidence intervals Ž short dashed line., empirical Q-Q plot Ž long dashed line.. that rapid reviews were faster in 994 than in 995. However, the diagonal is completely contained within the simultaneous band, so there is no overall evidence of a difference between the patterns of review times. The third example concerns times to breadown Ž in minutes. of an insulating fluid under three elevated voltage stresses, from data reported in Nair Ž 98., Table. It is important to determine whether the distribution of time to breadown changes with voltage. There are 60 uncensored observations at each voltage level Ž 34, 35 and 36 Kv.. As in Nair Ž 98. we use the 34 Kv measurements as a reference sample and put the breadown times on a log-scale. Figure 3 shows cross-sections of the 95% confidence regions for the multi-q plot at three values of the reference sample: t 0.4,.06 and.65. The confidence tube gives simultaneous coverage over the interval 0.4 t.65. The diagonal Ž t, t, t. runs above the pointwise confidence region at t.65 Ž top right plot. suggesting that increased voltage can reduce breadown time in the upper tail of the distribution. However, the diagonal falls completely inside the simultaneous tube Ž left column. so there is a lac of significant evidence for breadown time changing with voltage. In these examples, we computed the Lagrange multipliers in the system of equations 3. using the van WingaardenDeerBrent root finding algorithm Press, Teuolsy, Vetterling and Flannery Ž 99., page 359. The proof of Lemma 4. provides a constructive method to obtain the solution by repeated use of their algorithm. The thresholds c and Cˆ used in the ˆ

10 CONFIDENCE TUBES FOR QUANTILE PLOTS 357 FIG.. 95% confidence band Ž solid line. for the Q-Q plot based on the JASA time-to-first-review data, for 5 t 95 days; pointwise confidence intervals Ž short dashed line., empirical Q-Q plot Ž long dashed line.. confidence bands and tubes were computed by simulation of the Wiener processes on a fine grid. 4. Proofs. Here we present proofs of the theorems in Section. Some lemmas used in these proofs are given at the end of this section. PROOF OF THEOREM.. First we note that, by Lemma 4., below the system of equations in.3 with replaced by Ý has a unique solution for all. Define g : Ž D,. by g Ý log ž r / i i: Tit for,...,. Denote a g Ž 0. log SˆŽ t., where Sˆ is the KaplanMeier estimator of S F and b g 0 t n with as in 6. ˆ ˆ. Here t Q ŽF Ž t.. for,...,. Taylor series expansions of g and g, in conunction with Lemma 4. and the argument of Li Ž 995., proof of 5., page 0 yield Ž g Ž. g a a b b O Ž n., P

11 358 J. H. J. EINMAHL AND I. W. MCKEAGUE FIG. 3. Time to insulating fluid breadown Ž in log-scale.,36kv sample versus 35 Kv sample; cross-sections of the 95% simultaneous confidence tube Ž left column. and pointwise confidence regions Ž right column. at t 0.4,.06 and.65 in the 34 Kv reference sample Ž bottom row to top row, respectively.. uniformly in t,. Ignore the remainder term for the moment and consider the system of equations Ž 4.. b b a a for,...,, 0 with unnowns,...,. By Lemma 4.3 this system has as unique solution Ý Ž a a. i i i with the i as defined in the lemma. We now use this result to obtain an approximation for the. The remainder term in Ž 4.. consists of the remainders in the Taylor series Ž expansions of g and g, and both are of order O n. P. Attach these remainder terms to a and a, respectively, and apply Lemma 4.3. Note that

12 CONFIDENCE TUBES FOR QUANTILE PLOTS 359 Ž is a uniformly consistent estimator of, so b O n. ˆ P and i O Ž n., and it follows that P O Ž n. O Ž n. O Ž., P P P uniformly for t,. We also have that 4.3 b b OP n. Applying the Taylor series argument of Li Ž 995., page 0 to. and using Ž 4.3. then gives Ý P log RŽ t. b O Ž n.. Write the leading term above in the form Ý b Cw, where C is the -matrix with entries bb for i, c ' i i iý li li i b, for i. and w is the -vector with entries a log SŽ t. a log SŽ t. w. b b ' The proof is completed by noting that Ž./ ž W t wž t. D VŽ t.,,..., t ',..., where W,..., W are independent standard Wiener processes, and c d. i P i PROOF OF THEOREM.. Let us first simplify DV for this case. Note that for we have 0 Ž t. Ž t. Ž t. p p p D, Ž t. Ž t. Ž t. Ž t. ' ' p p p

