Working Paper 4-2 Statistics and Econometrics Series (4) July 24 Departamento de Estadística Universidad Carlos III de Madrid Calle Madrid, 26 2893 Getafe (Spain) Fax (34) 9 624-98-49 GOODNESS-OF-FIT TEST FOR RANDOMLY CENSORED DATA BASED ON MAXIMUM CORRELATION Ewa Strzalkowska-Kominiak and Aurea Grané () Abstract In this paper we study the goodness-of-fit test introduced by Fortiana and Grané (23) and Grané (22), in the context of randomly censored data. We construct a new test statistic under general right-censoring, i.e., with unknown censoring distribution, and prove its asymptotic properties. Additionally, we study a special case, when the censoring mechanism follows the well-known Koziol-Green model. We present an extensive simulation study on the empirical power of these two versions of the test statistic. We show the good performance of these statistics in detecting symmetrical alternatives and their advantages over the most famous Pearson-type test proposed by Akritas (988). Finally, we apply our test to the head-and-neck cancer data. Keywords: Goodness-of-fit; Kaplan-Meier estimator; Maximum correlation; Random censoring. AMS classification: 62E2, 62G, 62G2, 62N86. () A. Grané, Statistics Department, Universidad Carlos III de Madrid, C/ Madrid 26, 2893 Getafe (Madrid), Spain, e-mail: aurea.grane@uc3m.es, corresponding author. E. Strzalkowska-Kominiak, Statistics Department, Universidad Carlos III de Madrid, C/ Madrid 26, 2893 Getafe (Madrid), Spain, e-mail: ewa.strzalkowska@uc3m.es This work has been partially supported by Spanish grants MTM2-7323 (Spanish Ministry of Science and Innovation), MTM2-22392, ECO2-2576 (Spanish Ministry of Economy and Competitiveness).
Goodness-of-fi test for randomly censored data based on maximum correlation Ewa Strzalkowska-Kominiak Aurea Grané Statistics Department, Universidad Carlos III de Madrid. Abstract In this paper we study the goodness-of-fi test introduced by Fortiana and Grané (23) and Grané (22), in the context of randomly censored data. We construct a new test statistic under general right-censoring, i.e., with unknown censoring distribution, and prove its asymptotic properties. Additionally, we study a special case, when the censoring mechanism follows the well-known Koziol-Green model. We present an extensive simulation study on the empirical power of these two versions of the test statistic. We show the good performance of these statistics in detecting symmetrical alternatives and their advantages over the most famous Pearson-type test proposed by Akritas (988). Finally, we apply our test to the head-and-neck cancer data. Keywords: Goodness-of-fit Kaplan-Meier estimator; Maximum correlation; Random censoring. Introduction In many medical studies one encounters data which are not fully observed and so censored from the right. Let Y,..., Y n be the lifetimes of interest, coming from the continuous distribution function F and C,..., C n the censoring times from the distribution function G. In the context of right-censored data, for every i =,..., n, we observe X i = min(y i, C i ) and δ i = {Yi C i }, A denotes the indicator function being equal to if A is fulfille and otherwise. Even though the unknown distribution function F can be estimated by a well-known product-limit estimator, introduced by Kaplan and Meier (958), very often we may have sufficien information to know the shape Statistics Department, Universidad Carlos III de Madrid, C/ Madrid 26, 2893 Getafe, Spain. E-mails: E. Strzalkowska-Kominiak, ewa.strzalkowska@uc3m.es, A. Grané, aurea.grane@uc3m.es (corresponding author).
