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Stepwise likelihood ratio statistics in sequential studies
Author(s) -
Huang Wenzheng
Publication year - 2004
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1046/j.1369-7412.2003.05398.x
Subject(s) - likelihood ratio test , statistics , likelihood principle , mathematics , null hypothesis , statistic , statistical hypothesis testing , alternative hypothesis , sufficient statistic , score test , exponential family , sequential probability ratio test , simple (philosophy) , likelihood function , econometrics , maximum likelihood , quasi maximum likelihood , philosophy , epistemology
Summary. It is well known that in a sequential study the probability that the likelihood ratio for a simple alternative hypothesis H 1 versus a simple null hypothesis H 0 will ever be greater than a positive constant c will not exceed 1/ c under H 0 . However, for a composite alternative hypothesis, this bound of 1/ c will no longer hold when a generalized likelihood ratio statistic is used. We consider a stepwise likelihood ratio statistic which, for each new observation, is updated by cumulatively multiplying the ratio of the conditional likelihoods for the composite alternative hypothesis evaluated at an estimate of the parameter obtained from the preceding observations versus the simple null hypothesis. We show that, under the null hypothesis, the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1/ c . In contrast, under the composite alternative hypothesis, this ratio will generally converge in probability to ∞. These results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis. For illustration, we conduct two simulation studies, one for a normal response and one for an exponential response, to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches.