Abstracts of papers - Robert Davies

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Vere-Jones, D & Davies, R.B. (1966) A statistical survey of earthquakes in the main seismic region of New Zealand: Part 2 - Time series analyses. N.Z. J. Geol. Geophys. 9 251-284 (scanned copy - 2933kb) 

Time series analyses are carried out on earthquake data from the main seismic region of New Zealand for the years 1942-61. Origin times only are considered, the energies and exact positions of the shocks being largely ignored. The relevant statistical theory for the first and second order properties of the process is described, and simple probability models for earthquake occurrence are put forward. On the basis of these results, the data are examined for periodic and grouping effects. No significant periodic effects are found, either among the shallow shocks (depths up to 100 km) or among the deep shocks (depths 100 km or greater). Both components show strong evidence of grouping, and several alternative models to describe this effect are put forward and compared.


Mertz, D. B. & Davies, R.B. (1968) Cannibalism of the pupal stage by adult flour beetles: an experiment and a stochastic model. Biometrics 24 247-275

This paper describes an experiment involving adult flour beetles (genetic strain cIV-a of Tribolium castaneum) cannibalizing their own pupae and proposes a new stochastic model for this form of cannibalism. The model differs from earlier stochastic models of predation in that it is based on the hypothesis of predator satiation, and it gives a considerably better fit to the experimental data than do the earlier ones. In the experiment the percentage survival of pupae increased rapidly as the pupal density increased. This corresponds to an earlier finding that in self-limiting populations of cIV-a, large pupal populations are apt to be followed in time by sudden increases in adult numbers (i.e. outbreaks). The model shows that this would be expected for satiable adult predators. However, for insatiable predators, outbreaks would be unlikely. Apparently, cIV-a adults behave as if their appetites for pupae were satiable, but for most other Tribolium populations 'satiation' seems to be less important and outbreaks are uncommon.


Davies, R.B. (1969) Beta-optimal tests and an application to the summary evaluation of independent experiments. J. Roy. Statist. Soc. B 31 524-538.

A concept of optimality of a test, based on the speed with which its power function reaches a pre-assigned value, is introduced and conditions for a test to have this property are considered. The concept is applied to the weighted combination of independent normal test statistics; tests for the presence of effects and for the equality of effects are deduced.


Davies, R.B. (1973) Asymptotic inference in stationary Gaussian time-series. Adv. Appl. Prob. 5 469-497.

Conditions are given for the family of distributions of a stationary, discrete-time, Gaussian, vector-valued time-series with covariance structure given up to a finite number of parameters to satisfy the asymptotic differentiability conditions introduced by Le Cam (1969). (scanned copy - 2156kb)


Davies, R.B. (1973) Numerical inversion of a characteristic function. Biometrika 60 415-417.

A method is described for finding a bound on the error when a version of the usual characteristic function inversion formula is evaluated by numerical integration. The method is applied to the calculation of the distribution function of a quadratic form in normal random variables. (scanned copy - 317kb)
Some key words: Numerical inversion of characteristic function; Quadratic form in normal variables; Trapezoidal rule.


Davies, R.B. & Hutton, Bruce (1975) The effect of error in the independent variables in linear regression. Biometrika 62 383-391.

Suppose that the independent variables in a linear regression are subject to error. This paper is concerned with the bias introduced into the least squares estimators by these errors, first when they are regarded as fixed and second when they are regarded as random. Simple criteria are introduced for deciding whether the bias is likely to be serious. The paper also considers the effect of the occasional large error in either the dependent or independent variables. (scanned copy - 1225kb)
Some key words: Errors in data matrix; Least squares; Linear regression; Singularity of matrix.


Davies, R.B. (1977) Testing the hypothesis that a point process is Poisson. Adv. Appl. Prob. 9 724-746.

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found. (scanned copy - 1760kb)


Davies, R.B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 247-254.