of the distribution in question. Using such fully parametric models can lead to substantial gain in the efficien y of statistical procedures involving the distribution of the lifetimes F if only the parametric choice of that distribution is correct. This makes the goodness-of-fi test an important statistical tool, when dealing with (right-)censored data. It is clear, however, that the most famous tests for complete data, as Kolmogorov-Smirnov or Cramer-von Mises, are difficul to apply it the presence of censoring since the limit distribution depends on the censoring distribution G. The most recent overview of this kind of tests with randomly censored data is given by Balakrishnan et al (24). Some more classical approaches are to fin in Koziol and Green (976) and Akritas (988). While the firs one is based on the assumption that the distribution function G follows the so called Koziol-Green model and hence is more restrictive, the second one is a χ 2 test applied to general random censoring. For this reason, the Pearson-type goodness-of-fi test proposed by Akritas (988) is so far the best option for randomly censored data with unknown censoring distribution. Nevertheless, it requires a partition of the observations to the cells and an adequate choice of number of classes since the power of the test may vary depending on the degrees of freedom. In this work we propose a new goodness-of-fi test based on maximum correlation coefficient which has a normal limiting distribution and hence it is straightforward to apply. For this, we firs introduce the maximum correlation in a more general set-up. Let Y and Y 2 be two random variables with finit second order moments, joint cumulative distribution function (cdf) H and marginals F and F 2, respectively. Recall, that the Hoeffding representation of the correlation coefficien is given by ρ(f, F 2 ) = (H(x, y) F (x)f 2 (y))dxdy, σ σ 2 R 2 σ i denotes the standard deviation of Y i. Furthermore, the maximum correlation of the pair of random variables (Y, Y 2 ) is define as the correlation coefficien ρ + (F, F 2 ) corresponding to the bivariate cdf H + (x, y) = min(f (x), F 2 (y)), the upper Fréchet bound of H(x, y). The cdf H + (x, y) is a singular distribution, having support on the one-dimensional set {(x, y) R 2 : F (x) = F 2 (y)}, and the maximum correlation coefficien is given by ρ + (F, F 2 ) = ( F (p)f2 (p)dp µ µ 2 ), () σ σ 2 F i is the inverse of F i and µ i is the the mean of Y i. This maximum correlation, ρ + (F, F 2 ), is a measure of agreement between F and F 2, since ρ + = if and only if F = F 2 up to a scale and location change. In particular, Cuadras and Fortiana (993) proposed the statistic based on ρ + (F, F ) as a measure of goodness of fi of an iid sample Y,..., Y n with cdf F = F, to a given distribution F. The goodness-of-fi test based on maximum correlation was further studied by Fortiana and Grané (23) and Grané (22). As in the two latter publications, the present paper is devoted to testing uniformity, i.e. F = F U, a [, ] uniform distribution. As shown by Fortiana and Grané (23) the asymptotic approximation of 2
ρ + (F, F U ) is available, but convergence to its limiting law is rather slow. This led to definin Q = which equals one is F = F U. σ /2 ρ + (F, F U ) = 6 x(2f (x) )F (dx), (2) The goal of this paper is to study a test statistic based on Q when Y,..., Y n may not be fully observed and so censored from the right by censoring times C,..., C n. More precisely, we wish to test the hypothesis H : F = F U, F U is the cdf of a [, ] uniform random variable, based on the sample (X i, δ i ) i=,...,n, X i = min(y i, C i ), with X i [, ]. Nevertheless, our approach is not restricted to testing uniformity. We can also consider a more general null hypothesis F since, under H : F = F, we have that the transformed random variable F (Y ) follows a [, ] uniform distribution. That is, Ỹ = F (Y ) F U under the null hypothesis. Setting C = F (C) and since {Ỹi C i } = {Y i C i }, leads us to testing uniformity based on the iid sample ( X, δ ),..., ( X n, δ n ), X i = min(ỹi, C i ) and δ i = { Ỹ i C i }. Hence, testing for uniformity is equivalent to testing for a fully specifie continuous distribution. Even though it seems that we can extend the work of Fortiana and Grané (23) setting Q n = 6 x(2f n(x) )F n (dx), F n denotes the Kaplan-Meier estimator for censored data, it is far from being true. In contrast to the empirical distribution under completely observed data, the Kaplan-Meier estimator is biased (see Stute (994), for details). In Section 2 we show that such Plug- In estimator suffers from bias of this product-limit estimator and hence E(Q n ) = under H does not hold. To avoid this problem we propose to re-write Q in such a way, that it can be estimated by degenerated U-statistics. This leads to significan bias (and variance) reduction. In Section 3 we prove the asymptotic normality of the proposed estimator and in Section 4 we present our new goodness-of-fi test. In Section 5 we present an extensive simulation study. Finally, in Section 6 we adapt the test statistic to the case of composite null hypothesis and apply our test to the head-and-neck cancer data from Haghighi and Nikulin (24). 2 Test statistic In this section we propose our new statistic to test the goodness of fi for randomly censored data based on the modifie maximum correlation coefficient Recall that, under H : F = F U, the quantity Q = σ /2 ρ + (F, F U ) = 6 equals one. Hence in the following we prefer to work with Q = Q = 6 x(2f (x) )F (dx) x(2f (x) )F (dx) (3) 3
which equals zero if H is true. First, we defin a Plug-In estimator of Q by replacing F in (3) with the well-known Kaplan-Meier estimator. We obtain F n is define as follows Q n = 6 F n (x) = X i x x(2f n (x) )F n (dx), (4) [ ] δ i n k=. (5) {X k X i } It turns out, that under the null hypothesis and for finit samples the Plug-In estimator Q n suffers from significan bias and its convergence to the limiting distribution is very slow. To solve this problem, we propose to estimate Q with a U-statistic. For this, note that, if F is a continuous cdf and supp(f ) [, ], then Hence Q = = 2 (6x(2F (x) ) 2F (x))f (dx) = F (x)f (dx) =. [(6x 2)F (x) 6x( F (x))]f (dx) [(6x 2) {y x} 6x {y>x} ]F (dx)f (dy). (6) Now we may replace the unknown quantities by their estimators. For this we introduce the jumps of the Kaplan-Meier estimator by setting w in = F n (X i ) F n (X i ), F n (x ) is the left-continuous version of F n (x). Finally, the estimator of Q is given by Q n = w in w jn h(x i, X j ), (7) i= j i h(x, x 2 ) = (6x 2) {x2 x } 6x {x2 >x }. To illustrate the advantages of using Q n over the Plug-In estimator Q n, in panel (a) of Figure, we present the bias and variance of those estimators under the null hypothesis, that is, when the data comes from the [, ] uniform distribution. Additionally, in panels (b)-(c) of Figure, we compare the kernel density estimators of the standardized versions of Q n and Q n to that of the standard normal distribution. The standardization is done using the estimated asymptotic variances, discussed later on. Clearly, the U-statistic, Q n, exhibits much smaller bias (and variance) and its standardized version fit nicely the standard normal distribution for all the considered censoring rates. 4
Figure : Comparison between Q n and the Plug-In estimator Q n: Estimated bias (variance) based on 5 trials and kkernel densities for n = 2. (b) Kernel density of standardized Q n (a) Estimated bias (variance) % censoring Q n Q n n = 5.338 (.8) -. (.47) n =.5 (.32) -.4 (.22) 2% censoring Q n Q n n = 5. (.97) -.69 (.63) n =.47 (.74) -.26 (.26) 3% censoring Q n Q n n = 5 -.26 (.483) -.22 (.69) n = -.275 (.24) -.9 (.33).8.7.6.5.4.3.2. Normal % 2 % 3 % 5 4 3 2 2 3 4 5 (c) Kernel density of standardized Q n.8.7.6.5.4.3 Normal % 2 % 3 %.2. 