Optimal goodness-of-fit tests for recurrent event data

Transcription

1 Lifetime Data Anal (211) 17: DOI 1.17/s Optimal goodness-of-fit tests for recurrent event data Russell S. Stocker IV Akim Adekpedjou Received: 13 October 29 / Accepted: 14 February 211 / Published online: 5 March 211 Springer Science+Business Media, LLC 211 Abstract A class of tests for the hypothesis that the baseline intensity belongs to a parametric class of intensities is given in the recurrent event setting. Asymptotic properties of a weighted general class of processes that compare the non-parametric versus parametric estimators for the cumulative intensity are presented. These results are given for a sequence of Pitman alternatives. Test statistics are proposed and methods of obtaining critical values are examined. Optimal choices for the weight function are given for a class of chi-squared tests. Based on Khmaladze s transformation we propose distributional free tests. These include the types of Kolmogorov Smirnov and Cramér von Mises. The tests are used to analyze two different data sets. Keywords Goodness-of-fit tests Khmaladze s transformation Counting processes Effective age Martingales Stochastic integration 1 Introduction Recurrent event data is considered a special case of multivariate lifetime data where the events are assumed to be ordered and each event is of the same nature. It is observed in a variety of disciplines and areas such as biomedical, economics, engineering, public health, and the social sciences. Examples of such events include recurrent R. S. Stocker IV (B) Department of Mathematics, Indiana University of Pennsylvania, 21 South Tenth Street, Indiana, PA 1575, USA rstocker@iup.edu A. Adekpedjou Department of Mathematics and Statistics, Missouri University of Science and Technology, 3 W 13th Street, Rolla, MO 6549, USA akima@mst.edu

2 41 R. S. Stocker IV, A. Adekpedjou hospitalization of patients who have a chronic disease, the repeated failure of an electronic component, software crashes, employee absenteeism, and recessions. Therefore, developing statistical analysis methodology for this data is of the utmost importance. In reliability, recurrent events often represent failures in a repairable system. In this context many investigators have considered imperfect repair models. This includes the minimal repair model of Brown and Proschan (1983), that was extended to be age-dependent by Block et al. (1985). General classes of models have been proposed by Dorado et al. (1997), Last and Szekli (1998) and Peña and Hollander (24). Further, Hollander et al. (1992), Peña (21, 27) and Stocker and Peña (27) have presented estimation procedures and asymptotic properties for these general classes of models. We consider the class of imperfect repair models where after each time a unit fails a perfect repair is performed that brings the units back to a good-as-new state. Peña et al. (21) and Stocker and Peña (27) examine these types of models in the non-parametric and parametric settings. They give both estimators and asymptotic properties. For the class of models we consider, practitioners often assume a parametric form for the baseline intensity denoted as λ( ). This assumption plays a vital role when constructing statistical inference procedures. If the assumption is incorrect these procedures may lead to erroneous conclusions. If the assumption is correct the statistical inference procedures may have desirable properties. Therefore, it is important for practitioners to be able to evaluate their parametric assumption. In this paper we construct goodness-of-fit tests of the hypotheses, H : λ( ) C ={λ( ; θ) : θ } versus H a : λ( ) / C ={λ( ; θ) : θ }. The family of intensities could for example be those corresponding to the Weibull, gamma, or lognormal distributions. Presnell et al. (1994) proposed tests of the minimal repair assumption in the imperfect repair model. This assumption will not be of primary interest in this paper. Agustin and Peña (21) proposed goodness-of-fit tests for the Block et al. (1985) model and Agustin and Peña (25) proposed goodness-of-fit tests for an extended Block et al. (1985) model that allow for covariates. They used an intensity based adaptation of the Neyman (1937) smooth goodness-of-fit tests. The tests of Agustin and Peña (21) and Agustin and Peña (25) are extensions of the ideas given in Peña (1998a) and Peña (1998b). The proposed goodness-of-fits of Agustin and Peña (21) and Agustin and Peña (25) are constructed by embedding the baseline intensity function into a larger parametric family of intensity functions using a smooth transformation. This results in a score test and allows the use of counting process theory to obtain asymptotic properties. Hjort (199) considered a general class of weighted processes that compared a weighted difference between the nonparametric and parametric estimators of the cumulative intensity function in the univariate setting. He obtained the asymptotic properties of these processes both under an appropriate null hypothesis and under a Pitman sequence of alternative hypotheses. Using these results he was able to propose several different test statistics. He also obtained choices for the weight function