13 360 with J. H. J. EINMAHL AND I. W. MCKEAGUE as in Remar.. So 0 Ž t. 'p D, Ž t. Ž t. ' p where a aa, and hence DV 0 0 Ž t. / 'p ž W t W t t ' p ' p ' p t Ž t. 'p D WŽ Ž t.., Ž t. Ž t. ' p so that W Ž t. DV D. Ž t. It is well nown that Ž s. Ž s. ˆ P,,, and hence with some care it can be shown that ˆ Ž. Ž. l P l, l,. Setting c c Ž., Ž., this yields ˆc P c. Combining the above we obtain W Ž Ž t.. P Ž Q, B. P log RŽ t, Q Ž F Ž t.. ˆc for all t, ž / t, ž W Ž s. / s s, P sup c Ž t. P sup c, where we used, for the convergence statement, that the random variable in the last expression has a continuous distribution. Before continuing with the proofs of the theorems let us do some calcula- tions on DV of Theorem. in general. Note that D is symmetric and by

14 CONFIDENCE TUBES FOR QUANTILE PLOTS 36 Lemma 4.4 it is idempotent of ran. Thus we may diagonalize D Dt Ž. as follows: I 0 Ž 4.4. DŽ t. PŽ t. PŽ t., ž 0 0/ where Pt is orthogonal and I is the identity matrix of order. Put Zt Pt VŽ t.. Then Ž 4.5. DŽ t. VŽ t. VŽ t. DŽ t. DŽ t. VŽ t. VŽ t. DŽ t. VŽ t. I 0 ZŽ t. ZŽ t., 0 0 where the second equality follows since D is symmetric and idempotent. The covariance structure of the process ZŽ t. is given, for two values of t, say s t, by EŽ ZŽ s. ZŽ t.. Ž. Ž Ž ž Ž. Ž./ Ž 4.6. Ž s. Q FŽ s. Q FŽ s. PŽ s. diag,,..., PŽ t.. Ž t. Q F Ž t. Q F Ž t. First observe that FŽ QŽ FŽ t... FŽ t. Ž QŽ FŽ t... Ž t., F Q F Ž t. F Ž t. PROOF OF THEOREM.3. Ž. for,...,. This implies Dt, and hence Pt, does not depend on t. Thus the r.h.s. of Ž 4.6. reduces to Ž s. Ž t. It follows that the process Zt has the same distribution as the process I. W Ž t. W Ž t. ž t t /. Ž,...,, where the W s are independent standard Wiener processes, and hence by Ž 4.5., W Ž Ž t.. DV D Ý. Ž t. Now the proof of this theorem can be completed along the same lines as that of Theorem.. In this case, use continuity of the random variable W Ž s. sup Ý, s s,

