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Zbl 1070.62087
Banerjee, Moulinath; Wellner, Jon A.
Score statistics for current status data: comparisons with likelihood ratio and Wald statistics.
(English)
[J] Int. J. Biostat. 1, No. 1, Article 3, 27 p. (2005). ISSN 1557-4679

Summary: We introduce three natural "score statistics" for testing the hypothesis that $F(t_0)$ takes on a fixed value in the context of nonparametric inference with current status data. These three new test statistics have natural interpretations in terms of certain (weighted) $L_2$ distances, and are also connected to natural one-sided" scores. We compare these new test statistics with the analogue of the classical Wald statistic and the likelihood ratio statistic introduced by {\it M. Banerjee} and {\it J. A. Wellner} [Ann. Stat. 29, No.6, 1699-1731 (2001; Zbl 1043.62037)] for the same testing problem. Under classical "regular" statistical problems the likelihood ratio, score, and Wald statistics all have the same chi-squared limiting distribution under the null hypothesis. In sharp contrast, in this non-regular problem all three statistics have different limiting distributions under the null hypothesis. Thus we begin by establishing the limit distribution theory of the statistics under the null hypothesis, and discuss calculation of the relevant critical points for the test statistics. Once the null distribution theory is known, the immediate question becomes that of power. We establish the limiting behavior of the three types of statistics under local alternatives. We have also compared the power of these five different statistics via a limited Monte-Carlo study. Our conclusions are: (a) the Wald statistic is less powerful than the likelihood ratio and score statistics; and (b) one of the score statistics may have more power than the likelihood ratio statistic for some alternatives.
MSC 2000:
*62N03 Testing
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
62P10 Appl. of statistics to biology

Keywords: Brownian motion; contiguous alternatives; greatest convex minorant; likelihood ratio statistic; log-likelihood ratio; score statistic; Wald statistic

Citations: Zbl 1043.62037

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