3 Optimal goodness-of-fit tests for recurrent event data 411 that maximized local asymptotic power for a certain choice of test statistic. We will examine in this paper a similar class of weighted processes that are functions of both calendar and gap time. We obtain asymptotic results for these processes when they are treated as functions of gap time and propose test statistics. We will also obtain weight functions that optimize local asymptotic power for chi-squared tests. Our results will generalize those of Hjort (199) to the recurrent event data setting. Khmaladze (1981) investigated a transformation that led to goodness-of-fit test statistics whose null distributions were independent of the unknown parameters. Andersen et al. (1993), Sun (1997) and Sun et al. (21) utilized the idea of this transformation for models based on counting processes that are commonly utilized in lifetime data analysis situations. In our present situation we will investigate this transformation as it pertains to recurrent event data. Our resulting tests will generalize those for the univariate setting and will be based on a gap time scale approach. 2 Mathematical setting 2.1 General setting Let (, F, P) denote the common probability space on which all random entities are defined. We consider a study with n units. The ith unit is observed over a time period [,τ i, where τ i is a right-censoring random variable. We assume that τ 1,...,τ n are i.i.d. with common distribution function G(s) = P(τ i s). Fortheith unit we observe events at calendar times S i, < S i,1 < S i,2 < Associated with these calendar times are the interoccurrence times, T i,k = S i,k S i,k 1. We assume that the τ i s are noninformative about the T i, j s and they satisfy the independent censoring condition. The interoccurrence times are assumed to be i.i.d. random variables with a common distribution function F(t) = P(T i, j t). The cumulative hazard function is defined as (t) = λ(s)ds where λ(t) = f (t)/(1 F(t)) with the convention that / =. We define θ / θ and use to denote weak convergence. For a matrix A = (a i, j ) we define A 2 as AA t, A = sup i, j a i, j, and A as the generalized inverse of A. We also define the function δ i, j that is 1 if i = j and otherwise. We lastly assume that a set of regularity conditions that are given in Appendix A hold. 2.2 General class of models We describe the class of models using a stochastic process framework. Let F ={F s : s } be a filtration on the probability space (, F, P). Define for calendar time s the counting process N i (s) = j=1 I (S i, j s, S i, j τ i ) and the at-risk process Y i (s) = I (τ i s). The class of models of interest is obtained by postulating a compensator process with respect to the filtration F given as A i (s) = s Y i (v)λ (v S i,n i (v ) ) dv (1)

4 412 R. S. Stocker IV, A. Adekpedjou for i = 1,...,n. The process {M i (s) = N i (s) A i (s) : s } is a squareintegrable martingale with respect to the filtration F. The use of reformulated processes was introduced by Sellke (1988). It has also been employed in the papers of Peña et al. (2, 21, 27), and Stocker and Peña (27). This reformulation is necessary in order to obtain estimators and resulting asymptotic properties in the recurrent event setting. We first define the processes E i (s, t) = I (s S in t), i = 1,...,n, which i (s ) indicate whether at time s at most t time units have elapsed since the last failure. We then define the doubly-indexed processes, and N i (s, t) = s s E i (w, t)n i (dw); A i(s, t) = M i (s, t) = N i (s, t) A i (s, t) E i (w, t)a i (dw); for i = 1,...,n. Also, define the aggregated processes, N(s, t) = n i=1 N i (s, t); A(s, t) = n i=1 A i (s, t) and M(s, t) = n i=1 M i (s, t). By Proposition 1 of Peña et al. (2), A i (s, t) = Y i(s, w)λ(w)dw for i = 1,...,n, where Y i (s, t) = N i ((s τ i ) ) j=1 I (T i, j t) + I ((s τ i ) S i,n i ((s τ i ) ) t). Defining Y (s, t) = n i=1 Y i (s, t) and J(s, t) = I (Y (s, t)>), Peña et al. (21) established in the nonparametric setting that for fixed s, ˆ (s, t) = J(s,w)/Y (s,w) N(s, dw). Peña et al. (21) gave the asymptotic properties of ˆ (s, t), showed the lim s ˆ (s, t) = J(w)/Y (w)dn(w) ˆ (t), and established that n 1 Y (s, t) converges uniformly to a deterministic function Y (s, t). In the parametric setting we obtain a score process U(θ; s, t ) and modified maximum likelihood estimators as shown in Stocker and Peña (27). Asymptotic properties are established in the following theorem where θ is the true value of the parameter and s and t are maximum observation calendar and gap times. Theorem 1 If conditions (A) (E) hold for s R + then, as n, there exists a sequence of solutions { ˆθ n (s, t )} to the sequence of ML estimating equations U n (θ; s, t ) = such that ˆθ n (s, t ) p θ and n( ˆθ n (s, t ) θ ) d N ( ), 1 (θ ; s, t ). Furthermore, n 1 I n ( ˆθ n ; s, t ) is a consistent estimator of (θ ; s, t ), where I n (θ ; s, t ) is the observed Fisher information based on n units. The proof is given in Appendix A.

5 Optimal goodness-of-fit tests for recurrent event data Class of weighted empirical processes The goodness-of-fit tests we propose are based on the class of weighted empirical processes W n (s, t) defined as W n (s, t) = n K n (s,w; ˆθ)[ ˆ (s, dw) (dw; ˆθ(s, t)) where (t; ˆθ(s, t)) = J(s, t) (t; ˆθ(s, t)) and K n (s, t; ˆθ) is a weight process that sometimes can be chosen to optimize local asymptotic power. 3.1 Asymptotic properties For n = 1, 2,..., consider the sequence of hypotheses, versus H : λ( ) C ={λ( ; θ) : θ } H 1n : λ( ) C ={λ( ; θ)(1 + γ/ nφ( ; θ)) : θ }. H 1n represents a Pitman sequence of alternative hypotheses that approaches H as n. For notational convenience we define B n (s, t; θ) K n (s,w; ˆθ)J(s,w) θ log λ(w; θ)λ(w; θ)dw and b(s, t; θ) k(s,w; θ ) θ log λ(w; θ)λ(w; θ)dw. Theorem 2 Under the sequence of alternatives, H 1n, as n, t n K n (s,w; ˆθ)[ ˆ (s, dw) (dw; ˆθ(s, t)) W (s, t) + γ b t (s, t; θ ) 1 (θ ; s, t ) [ t k(s,w; θ )φ(w; θ )λ(w; θ )dw ( t y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw )

6 414 R. S. Stocker IV, A. Adekpedjou in D[, t, where the process, {W (s, t) : t [, t } is a zero mean Gaussian process with covariance function, Cov(W (s, t 1 ), W (s, t 2 )) = t 1 t 2 k 2 (s,w; θ ) λ(w; θ )dw b t (s, t 1 ; θ ) 1 (θ ; s, t )b(s, t 2 ; θ ). (2) A proof of Theorem 2 is given in Appendix C. By setting γ = we obtain the following corollary, Corollary 1 Under the null hypothesis, H, as n, n t K n (s,w; ˆθ)[ ˆ (s, dw) (dw; ˆθ(s, t)) W (s, t) in D[, t, where {W (s, t) : t [, t } is the same zero mean Gaussian process given in Theorem Goodness-of-fit test statistics Goodness-of-fit tests can be constructed based on Corollary 1. For example, a practitioner could use test statistics of the Kolmogorov Smirnov and Cramér von Mises form given respectively as Z (n) 1 (s, t ) = sup Wn (s, t) or Z (n) t [,t 2 (s, t ) = W 2 n (s,w)dw. They are functions of gap time and their null distributions are usually mathematically intractable. The null distribution can be obtained when the null hypothesis is fully specified and under a certain choice for the weight function when the model is dependent on a one-dimensional parameter Fully specified case In the fully specified case, we are testing the hypotheses H : λ( ) = λ ( ) versus H 1 : λ( ) = λ ( ). Using Corollary 1 we have under H that Cov(W (s, t 1 ), W (s, t 2 )) = t 1 t 2 k 2 (s,w) λ (w)dw.

7 Optimal goodness-of-fit tests for recurrent event data 415 Therefore, W n (s, t) converges weakly to a time-transformed Brownian motion process, {B(σ 2 (s, t)), t } where σ 2 (s, t) = Var(W (s, t)). Let ˆσ 2 (s, t ) be a consistent estimator of σ 2 (s, t ). Test statistics can be constructed such as Z (n) 3 (s, t ) = sup t [,t W n(s, t) d sup t [,t B(σ 2 (s, t)) ˆσ 2 (s, t ) σ 2 (s, t ) d = sup B(s, t). t [,1 Critical points can be obtained using well known facts about the distribution of the functional sup t [,1 B(s, t). Other test statistics could also be considered whose asymptotic distribution is equal to that of a functional of a Brownian motion process. Upper quantiles could be obtained either mathematically or through simulation One unknown parameter If one parameter is unknown such as when testing for exponentiality we can for an appropriate choice of the weight function apply a time transformation to obtain a test based on a functional of a Brownian bridge process. By Corollary 1, the process {W n (s, t), t } converges weakly on D[, t, to the zero mean Gaussian process {W (s, t), t } with Cov(W (s, t 1 ), W (s, t 2 )) = t 1 t 2 k 2 (s,w; θ ) λ(w; θ )dw b(s, t 1; θ )b(s, t 2 ; θ ) σ(θ ; s, t. ) Choosing K n (s, t; ˆθ) = n 1 Y (s, t) θ log λ(w; ˆθ) p y(s, t) θ log λ(w; θ ) leads to Cov(W (s, t 1 ), W (s, t 2 )) being equal to [ b(s, b(s, t t1 t 2 ; θ ) ; θ ) b(s, t ; θ ) b(s, t 1 ; θ )b(s, t 2 ; θ ) b(s, t ; θ )b(s, t ; θ ) Therefore the process {W (s, t), t } is a time-transformed Brownian bridge process, {σ(θ ; s, t )B (s,α(t)), t 1} where α(t) = b(s, t; θ )/σ (θ ; s, t ) and {B (s, t), t 1} is a Brownian bridge process. We could then construct properly standardized test statistics that are functionals of a Brownain bridge process. Upper quantile points can then be obtained utilizing well known facts about the Brownian bridge process or via simulation..

