© 1987 by Biometrika Trust
Hypothesis testing when a nuisance parameter is present only under the alternative
Applied Mathematics Division, DSIR Wellington, New Zealand
We wish to test a simple hypothesis against a family of alternatives indexed by a one-dimensional parameter, 
. We use a test derived from the corresponding family of test statistics appropriate for the case when is given. Davies (1977) introduced this problem when these test statistics had normal distributions. The present paper considers the case when their distribution is chi-squared. The results are applied to the detection of a discrete frequency component of unknown frequency in a time series. In addition quick methods for finding approximate significance probabilities are given for both the normal and chi-squared cases and applied to the two-phase regression problem in the normal case.
Key Words: Chi-squared process • Frequency component • Hypothesis test • Maximum • Nuisance parameter • Quick test • Two-phase regression • Time series • Upcrossing
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