5 4 3 2 2 3 4 3 Asymptotic properties In this section we study the asymptotic properties of our test statistic Q n. First we consider Q n under general censoring mechanism, that is, without assuming any shape of the distribution function G(x) = P (C x), which is the cdf of the censoring times. Then we apply the results to the special case of Koziol-Green model. Recall that F (x) = P (Y x) is the cdf of the lifetimes of interest. We need following assumptions: A : A2 : F (du) G(u) < ϕ(u) C /2 (u)f (du) < ϕ is a score function, C(x) = x Theorem. Under A and A2, we have n( Qn Q ) N (, σ 2 ), σ 2 = [ ϕ 2 (x) 2 G(x) F (dx) ϕ(x)f (dx)] G(dy) and F is continuous with support in [, ]. ( G(y)) 2 ( F (y)) 5 [ x ] 2 ( F (x))g(dx) ϕ(y)f (dy) ( H(x)) 2
and ϕ(x) = 2xF (x) 6x 2 2 x yf (dy) + 6 yf (dy). Proof. In view of (7), we can write Q n in the following way Q n = h(x, y)f n (dx)f n (dy), (8) h(x, y) = (6x 2) {y<x} 6x {y>x}. In the fis step of the proof we write Q n as a sum of four terms as follows Q n = Q + Q 2n + Q 3n + Q 4n, Q = Q 2n = Q 3n = Q 4n = h(x, y)f (dx)f (dy) h(x, y)(f n (dx) F (dx))f (dy) h(x, y)(f n (dy) F (dy))f (dx) h(x, y)(f n (dx) F (dx))(f n (dy) F (dy)). By (6) and since F is continuous, we have that Q = Q. As to Q 2n + Q 3n, we obtain ϕ(x) = Q 2n + Q 3n = h(x, y)f (dx) + = 2xF (x) 6x 2 2 ϕ(x)(f n (dx) F (dx)), x h(x, y)f (dy) yf (dy) + 6 yf (dy). It remains to show that Q 4n = o P (n /2 ). For this, set τ H = inf{t : H(t) = }, H(t) = P(X t) is the cdf of the observed sample. Since supp(f ) [, ] and G fulfill assumption A, we have that τ H =. Moreover, by definitio of h(x, y), we can show that Q 4n = 2 x(f n (x) F (x))(f n (dy) F (dy)) 2(F n () F ()) 2 =: Q a 4n + Q b 4n. 6
Now, we may consider the two terms, Q a 4n and Q b 4n, separately. According to Theorem 2 (7) in Ying (989) and under A, the process n(f n F ) converges weakly to a Brownian process. See, also equation () in Wellner (27). More precisely, n(fn F ) ( F )B(C), in D[, τ H], D[, τ H] denotes the Skorohod space. Furthermore, since F is continuous and D is a set of uniformly bounded functions, we have that n(f n F ) D with probability exceeding ε for every ε >. Additionally, x [, ] and sup x [,τ H] F n (x) F (x) almost surely. Hence, using Theorem 2. in Rao (962) with g(x) = n(f n (x) F (x))x, we obtain n Qa 4n = 2 g(x)(f n (dy) F (dy)) = o P (). Additionally, under A, n Q b 4n = o P (). Finally, we obtain the following representation Q n = Q + ϕ(x)(f n (dx) F (dx)) + o P (n /2 ). The asymptotic normality is now a direct consequence of Stute (995). More precisely, under A and A2, we obtain This completes the proof. Consequently, we have n ϕ(x)(f n (dx) F (dx)) N (, σ 2 ). Corollary. Under H, A and A2, we have n Qn N (, σ 2 ). It is to see, that the variance under H would not simplify since it does depend on the distribution function of the censoring times G, which is unknown. Nevertheless, under the Koziol-Green model, we have an explicit expression for σ 2. First, recall that G follows a Koziol-Green model if G(x) = ( F (x)) β, β is an unknown parameter. We can see, however, that p := P (Y > C) = β β + and p = ( G(x))F (dx). Hence β can be easily estimated using Kaplan-Meier estimators for F and G. Finally, it is easy to check that the assumptions A and A2 are fulfille under the Koziol-Green model with β (, ), that is, if the censoring is not heavier than 5%, which is a very reasonable assumption. So, as a consequence of Corollary, we get the following result. 7
