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The random weight function, WG(t) is a predictable stochastic process, which converges uniformly in probability to a nonnegative bounded function on (0, to). Note that when WG(t) = 1, we have the likelihood function in (1.7). Typically, for right censored data, WG(t) = F(t), the left continuous version of the Kaplan Meier estimator of the survival function.

Now define 0w, n = (aw,n, Pw,n, Awn) to be the maximizer of WLn(0) over 0 e &n. This estimator is obtained by following the steps of the EM algorithm of Dupuy and Mesbah. (2002) for the non-weighted likelihood function. Note that the weights themselves do not depend on the parameters, thus ensuring that all the steps for the EM algorithm for the weighted likelihood function go through as for they do for the non-weighted likelihood function.

Since the test statistic is a function of pw,n, we first present the following asymptotic results for the weighted parameter pw,n '■

Theorem 1. Under the model (1.7), the vector yfn (fiw,n — f3o)converges in distribution to a bivariate normal distribution with zero mean and a covariance

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