Noncentrality Parameter

Fig. 1. Power of Qopt versus Q3 for Different K

Noncentrality Parameter

Fig. 1. Power of Qopt versus Q3 for Different K

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Appendix I Asymptotic Normality of WLW Method under General Contiguous Alternatives

Let Yki(t) = 1(Xki > t) and Nki(t) = 1(Xki < t,Aki = 1), and define the functions S{k](P,t), Vk(p,t), s^)(^,t) and vk(p,t) for r = 0,1, 2 by

Define the martingale Mki(t) as

First consider f3, the maximizing solution of equation (2). Under the regularity conditions, it can be shown that f3 converges to f3, the solution of the following equation:

Under contiguous alternatives, the above equation becomes

which has its solution at f = 0. Let /3(n) denote the estimate of f when the alternatives are the sequence of general contiguous alternatives defined in Section 3. Using the above argument, it also follows that f(n) 0 as n — to.

To determine the asymptotic distribution of ^Jn3(n under the general contiguous alternative, we first derive the asymptotic distribution of the score statistic U (f(n)), where fj(n) =

ELiT Vk (0, t)sk (0, t)Xk(t)dt Rewriting U (P{n)), we have

The first term of right hand side of the second equal sign is martingale. It can be shown that this term converges to the N(0,1'V 1) distribution and that the second term converge to 0 in probability as n — to. Thus, as n — to, n

Tests for Multivariate Survival Data 317 n-2 U (3(n)) —► N(0, 1V1) , where V is the KxK matrix with (k, k) element vk (3k ) = / Vk (3k ,t)sf\3k,t)Ak (t)dt, Jo and with (k, w) element (k = w)

vkw(3k,3w) = E[f °{z— ^l3^ }dMki][i°{zi— sWyw,t] }dMWi] (k = w). Jo sf(3k,t) Jo sW0)(3w ,t)

Letting

1 EK=i fo°° Vk(0,t)sk0)(0,t)Ak(t)dt , and noting that 3(n) — 0, it follows that

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