Suppose that the distribution of a random variable representing the outcome of an experiment depends on two parameters \xi and \theta and that we wish to test the hypothesis \xi = 0 against the alternative \xi > 0. If the distribution does not depend on \theta when \xi = 0, standard asymptotic methods such as likelihood ratio testing or C(\alpha) testing are not directly applicable. However, these methods may, under appropriate conditions, be used to reduce the problem to one involving inference from a Gaussian process. This simplified problem is examined and a test which may be derived as a likelihood ratio test or from the union-intersection principle is introduced. Approximate expressions for the significance level and power are obtained. (scanned copy - 1067kb)
Some key words: C-alpha. test; Hypothesis testing; Likelihood ratio test; Maximum of Gaussian process ; Simple hypothesis ; Union-intersection principle.


Davies, R.B. (1985) Asymptotic inference when the amount of information is random. Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Volume 2. Eds L.M. LeCam & R.A. Olsen. 841-864. Wadsworth, Belmont.

Some results of asymptotic statistical decision theory are extended to allow for the situation in which the Fisher information should be treated as random. They are applied to parameter estimation and hypothesis testing for the supercritical Galton-Watson process and to sequential analysis.
Key Words and Phrases: asymptotic inference, conditional inference, contiguity, curved exponential family, Galton-Watson process, information matrix, sequential estimation, stochastic process estimation. (scanned copy - 2394kb)
AMS 1980 Subject Classifications: primary 62F12; secondary 62FO5, 62L12, 62MO5, 62MO9.


Davies, R.B. & Harte, D.S. (1987) Tests for Hurst effect. Biometrika 74 95-101.

We consider the power of tests for distinguishing between fractional Gaussian noise and white noise of a first-order autoregressive process. Our tests are based on the beta-optimal principle (Davies, 1969), local optimality and the rescaled range test. (scanned copy - 868kb)
Some key words: Autoregressive process; Beta-optimal test; Fractional Gaussian noise; Hydrological series; Locally optimal test; Long-term dependence; Rescaled range; Self-similar process; Simulation.


Davies, R.B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74 33-43.

We wish to test a simple hypothesis against a family of alternatives indexed by a one-dimensional parameter, \theta. We use a test derived from the corresponding family of test statistics appropriate for the case when \theta 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. (scanned copy - 1313kb)
Some key words: Chi-squared process; Frequency component; Hypothesis test; Maximum; Nuisance parameter; Quick test; Two-phase regression; Time series; Up crossing.


Davies, R.B. (2001) Integrated processes and the discrete cosine transform.  Probability, statistics and seismology, A Festschrift for David Vere-Jones. Ed D.J.Daley. J. Appl. Probab. Special Volume 38A. 105-121.

A time-series consisting of white noise plus Brownian motion sampled at equal intervals of time is exactly orthogonalised by a discrete cosine transform (DCT-II). This paper explores the properties of a version of spectral analysis based on the discrete cosine transform and its use in distinguishing between a stationary time-series and an integrated (unit root) time-series. (preprint - 298kb)
Keywords: Beta-optimal test; Brownian motion; DCT-II; discrete cosine transform; integrated process; random walk; spectrum; time-series; unit root.


Davies, R.B. (2002) Hypothesis testing when a nuisance parameter is present only under the alternative - linear model case. Biometrika 89 484-489.

The results of Davies (1977, 1987) are extended to a linear model situation with unknown residual variance. (preprint - 166kb)
Some key words: Change point; F-process; Frequency component; Hypothesis test; Nuisance parameter; t-process; Two-phase regression; Up-crossing.


Davies, R.B., Withers, C.S. & Nadarajah, S. (2011) Confidence intervals in a regression with both linear and non-linear terms. Electron. J. Statist. 5, 603-618.

We present a simple way for calculating confidence intervals for a class of scalar functions of the parameters in least squares estimation when there are linear together with a small number of non-linear terms. We do not assume normality.  (Open access - 1106KB)
Keywords: Confidence interval; estimation; optimization; two-phase regression 

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