8 416 R. S. Stocker IV, A. Adekpedjou Chi-squared tests We can also consider tests of the chi-squared form. They were studied in the univariate setting in Hjort (199) and in Andersen et al. (1993, pp ). To construct these tests we begin by partitioning the gap time scale, [, t,as t < t 1 < < t m = t. Within this partition we have m cells, I i = (m i 1, m i. Consider the processes Q n,i = W n,i (s, t i ) = n I i [ K n (s,w; ˆθ) ˆ (s, dw) (dw; ˆθ(s, t )). Using Corollary 1 we have that the vector Q n with elements Q n,i converges in distribution to G, a zero mean multivariate normally distributed random vector with a covariance matrix that can be calculated by Eq. 2. We observe that is of the form D C t 1 (θ ; s, t )C where the matrix D is a m m diagonal matrix with entries d i, j = δ i, j I i k 2 (s,w; θ ) λ(w; θ )dw and C = (c 1,...,c m ) is a p m matrix where c i = k(s,w; θ ) θ log λ(w; θ )λ(w; θ )dw. I i Let ˆ be a consistent estimator of. The test statistic Xn 2 = Qt n ˆ Q n converges in distribution to a chi-squared distributed random variable, χr 2. The degrees of freedom r, is equal to the rank of. The tests rejects H when Xn 2 >χ2 r,α. If K n (s, t; ˆθ) = 1 then Q n,i = n I i [ ˆ (dw) J(s, w)λ(w; ˆθ)dw. Inthis case, Xn 2 compares the observed minus the expected intensities over the m gap time intervals. If the weight function is K n (s, t; ˆθ) = n 1 Y (s, t) then Q n,i = n 1/2 I i J(s,w)[N(s, dw) Y (s, w)λ(w; ˆθ)dw. Therefore the test statistic Xn 2 compares the observed number of recurrent events to the expected number of events across gap time. Let a = (a 1,...,a p ) t be a p 1 vector of constants. Consider the weight function K n (s, t) = n 1 Y (s, t)a t θ log λ(w; ˆθ). This is a natural extension of the weight function considered in Sect For this choice of weight function, Q n,i = n 1/2 I i J(s,w)a t θ log λ(w; θ)[n(s, dw) λ(w; ˆθ)dw. Therefore by the definition of the MLE, m i=1 Q n,i =. Thus Xn 2 is approximated by a χ 2 distributed random variable with m 1 degrees of freedom. 3.3 Chi-squared optimal tests Hjort (199) studied how to choose the weight function K n (s, t; ˆθ) to optimize local asymptotic power in the univariate setting. We obtain similar results for our tests in the recurrent event setting.

9 Optimal goodness-of-fit tests for recurrent event data Fully specified cased We consider testing the null hypothesis, H : λ( ) = λ ( ) where λ ( ) is fully specified. We begin by considering a chi-square test statistic based on only one cell, (a, b. A natural choice for a test statistic is Z (n) 4 (a, b) = W n(s, b) W n (s, a) ˆσ 2 (s, b) ˆσ 2 (s, a). We would reject the null hypothesis if Z (n) 4 (a, b) >z α/2. The local asymptotic power under the sequence of alternatives given in Eq. 3.1 is P(χ 2 1 (γ 2 τ(a, b) >z 2 α/2 ), where the non-centrality parameter is [ b 2 γ 2 τ(a, b =γ 2 a k(s,w; θ )φ(w; θ )λ(w; θ )dw b a k2 (s,w; θ )/y(s, w)λ(w; θ )dw. This expression is obtained utilizing Theorem 2 and the fact that ˆσ 2 (s, a) σ 2 (s, a). This result is similar to that given by Hjort (199) except in a gap time scale formulation. An application of the Cauchy Schwarz inequality shows that K n (s, t) = n 1 Y (s, t)φ(t; ˆθ) maximizes the local asymptotic power. Observe that this choice is independent of the cell that is chosen. Hjort (199) gives a similar result in the univariate setting. Consider when the number of cells is more than one. The test statistic is Xn 2 = Q t n ˆ Q n given in Sect The local asymptotic power is P(χ 2 m (γ 2 m i=1 τ(t i 1, t i )>χ 2 α,m ). Extending the argument given in the one cell case leads to K n(s, t) = n 1 Y (s, t)φ(t; ˆθ) optimizing the local asymptotic power Parametric case Consider the sequence of hypotheses given in Eq. 3.1 where λ( ; θ) is dependent on an unknown parameter θ.let and d(a, b) = C(a, b) = b a b a k 2 (s,w; θ ) λ(w; θ )dw, k(s,w; θ ) θ log λ(w; θ )λ(w; θ )dw, p b N(a, b) = y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw. a

10 418 R. S. Stocker IV, A. Adekpedjou A natural test statistic based on the interval I = (a, b is Z (n) 5 (a, b) = W n (s, b) W n (s, a) [ ˆd(a, b) Ĉ t (a, b) ˆ 1 (θ ; s, t )Ĉ(a, b) 1/2 where ˆd(a, b) and Ĉ(a, b) are consistent estimators of d(a, b) and C(a, b) respectively. Then by an application of Theorem 2, the local asymptotic power is P(χ 2 1 (γ 2 τ (a, b) >z 2 α/2 ) where τ (a, b [ b a k(s,w; θ )φ(w; θ )λ(w; θ )dw C t (a, b) 1 (θ ; s, t )N(, t ) = d(a, b) C t (a, b) 1 (θ ; s, t )C(a, b) 2. Let c (s, t ) = (a,b) c (a,b) c 1 ( θ log λ(w; θ )) 2 λ(w; θ )dw y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw. Theorem 3 Consider the hypotheses give in Eq The weight function K n (s, t; ˆθ)= n 1 Y (s, t)[φ(t; ˆθ) ( cˆ (s, t)) t θ log λ(t; ˆθ) maximizes the local asymptotic power of the test based on the test statistic Z (n) 5 (a, b). The proof is similar to that of Hjort (199). The choice of the weight function that optimizes the local asymptotic power depends on the interval that is chosen. Now, consider the case where the number of cells, m > 1. An application of Theorem 2 gives an expression for the local asymptotic power given as P ( χm 2 (γ 2 τ )>χα,m 2 ) where τ = e t e. The elements of the vector e are e i = k(s,w; θ )φ(w; θ )λ(w; θ )dw I i c i t 1 (θ ; s, t ) y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw. Since the optimal choice of the weight function for the test based on a single cell depends on the choice of the interval extending the argument to the full test does not

