SUMMARY We wish to test a simple hypothesis against a family of alternatives indexed by a one-dimensional parameter, 0. We use a test derived from the corresponding family of test statistics appropriate for the case when 0 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.

Hypothesis testing when a nuisance parameter is present only under the alternative

A problem of optimal stopping is formulated and simple rules are proposed which are asymptotically optimal in an appropriate sense. The problem is of central importance in quality control and also has applications in reliability theory and other areas.

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Procedures for reacting to a change in distribution

In attempting to estimate the parameters of a linear regression system obeying two separate regimes, it is necessary first to estimate the position of the point in time at which the switch from one regime to the other occurred The suggested maximum likelihood estimating procedure is based upon a direct examination of the likelihood function An asymptotic and a small-sample test are suggested for testing the hypothesis that no switch occurred against the single alternative that one switch took place The procedure is illustrated with a sampling experiment in which the true switching point is correctly estimated * I am indebted to Professors F Anscombe, R Dorfman and F Stephan and the referees of this paper for criticism and helpful suggestions I am also indebted to Professor John S Chipman for originally suggesting the problem The responsibility for errors is, of course, mine

The Estimation of the Parameters of a Linear Regression System Obeying Two Separate Regimes

https://projecteuclid.org/download/pdf_1/euclid.bsmsp/1200512162

Tests of Separate Families of Hypotheses

The classical method of studying a cumulative sum control scheme of the decision interval type has been to regard the scheme as a sequence of sequential tests, to determine the average sample number for these component tests and hence to study the average run length for the scheme. A different approach in which the operation of the scheme is regarded as forming a Markov chain is set out. The transition probability matrix for this chain is obtained and then the properties of this matrix used to determine not only the average run lengths for the scheme, but also moments and percentage points of the run-length distribution and exact probabilities of run length. The method may be used with any discrete distribution and also, as ani accurate approximation, with any continuous distribution for the random variable which is to be controlled. Examples are given for the cases of a Poisson random variable and a normal random variable.

E. S. Page

Papers

A test for a change in a parameter occurring at an unknown point

On problems in which a change in a parameter occurs at an unknown point

Tables for wald tests for the mean of a normal distribution