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A Simple Derivation of r * for Curved Exponential Families
Author(s) -
Jensen Jens Ledet
Publication year - 1997
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00047
Subject(s) - mathematics , exponential family , statistic , order statistic , simple (philosophy) , statistics , likelihood ratio test , exponential function , test statistic , independence (probability theory) , asymptotic distribution , series (stratigraphy) , statistical hypothesis testing , combinatorics , mathematical analysis , paleontology , philosophy , epistemology , estimator , biology
For curved exponential families we consider modified likelihood ratio statistics of the form r L =r+ log( u/r)/r , where r is the signed root of the likelihood ratio statistic. We are testing a one‐dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u , and include a discussion of the case where some coordinates of the underlying variable are lattice.