Corollary 2. Under Koziol-Green model with β (, ) we have that, under H, n Qn N (, σ 2 ), σ 2 = β 4 + 4β 3 7β 2 + 38β 24 (β )(β 2)(β 3)(β 4)(β 5). 4 Goodness-of-fi test Once the test statistic is proposed and its limiting distribution is established, we are in the position to defin the goodness-of-fi test. For this we estimate the asymptotic variance σ 2 using the Plug- In principle. That is, we replace the unknown quantities with its estimators. First, we defin distribution function of the observed times H(x) = P (X x) and set H n (x) = n n i= {X i x} as its empirical counterpart. Moreover, let and H (x) = P(X x, δ = ) = H (x) = P(X x, δ = ) = x x ( F (u))g(du) ( G(u))F (du) be the subdistributions related with the observed censored and uncensored lifetimes. Their estimators are define as follows and Hence σ 2 n = n n i= H n(x) = n {Xi x}( δ i ) i= H n(x) = n {Xi x}δ i. i= [ ϕ 2 n(x i ) ( G n (X i )) δ 2 i n i= δ i ( H n (X i )) 2 ϕ n (x) = 2xF n (x) 6x 2 2 n and G n is a Kaplan-Meier estimator given by i= G n (x) = X i x [ n j= i= ] 2 ϕ n (X i ) G n (X i ) δ i ϕ n (X j ) G n (X j ) δ j {Xj X i } X i δ i G n (X i ) {X i x} + 6 n [ ] δ i n k=. {X k X i } i= ] 2, X i δ i G n (X i ). the 8
Finally, we set T n := n Qn σ 2 n. (9) We reject H at level α if T n Φ (α/2) or T n Φ ( α/2), Φ is the inverse of the standard normal cdf. Additionally, under the Koziol-Green model and in view of Corollary 2, we defin T n := n Qn ˆσ 2, () and ˆσ 2 = ˆβ 4 + 4 ˆβ 3 7 ˆβ 2 + 38 ˆβ 24 ( ˆβ )( ˆβ 2)( ˆβ 3)( ˆβ 4)( ˆβ 5) ( ˆβ = ( G n (x))f n (dx)). As before we reject H at level α if T n Φ (α/2) or T n Φ ( α/2). 5 Simulation study Here we conduct an extensive simulation study to show the behavior of our test. In the following subsection we consider only the null hypothesis, while in Subsection 5.2 we include the power study under different alternatives. In both subsections we compare our method with the Pearsontype goodness-of-fi test proposed by Akritas (988). Following the notation of the previous section, we denote by T n and Tn our test statistics for the general censoring and under the Koziol-Green model, respectively. See, equations (9) and () for details. Moreover, we denote by A (nc) the χ 2 test proposed by Akritas (988), nc denotes the number of cells. 5. Null hypothesis In this section we present the results of the proposed methods under the null hypothesis and at 5% significanc level. As mentioned before, we consider our tests T n and Tn, together with the test presented by Akritas (988). Following the latter work, we consider nc = 2 and nc = 5 and denote these tests by A (2) and A (5), respectively. The results are based on 5 trials. As it is to see in the Table, our test T n and the one from Akritas hold very well the significanc level. As to the one based on Koziol-Green model, it holds the 5% level when censoring is low but for more than 2% of missing data, the variance σ 2 does not captures the variability of our Q n correctly and hence the significanc level is slightly overestimated for heavy censoring. 9
Table : Null Hypothesis % censoring 2% censoring 3% censoring T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn n = 5.58.552.538.578.45.56.558.53.496.596.56.648 n =.48.582.548.532.478.53.492.56.52.568.528.78 n = 2.468.524.574.498.522.58.58.64.494.522.468.666 5.2 Power study To study the power of our test we consider two different models: Model : To test the uniformity (H : F = F U ) we choose three parametric families of alternative probability distributions with support on [, ]. (a) Lehmann alternatives, F θ (x) = x θ, x, θ > ; for θ = we have F θ = F U. (b) compressed uniform alternatives, F θ (x) = x θ 2θ, θ x θ, θ /2; and for θ = we have F θ = F U. (c) centered distributions having a U-shaped density for θ (, ), or wedge-shaped density for θ > F θ (x) = for θ = we have F θ = F U. { 2 (2x)θ, x /2 (2( 2 x))θ, /2 x Model 2: An exponentiality test (with parameter λ = ), the alternatives are Weibull distributions with parameters and θ. More precisely, F θ (x) = e xθ, θ = gives us the exponential distribution of the null hypothesis. Additionally, the censoring variable, C, is generated under the Koziol-Green model. That is, G(x) = ( F (x)) β, β = p and p = P(X > C) is the censoring level. p In the following figure and tables we present the power study at a 5% significanc level. Panels (a)- (c3) of Figure 2 contain the power of the test for Model and panels (d)-(d3) of Figure 2 contains the power under Model 2, for different sample sizes (n = 5,, 2) and one censoring level of 2%. All those figure are based on 2 trials. Moreover, Tables 2 are based on 5 trials and show the power under alternatives for two different sample sizes (n =, 2), censoring levels p =.,.2,.3 and different values of the parameters θ. In particular, for Model 2, we choose