15 36 J. H. J. EINMAHL AND I. W. MCKEAGUE which follows from a property of Gaussian measures on Banach spaces, namely that the measure of a closed ball is a continuous function of its radius; apply, for example, Paulausas and Racausas ˇ Ž 989., Chapter 4, Theorem. to the Gaussian measure induced by the process Ž. s W s,..., W s on the Banach space of -valued continuous functions on Ž., Ž. endowed with the supremum norm. PROOF OF THEOREM.4. This theorem can be proved along the lines of the previous two. We only note that now the r.h.s. of Ž 4.6., with s t, reduces to the identity matrix I. Thus ZŽ. is a -vector of independent standard normal random variables. Hence from Ž 4.5. we find that DV, evaluated at, has a distribution. PROOF OF THEOREM.5. In order not to overdo the notation we restrict ourselves to proving this theorem for 3; for 3 the proof is essentially the same. W.l.o.g. we tae 3. Because the denominator of the lielihood ratio does not depend on t Ž t,..., t., we only consider the expression 3 N r i ŁŁ i Ž i. i log h h, with the h Ž 0,. defined by h F Ž T. Ž F Ž T.. i i i, i. Setting z i logž h., this becomes i 3 N ŁŁ i i i i Ž. Ž. log exp z exp z r 3 N Ý Ý i Ž i. Ž i. 4 z r log expž z. g Ž z., i with z Ž z,..., z, z,..., z, z,..., z. N N 3 3N. Observe that g is a con- 3 vex function. N N N Now g z, z, 0 3, has to be minimized under the constraints 4.7 z z and z z. Ý Ý Ý Ý i i i 3i i: Tit i: Tit i: Tit i: T3it3 Žl. Žl. Solutions of 4.7 for t t that minimize g z, are denoted with z, l,, respectively. Ž. Ž. For t t, t, define the function Ý Ž i i. Ý Ž 3i 3i. fž x. xz Ž. Ž x. z Ž. xz Ž. Ž x. z Ž. i: Tit i: T3it3 Ž. for 0 x. Since t t, we easily see that fž Similarly, using Ž. t t, we obtain f 0. Thus there exists an x 0, 3 3 such that fž x*. 0. Define z* x*z Ž. Ž x*. z Ž.,..., x z Ž. Ž x*. z Ž.. Ž 3 N 3 N. 3 3

16 CONFIDENCE TUBES FOR QUANTILE PLOTS 363 Then trivially the two equations in Ž 4.7. are satisfied for z z* and t t*. Also because g is convex, g Ž z*. x*g Ž z Ž.. Ž x*. g Ž z Ž... Žl. This implies, since log RŽ t. C ˆ, l,, that log RŽ t*. C ˆ, that is, t* T. The proof of Theorem.6 is similar to, but easier than, the previous proof. Moreover it is a straightforward extension of the proof of Theorem in Li, Hollander, McKeague and Yang Ž Therefore we will omit the proof here. We conclude by proving the four lemmas that we used earlier. LEMMA 4.. The system of equations 3.,with unnowns,...,, has a unique solution for all provided D 0 for,...,. PROOF. Define f : Ž D,. Ž 0,. by Ž 4.8. f Ł i: T t ž r / i i for,...,. We need to show that the system of equations Ý 4.9 f f,,..., has a unique solution. Note that f is continuous, strictly increasing, and vanishes as D. It then follows that there is a unique solution to Ž 4.9. when, because the decreasing function f Ž. must cross the increasing function f Ž. at exactly one value of Ž D, D.. Now consider 3. For each fixed D and 3,...,, there exists a unique Ž. such that f Ž. f Ž.. Each of these s is strictly increasing as a function of because f and f are strictly increasing. Now consider the equation Ý 3 Ž 4.0. f Ž. f Ž.. The l.h.s. of Ž 4.0. is defined whenever D D, where D is the unique solution to Note that D D because Ý 3 D D D. Ý D Ž D. D 0 D. Ý 3 Moreover, as a function of Ž D, D., the l.h.s. of Ž 4.0. is strictly decreasing and vanishes as D ; the r.h.s. is strictly increasing and