11 Optimal goodness-of-fit tests for recurrent event data 419 follow like in the fully specified case discussed in Sect Over very small intervals we could approximate c (s, t ) with c (s, t ) = 1 (θ ; s, t ) y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw. This leads to a reasonable choice for the weight function being K n (s, t; ˆθ) = Y (s, t) [φ(t; ˆθ) (ĉ n (s, t )) t θ log λ(t; ˆθ). 4 Khmaladze s transformation Chi-square tests require the researcher to make a subjective choice for the partition of the gap time scale. This is often a criticism of these tests and has led researchers to consider alternative methodology. Khmaladze (1981, 1988), and Khmaladze (1993) utilized a transformation approach to deal with the mathematical intractability of the null distribution of tests statistics when q > 1. Andersen et al. (1993) discuss the use of this transformation in the univariate case for lifetime data and Sun (1997) and Sun et al. (21) utilize this approach for multivariate counting process models. In this section we will examine this transformation in the renewal process context. From the proof of Theorem 2 we have that under H, W n (s, t) = k(s,w; θ ) M(s, dw) n b(s, t; θ ) t 1 (θ ; s, t ) k(s,w; θ ) U(s, dw) b(s, t; θ ) t 1 (θ ; s, t ) M(s, dw) θ log λ(w; θ ) + o p (1) n θ log λ(w; θ )U(s, dw) W (s, t). U(s, t) is a zero mean Gaussian process and the weak convergence is on D[, t. The covariance of W (s, t) is given in Eq. 2. The idea is to transform the original test statistic so that the resulting statistic will converge weakly to a zero mean Gaussian process whose covariance function is of the form

12 42 R. S. Stocker IV, A. Adekpedjou t 1 t 2 k 2 (s,w; θ ) λ(w; θ )dw. (3) This will provide a mathematically tractable null distribution. We begin by defining the filtrations, G = σ θ log λ(w; θ )U(s, dw) and G t = σ v k(s,w; θ ) U(s, dw); v t. Using an approach similar to Andersen et al. (1993) we can show that the compensator of W (s, t) with respect to the filtration G Gt is D(s, t; θ ) = D 1 (s, t; θ ) D 2 (s, t; θ ) where and D 1 (s, t; θ ) = k(s,w; θ ) ( θ log λ(w; θ )) T z D 2 (s, t; θ ) = 1 (θ ; s, t )b(s, t; θ ) t z ( θ log λ(w; θ )) 2 y(s, w)λ(w; θ )dw θ log λ(w; θ )dw y(s, w)λ(w; θ )dw 1 θ log λ(w; θ )U(s, dw; θ ). Utilizing counting process and stochastic integration theory we have that V (s, t) = W (s, t; θ ) D(s, t; θ ) is a zero-mean Gaussian process whose quadratic variation process is that given in Eq. 3. In the processes D 1 (s, t; θ ) and D 2 (s, t; θ ) we replace k(s,w; θ ),, and log λ(w; θ ) with K n (s,w; ˆθ),n 1 Y (s,w) and log λ(w; ˆθ) respectively. This results in the processes D 1 (s, t; ˆθ) and D 2 (s, t; ˆθ). We can show by an application of MLE that D(s, t; ˆθ) = D 1 (s, t; ˆθ) D 2 (s, t; ˆθ) = D 1 (s, t; ˆθ). Therefore the empirical process W n (s, t; ˆθ) W n (s, t; ˆθ) D(s, t; ˆθ) = W n (s, t; ˆθ) D 1 (s, t; ˆθ).

13 Optimal goodness-of-fit tests for recurrent event data 421 Define [ nk n (s,w; ˆθ) A n (s, t; ˆθ) = ( θ log λ(w; ˆθ)) T Y (s,w) ( 2 θ log λ(v; ˆθ)) y(s, v)λ(v; ˆθ)dv z y(s, w)λ(w; ˆθ)dw θ log λ(t; ˆθ). 1 Similarly to A n (s, t; ˆθ) we define A(s, t; θ ) where K n (s,w; ˆθ) is replaced by k(s,w; θ ), n 1 Y (s,w)is replaced by and λ(w; ˆθ) is replaced by λ(w; θ ). Theorem 4 Under regularity conditions A C and F G, the sequence of processes {W n (s, t) : t [, t } converges weakly on the Skorohod space D[, t to a zero mean Gaussian process with covariance process given in Eq. 3. The proof of this theorem is similar to that of Theorem 5.1 in Sun (1997) and will be omitted. 4.1 Test statistics Based on our transformed statistic, W (s, t), we can now define test statistics whose asymptotic null distribution can be obtained when q > 1. Let τ 2 (s, t) be a consistent estimator of the covariance process given in Eq. 3. For example this could be ˆτ 1 (s, t) = ( ) t n 1 Kn 2(s,w; ˆθ)/Y (s,w) N(s, dw) or ˆτ 2 (s, t) = Kn 2(s,w) N(s, dw). Y (s,w)(y(s,w) N(s, w)) Let {B(s,τ 2 (s, t)), t } be a time transformed Brownian bridge process. A test based on a Kolmogorov Smirnov test statistic can be constructed by noting that T K = sup t t W n (s, t) ˆτ(s, t ) d sup t t B(s,τ2 (s, t)) τ(s, t ) d = sup B(s, t) (4) t 1 where B(s, t) is a Brownian motion on [, 1. A test would then reject the null hypothesis when T k >w α where w α is the upper percentile of sup t 1 B(s, t). A Cramér von Mises test can be constructed by noting that T C = (W n (s,w))2 K 2 n (s,w, ˆθ)dw ˆτ 2 (s, t ) 1 d B 2 (s,w)dw.

14 422 R. S. Stocker IV, A. Adekpedjou The test rejects if T C > c α, where c α is the upper percentile of the distribution of 1 B2 (s,w)dw. 5 Application to real data We now apply the chi-square tests to the data sets of Proschan (1963) and Kumar and Klefsjö (1992). As described in Proschan (1963), the times of successive failures were kept for the air condition system of a fleet of Boeing 72 jet airplanes. Proschan (1963) used this data set in helping to explain an observed decreasing failure rate. Kumar and Klefsjö (1992) analyzed the failure times of the hydraulic subsystems of six load-haul-dump (LHD) machines. These machines as described in Kumar and Klefsjö (1992) are used to pick up ore or waste rock from the mining points and for dumping it into trucks or ore passes. For both data sets we assumed a Weibull form for the hazard function given as λ(s; θ 1,θ 2 ) = (θ 2 /θ 1 )(s/θ 1 ) θ 2 1. For the failure time data of the air conditioning systems given in Proschan (1963) we obtained that ˆθ 1 = and ˆθ 2 = When K n (s, t) = 1 and K n (s, t) = Y (s,w)/n we obtained test statistics of and Both of these test statistics result in p values that are less than.1. These results are confirmed when considering Fig. 1. Λ^(t) Fig. 1 Graphs of the parametric and non-parametric estimators for the cumulative hazard function of the data given by Proschan (1963) t

15 Optimal goodness-of-fit tests for recurrent event data 423 Λ^(t) t Fig. 2 Graphs of the parametric and non-parametric estimators for the cumulative hazard function of the data given by Kumar and Klefsjö (1992) For the failure time data of the LHD machines given in Kumar and Klefsjö (1992) we obtained that ˆθ 1 = and ˆθ 2 = When K n (s, t) = 1 and K n (s, t) = Y (s,w)/n we obtained test statistics of and The corresponding p values are.38 and.78. Therefore we would assert that a Weibull hazard is not the correct choice for λ( ). These results are confirmed when considering Fig. 2. Acknowledgements We would like to thank the editors and referees for their constructive criticism. Their comments and suggestions helped to improve this manuscript. Appendix A: Regularity Conditions (A) There exists a neighborhood of θ such that for all t [, t the first, second, and third order partial derivatives with respect to θ of λ(t; θ)and log λ(t; θ)exist and are continuous in θ. The log-likelihood equation may be differentiated three times with respect to θ and the order of integration and differentiation may be interchanged. (B) There exists finite functions σ jk (θ; s, t ) defined on, a neighborhood of θ, such that for all j, k, as n, 1 n n i=1 { }{ } Y i (s,w) log λ(w; θ ) log λ(w; θ ) λ(w; θ )dw. θ j θ k converges in probability to σ jk (θ ; s, t ). (C) The q q matrix (θ ; s, t ) with entries σ jk (θ ; s, t ) is positive definite.

16 424 R. S. Stocker IV, A. Adekpedjou (D) For all j, k, asn, 1 n n i=1 { 2 } 2 Y i (s,w) log λ(w; θ ) λ(w; θ )dw θ j θ k converges in probability to a finite quantity. (E) There exists predictable processes G 1 (t) and G 2 (t) and a neighborhood of θ such that for all t [, t and for all j, k, l, sup 3 λ(t; θ) θ j θ k θ G 1(t) and sup 3 log λ(t; θ) l θ j θ k θ G 2(t). l θ θ Moreover, n 1 n i=1 G 1(w)Y i (s,w)dw and n 1 n i=1 G 2(w)Y i (s, w)λ(w; θ )dw both converge in probability to finite quantities as n.also,foranyɛ>, as n, 1 n n i=1 Y i (s,w)g 2 (w)i { } n 1 G 2 (w) >ɛ λ(w; θ )dw p. (F) K n (s, t; ˆθ)is F-predictable for each t with total variation bounded by a constant independent of n. Furthermore, there exists a function k(s, t; θ ) so that p sup K n (s, t; ˆθ) k(s, t; θ ) (s,t) [,s [,t as n. (G) A n (s, t; ˆθ) is left continuous with total variation bounded by a constant independent of n. Furthermore, sup (s,t) [,s [,t A p n(s, t; ˆθ) A(s, t; θ ) as n. Appendix B: Proof of Theorem 1 Proof For notational convenience we set ˆθ = ˆθ(s, t ). Let R n (θ; s, t) = θ t I n (θ; s, t). By an application of Taylor s Theorem, U n (θ; s, t ) = U n (θ ; s, t ) (θ θ ) t I n (θ; s, t ) 1 2 (θ θ ) t R n (θ ; s, t )(θ θ ) where θ lies in the line segment joining θ and θ. To establish the existence of a sequence of consistent solutions to U n (θ; s, t ) = it suffices to show that as n, (I) n 1 U n (θ ; s, t ) = O p (n 1/2 );