θ = + Hn.5 and H { 4, 2, 2, 4}. Both, tables and figures include a comparison to the Pearson-type test proposed by Akritas (988). As before, we use the number of cells (nc) equal to 2 and 5..9.9.9.8.8.8.7.7.7.6.6.6.5.5.5.4.4.4.3.3.3.2.2.2....5 2 2.5.5 2 2.5.5 2 2.5 (a) Model a), n = 5 (a2) Model a), n = (a3) Model a), n = 2.9.9.9.8.8.8.7.7.7.6.6.6.5.5.5.4.4.4.3.3.3.2.2.2....2.4.6.8..2.4.6.8.2.2.4.6.8..2.4.6.8.2.2.4.6.8..2.4.6.8.2 (b) Model b), n = 5 (b2) Model b), n = (b3) Model b), n = 2.9.9.9.8.8.8.7.7.7.6.6.6.5.5.5.4.4.4.3.3.3.2.2.2....5.5 2.5.5 2.5.5 2 (c) Model c), n = 5 (c2) Model c), n = (c3) Model c), n = 2.9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2..5.6.7.8.9..2.3.4.5.5.6.7.8.9..2.3.4.5.5.6.7.8.9..2.3.4.5 (d) Model 2, n = 5 (d2) Model 2, n = (d3) Model 2, n = 2 Figure 2: Power study for Model (a c3) and Model 2 (d d3) for three different sample sizes and censoring rate p =.2. T n (solid line), A (5) (dashed line) and A (2) (dash-dotted line). Concerning the uniformity test (Model ), it is clear that for alternatives b) and c) our test outperforms
the one proposed by Akritas. Additionally, our test does not depend on the number of cells and the choice of cell boundaries. The influenc of the number of cells is made obvious in panels (a)-(c3) of Figure 2. While A (2) gives better results than A (5) in alternative a), the opposite can be observed in alternatives b) and c). Regarding the exponentiality test (Model 2), we get better results than the competitive test of Akritas (988) when the alternative is Weibull with parameter θ >. For θ <, our test reaches the high power of the Pearson-type test for big sample sizes. Notice, however, that in Model 2 and for all parameters θ, the test statistic under the Koziol-Green model, Tn, gives very good results independently on the sample size. Table 2: Power study for Model and Model 2. Model, Alterative a) n = p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.5.5277.77.84.5547.476.722.89.567.3957.793.826.457 θ = 2.9847.9997.9997.9873.9997.97.998.9995.952 θ = 2.5.9997.9237.999.9997.967 n = 2 p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.5.84.9823.9887.854.82.983.9893.82.757.9833.99.757 θ = 2.9997.9997.995.999 θ = 2.5.9983.9997 Model, Alterative b) n = p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.5.5987.3897.294.5473.537.327.2297.544.4563.253.83.537 θ =..9837.85.9987.9957.943.763.995.94.9.59.982 θ =.5.9963.9797.994.95 n = 2 p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.5.933.757.582.883.856.63.425.833.7683.487.3287.7973 θ =..9933.957.9977.9987.8977 θ =.5.9993 Model, Alterative c) n = p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.75.6587.3623.327.683.5337.34.2877.67.349.367.2823.533 θ =.25.493.39.323.5347.423.2667.2757.4957.374.2323.2423.472 θ =.5.953.85.7987.9643.8787.7287.7353.9333.797.672.68.9 n = 2 p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn θ =.75.9337.727.667.9423.8933.6783.586.967.737.6363.5423.8223 θ =.25.792.553.5537.83.728.4823.4993.7773.6243.4277.4437.733 θ =.5.9993.987.975.9993.998.974.9623.9993.963.9433.9233.9937 Model 2. Power study for θ = + Hn.5 n = p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn H = 4.972.996.9872.994.9346.9928.9754.9844.7752.9822.9564.9328 H = 2.558.4828.4746.694.4236.48.3966.5528.2354.3684.35.4364 H = 2.4784.32.38.4954.4278.268.2736.473.39.294.232.4578 H = 4.9594.8464.8296.9598.96.7564.743.942.824.6572.6478.956 n = 2 p =. p =.2 p =.3 T n A (5) A (2) Tn T n A (5) A (2) Tn T n A (5) A (2) Tn H = 4.9876.9938.9856.994.974.982.964.9864.8664.9682.936.949 H = 2.572.439.446.5976.4636.387.388.532.2892.337.3338.4268 H = 2.58.332.3354.5284.4562.2768.288.4962.3998.232.24.4736 H = 4.9686.8946.86.9676.9446.826.783.955.8764.733.732.936 2