17 364 J. H. J. EINMAHL AND I. W. MCKEAGUE vanishes as D. Thus 4.0 holds for some unique D, D. Now set Ž. for 3,...,. It is then clear that Ž,...,. is the unique solution to Ž LEMMA 4.. Suppose n n p 0 for,...,. Let t Q ŽF Ž t.. for,..., and t Ž t,..., t.. Then Ž t. O Ž n. uniformly over,. P PROOF. Write the value of each side of 3. as p when t has the above form. By Li Ž 995., page 0, if 0 then n n / logž p. A ˆ Ž t., where Aˆ is the NelsonAalen estimator of A, the cumulative hazard function corresponding to F, and if 0 then the above inequality reverses. Thus for any pair, with 0, 0 Ž l l such pairs always exist, if not all the s are 0, since Ý 0. we have and hence ž / n nl Aˆ t Aˆ Ž. lž t l. ž, n n l l n Aˆ Ž t. n Aˆ Ž t. Aˆ Ž t. Aˆ Ž t. n n. l l l l l l l Note that A Ž t. A Ž t. and A Ž t. is bounded away from 0 if t. Thus by the uniform convergence of the NelsonAalen estimators A ˆ, we have that ˆ for any 0 and n sufficiently large, A Ž t. A Ž t. for all t, with probability at least, similarly for A ˆ. It then follows that 0 Ž n n. A Ž t. Aˆ Ž t. Aˆ Ž t. n n, l l l l l l with probability, for n sufficiently large. Finally, using the fact that ˆ Ž. Ž A t A t O n uniformly over,, we find that O n. P P for all,...,, uniformly for t,. LEMMA 4.3. where The system of equations Ž 4.. has solution Ý Ž ai a. i, i i 0 Ł l 0 Ý Ł l li, l ž li / i b and b. The solution is unique when all the b s are positive. l

18 CONFIDENCE TUBES FOR QUANTILE PLOTS 365 PROOF. The coefficient of a in Ý is Ý i Ý 0, i similarly for the coefficients of a,..., a. Thus Ý a in b,is and the coefficient of a in b is Ł b b Ý 0 l l Ý Ł b b i 0 l i i li so the coefficient of a in b b is Ł Ý Ł b b. 0 l 0 l l i li 0. The coefficient of The same argument shows that the coefficient of a in b b is. The coefficient of a, with q 3, in b b is q ž Ł Ł / b b b b b b 0. q q 0 l l l, lq l, lq This shows that b b a a and the same argument shows that all the other equations in Ž 4.. are satisfied. Ž. LEMMA 4.4. The -matrix D D t is idempotent, that is, D D and of ran. PROOF. Setting v t p, we have Ý i Łl vl dii, Ý Ł v q l q l ' i l i, l l vv Ł v di, i. Ý Ł v q l q l Because of the various symmetries it suffices to show that 4. d di and d d i, d i, i i Ý Ý for the idempotency of D. For the first equality we need to show that ÝŁ ÝŁ ÝŁ Ý Ł v v v vv v. l l l l l i li l l, l

19 366 J. H. J. EINMAHL AND I. W. MCKEAGUE Writing C Ý Ł v, this reduces to l l C C Łvl C v Ý v Ł v l, l l, l or, subtracting C on both sides, ÝŁ Ł Ý Ł v v v v v, l l l l l l, l which is easily seen to be true. For the second equality in 4. we have to show that ' vv Łvl Ý Łvl Ý Łvl ' vv Łv ž l l3 li l l3 / i ž ÝŁvl ' vv Łv /ž l l l3 / ' Ý Ł Ł v v v v v. i3 Dividing both sides by ' vv Ł l3 v l yields i l l l, li l, li ÝŁ ÝŁ ÝŁ ÝŁ v v v v, l l l l i li l l i3 li which is obviously true. This establishes the idempotency of D. For the second statement in the lemma, note that the ran of an idempotent matrix is equal to its trace. It is easily seen that the trace of D is. Acnowledgments. We than the reviewers for many constructive suggestions and Myles Hollander for providing the JASA time-to-first-review data. John Einmahl thans the Department of Statistics, Florida State University, for their warm hospitality during the writing of the article. REFERENCES ALY, E.-E. A. A. Ž Quantile-quantile plots under random censorship. J. Statist. Plann. Inference BEIRLANT, J. and DEHEUVELS, P. Ž On the approximation of P-P and Q-Q plot processes by Brownian bridges. Statist. Probab. Lett DEHEUVELS, P. and EINMAHL, J. H. J. Ž 99.. Approximations and two-sample tests based on P-P and Q-Q plots of the KaplanMeier estimators of lifetime distributions. J. Multivariate Anal DOKSUM, K. A. Ž Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist DOKSUM, K. A. Ž Some graphical methods in statistics. A review and some extensions. Statist. Neerlandica

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