17 Optimal goodness-of-fit tests for recurrent event data 425 (II) n 1 I n (θ; s, t ) p (θ ; s, t ); and (III) there exists a constant B not depending on θ such that for every θ, { } lim P 1 n n R n(θ; s, t ) B 1. Under regularity conditions (A) and (C) and using Proposition 2 of Peña et al. (21) we can apply Theorem 1 of Peña et al. (2) to establish that as n,the process {n 1/2 U n (θ ; s, t) : t [, t } converges weakly on D[, t to a zero mean Gaussian process. This verifies (I). To verify (II) we observe that 1 n I n(θ ; s, t ) = 1 n Y (s,w)( θ log λ(w; θ )) 2 λ(w; θ )dw [ 2 θ θ t log λ(w; θ ) M(s, dw; θ ). By condition (D) and Theorem 1 of Peña et al. (2) the second term is O p (n 1/2 ). By condition (B) the first term converges in probability to (θ ; s, t ). An application of Slutsky s Theorem establishes (II). To prove (III), by condition (E) we observe that sup 1 n R n(θ; s, t ) 1 n θ = 1 n + 1 n G 2 (w)n(s, dw) + 1 n G 2 (w)m(s, dw; θ )+ 1 n G 1 (w)y (s,w)dw G 1 (w)y (s,w)dw G 2 (w)y (s, w)λ(w; θ )dw By condition (E), the last two terms are O p (1). By Theorem 1 of Peña et al. (2) and condition (E) the first term is O p (n 1/2 ). Therefore sup θ n 1 R(θ; s, t ) is O p (1). This verifies (III). To prove the asymptotic normality result we have by an application of Taylor s Theorem that, = 1 n U n ( ˆθ; s, t ) = 1 n U n (θ ; s, t ) 1 n I(θ ; s, t ) n( ˆθ θ )

18 426 R. S. Stocker IV, A. Adekpedjou where θ lies in the line segment joining ˆθ and θ. By another application of Taylor s Theorem, 1 n I n(θ ; s, t ) = 1 n I n(θ; s, t ) 1 n R( ˇθ; s, t )(θ θ ) (5) where ˇθ lies in the line segment joining θ and θ.asn,eq.5 is (θ ; s, t ) + o p (1). The event { (θ ; s, t ) + o p (1) is invertible} converges in probability to 1 as n. Thus, n( ˆθ θ ) = 1 (θ ; s, t ) 1 n U(θ ; s, t ) + o p (1). An application of Theorem 1 of Peña et al. (2) establishes the desired result. Lastly, we observe that n 1 I n ( ˆθ n ; s, t ) is a consistent estimator of (θ ; s, t ) since Eq. 5 is (θ ; s, t ) + o p (1). Appendix C: Proof of Theorem 2 Proof In the proof of Theorem 2 we will establish the following: (I) The weighted empirical class of process can be expressed as, W n (s, t) = n 1/2 +γ K n (s,w; ˆθ)J(s,w) n 1 M(s, dw; θ ) Y (s,w) K n (s,w; ˆθ)J(s, w)φ(w; θ )λ(w; θ )dw B t n (s, t; θ ) n( ˆθ θ ) + o p ( n). (II) The standard score process can be expressed as, n 1/2 U n (θ; s, t ) = n 1/2 θ log λ(w; θ)m(s, dw; θ) +γ n 1 Y (s, w)φ(w; θ) θ log λ(w; θ)λ(w; θ)dw.

19 Optimal goodness-of-fit tests for recurrent event data 427 (III) n 1 I n (θ, s, t ) p (θ ; s, t ) where the entries of (θ ; s, t ) are given as y(s,w)( θ log λ(w; θ )) 2 λ(w; θ )dw. Results (I) (III) along with an application of Theorem 1 given in Peña et al. (2) establishes the result. To verify (I) we observe that W n (s, t) = n = n +γ n K n (s,w; ˆθ)[ ˆ (s, dw) (dw; ˆθ) K n (s,w; ˆθ)J(s,w) M(s, dw; θ ) Y (s,w) K n (s,w; ˆθ)J(s, w)φ(w; θ )λ(w; θ )dw K n (s,w; ˆθ)J(s, w)(λ(w; ˆθ) λ(w; θ ))dw. (6) Observing that λ(w; ˆθ) λ(w; θ ) = ( θ λ(w; θ )) t ( ˆθ θ ) + o p (1) establishes (I). We establish (II) by observing that, U n (θ; s, t ) = θ log λ(w; θ)[n(s, dw) Y (s, w)λ(w; θ)dw = θ log λ(w; θ)[m(s, dw; θ) + Y (s, w)λ(w; θ)(1 + γ/ nφ(w; θ))dw Y (s, w)λ(w; θ)dw = θ log λ(w; θ)m(s, dw; θ) +γ/ n Y (s, w)φ(w; θ) θ log λ(w; θ)λ(w; θ)dw. Multiplying by n 1/2 establishes (II).