6 Further extensions and Application 6. Composite Null Hypothesis So far, our test T n has been designed to test a fully specifie null hypothesis. It does strongly base on the fact that the transformed lifetime F (X) is [, ] uniformly distributed under H : F = F. In this section we consider a more general case, when the distribution function to be tested depends on an unknown parameter λ. Let now consider the following null hypothesis H : F {F λ : λ R d }. In this case firs we need to estimate the parameter λ using, e.g., a maximum-likelihood estimator ˆλ. Clearly, if F λ is twice differentiable in λ and the estimator ˆλ is n consistent, by the Taylor expansion we have that Fˆλ(X) = U + O P (n /2 ), U = F λ (X) U[, ] under the null hypothesis H. The test statistic Q n should still admit a normal limit but the error term enters the variance of our test statistic and hence the asymptotic variance given in Theorem is no longer valid. Even though the theoretical properties of our test in the case of such a composite hypothesis are beyond the scope of this paper, to test this kind of hypothesis we propose a modifie test with jacknife estimator of the variance, which does take into account the estimation of parameters and works very well in practice. We proceed as follows:. Based on the sample X,..., X n, fin the maximum-likelihood estimator (MLE) ˆλ. 2. Defin the pseudo-values X i = Fˆλ(X i ) for i =,..., n. 3. Based on the sample X,..., X n, compute the test statistic Q n define in (7). 4. Compute the jacknife estimator of the variance following the steps For every i =,..., n, choose the subsample X,..., X i, X i+,..., X n and compute the MLE ˆλ ( i). Defin the pseudo-values X j = Fˆλ( i)(x j ) for j =,..., i, i +,..., n. Based on the the sample X,..., X i, X i+,..., X n, compute the test statistic Q ( i) n. Set Q n = n 5. Defin the test statistic n i= Q ( i) n. nv n ( Q n ) = (n ) J n := ( i= n Qn nv n ( Q. n ) Q ( i) n Q n ) 2, 3
6. Reject H, if J n Φ (α/2) or J n Φ ( α/2). To check the behavior of this new jacknife-test, J n, we study the hypothesis H : F {exp(λ) : λ (, )}, the alternatives come from the Weibull distribution. Our simulated sample comes from exp(λ = ) but λ is estimated using Maximum-Likelihood method. In Figure 3 we compare out test T n from Section 4 with the test based on J n..9.8.7.6.5.4.3.2..6.7.8.9..2.3.4.5.9.8.7.6.5.4.3.2..6.7.8.9..2.3.4.5.9.8.7.6.5.4.3.2..6.7.8.9..2.3.4.5 Figure 3: Power study: J n (solid line) and T n (dashed line), n = 5 (left), n = (middle) and n = 2 (right), p =.2 6.2 Real Data Example We illustrate the use of our test on the head-and-neck cancer data from Haghighi and Nikulin (24). These authors fitte the Power Generalized Weibull distribution F (x, σ, v, γ) to the data. We use here the truncated data set with Survival Times (in months) smaller than 2. It was motivated by the boxplot in Figure 4. This gives us 44 observations with around % censoring rate. We perform a goodness-of-fi test for the before-mentioned Power Generalized Weibull distribution F a (x, σ, v, γ) = F (x, σ, v, γ). Additionally, we also consider the Weibull distribution F b (x, σ, v) = F (x, σ, v, ) and the Exponential distribution F c (x, σ) = F (x, σ,, ), F (x, σ, v, γ) = exp ( ( + (x/σ) v ) /γ). First, we fitte the parameters using MLE under random censoring obtaining the estimators (ˆσ, ˆv, ˆγ) and the following distributions F a (x, 4.63,.82,.9), F b (x,.44, 8.45) and F c (x, 8.33). Then we applied our test J n and obtained the following p-values: p a =.86, p b =.88 and p c =. for Power Generalized Weibull, Weibull and Exponential, respectively. Hence, the results of the test confir what it is to see in Figure 4, that both Power Generalized Weibull and Weibull fi the data very well while the Exponential distribution is not adequate to describe the head-and-neck cancer data. 4
F a (x, ˆσ, ˆv, ˆγ) F b (x, ˆσ, ˆv) F c (x, ˆσ) χ 2
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