20 428 R. S. Stocker IV, A. Adekpedjou To verify (III) we begin by observing that the information is I n (θ; s, t ) = = = [ 2 Y (s,w) λ(w; θ) dw θ θt [ 2 Y (s,w) λ(w; θ) dw θ θt γ/ n + +γ/ n [ 2 log λ(w; θ) θ θt N(s, dw) [ 2 log λ(w; θ) M(s, dw; θ) θ θt [ 2 Y (s,w) log λ(w; θ) λ(w; θ)(1 + γ/ nφ(w; θ))dw θ θt [ 2 log λ(w; θ) θ θt M(s, dw; θ) [ 2 Y (s, w)φ(w; θ) λ(w; θ) dw θ θt Y (s,w)( θ log λ(w; θ)) 2 λ(w; θ)dw Y (s, w)φ(w; θ)( θ log λ(w; θ)) 2 λ(w; θ)dw. By Theorem 1 of Peña et al. (2), the first term is O p ( n) at θ = θ. The second and fourth terms are o p (1). Therefore 1 n I n(θ ; s, t ) p (θ ; s, t ). Thus (III) has been established. Now by an application of Taylor s Theorem we have that, 1 n U n ( ˆθ; s, t ) = 1 n U n (θ ; s, t ) 1 n I n(θ ; s, t ) n( ˆθ θ ), where θ lies on the line segment joining ˆθ and θ. Therefore we have that

21 Optimal goodness-of-fit tests for recurrent event data 429 n( ˆθ θ ) = 1 (θ ; s, t ) 1 θ log λ(w; θ )M(s, dw; θ ) n + γ y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw + o p (1). Using Eq. 6 we have W n (s, t) = 1 n +γ t K n (s,w; ˆθ)J(s,w) n 1 M(s, dw; θ ) Y (s,w) K n (s,w; ˆθ)J(s, w)φ(w; θ )λ(w; θ )dw B t n (s, t; θ ) 1 (θ ; s, t ) 1 θ log λ(w; θ )M(s, dw; θ ) n + γ = 1 n n 1 Y (s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw + o p (1) ( K n (s,w; ˆθ)J(s,w)I (w t) n 1 Y (s,w) ) B t n (s, t; θ ) 1 (θ ; s, t ) θ log λ(w; θ ) M(s, dw; θ ) +γ K n (s,w; ˆθ)J(s, w)φ(w; θ )λ(w; θ )dw B t n (s, t; θ ) 1 (θ ; s, t ) n 1 Y (s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw +o p (1) We recall that J(s, t) p 1 and we assumed that K n (s, t; ˆθ) p k(s, t; θ ) and n 1 Y (s, t) p y(s, t). An application of Theorem 1 of Peña et al. (2) establishes that the process {W n (s, t), t [, t } converges weakly on D[, t to a Gaussian process whose mean process is

22 43 R. S. Stocker IV, A. Adekpedjou μ(s, t) = γ k(s,w; θ )φ(w; θ )λ(w; θ )dw b t (s, t; θ) 1 (θ ; s, t ) y(s, w)φ(w; θ ) θ log λ(w; θ )λ(w; θ )dw. The covariance function is Cov(W (s, t 1 ), W (s, t 2 )) = k(s,w; θ )I (w t 1 ) b t (s, t 1 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ ) [ k(s,w; θ )I (w t 2 ) b t (s, t 2 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ ) y(s, w)λ(w; θ )dw = t 1 t 2 k 2 (s,w; θ ) λ(w; θ )dw k(s,w; θ )b t (s, t 2 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ )λ(w; θ )dw k(s,w; θ )b t (s, t 1 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ )λ(w; θ )dw b t (s, t 1 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ ) b t (s, t 2 ; θ ) 1 (θ ; s, t ) θ log λ(w; θ )y(s, w)λ(w; θ )dw. The second term is 1 k(s,w; θ ) θ log λ(w; θ )λ(w; θ )dw 1 (θ ; s, t )b(s, t 2 ; θ ). This is simply b t (s, t 1 ; θ ) 1 (θ ; s, t )b(s, t 2 ; θ ). Similarly we can establish that the third term is b t (s, t 1 ; θ ) 1 (θ ; s, t )b(s, t 2 ; θ ). The last term can be expressed as b t (s, t 1 ; θ ) 1 (θ ; s, t ) (θ ; s, t ) 1 (θ ; s, t )b(s, t 2 ; θ )

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