Showing papers in "Biometrika in 1972"

An approach to the probability distribution of cusum run length

Abstract: 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.

Separate sample logistic discrimination

Abstract: The problem of discrimination when all or most of the observations are qualitative is discussed. The method of logistic discrimination introduced by Cox (1966) and Day & Kerridge (1967) is extended to the situation where separate samples are taken from each population, using the results of Aitchison & Silvey (1958) on constrained maximum likelihood estimation. The method is further extended to discrimination between three or more populations. The properties of logistic discrimination are investigated by simulation and the method is applied to the differential diagnosis of kerato-conjunctivitis sicca.

The multiple recapture census for closed populations and incomplete 2k contingency tables

Abstract: SUMMARY The multiple recapture census for closed populations is reconsidered, assuming an underlying multinomial sampling model. The resulting data can be put in the form of an incomplete 2k contingency table, with one missing cell, that displays the full multiple recapture history of all individuals in the population. Log linear models are fitted to this incomplete contingency table, and the simplest plausible model that fits the observed cells is projected to cover the missing cell, thus yielding an estimate of the total population size. Asymptotic variances for the estimate of the population size are considered, and the techniques are illustrated on a population of children possessing a common congenital anomaly.

Empirical Bayes on vector observations: An extension of Stein's method

Abstract: SUMMARY The statistician is considering several independent normal linear models with identical structures, and desires to estimate the vector of unknown parameters in each of them. An estimator is constructed which dominates the usual Gauss-Markov estimator in terms of total squared error loss. This estimator is shown to have good efficiency in the Bayesian situation where the parameter vectors themselves have a normal prior distribution. A practical example is given.

Asymptotic behaviour of temporal aggregates of time series

Abstract: SUMMARY The problem of modelling and forecasting temporal aggregates of time series is discussed. Let zt be the basic series and XT be the m-component temporal aggregates. Forecasts of future XT may be constructed from data on (i) zt or (ii) XT. It is shown that, for large m, there is no gain in using the basic data if ztis stationary, but considerable gain can be obtained when Zt is nonstationary.

Studies in the History of Probability and Statistics. XXIX The discovery of the method of least squares

Abstract: SUMMARY The circumstances in which the discovery of the method of least squares took place and the course of the ensuing controversy are examined in detail with the aid of correspondence. Some conclusions are drawn about the attitudes of the main participants and the nature of historical research in statistics.

The distribution of a statistic used for testing sphericity of normal distributions

Abstract: SUMMARY The joint distribution of the sum of the rth powers (r = 1, ..., p - 1) and the product of all the latent roots of a p x p Wishart matrix is obtained and used to derive the null distributions of the likelihood ratio test criterion and the locally most powerful invariant test criterion for detecting deviations of the variances and covariances of a p-variate normal distribution from proportionality to specified numbers. The sphericity of the distribution is a special case. Explicit expressions are given for the null distributions in the trivariate case. In the bivariate case the two test criteria coincide and their null distribution has been known. The distribution of the locally most powerful test criterion being complicated for values of p larger than three, some approximations are fitted by the method of moments and compared.

Asymptotic normality and efficiency for certain goodness-of-fit tests

Abstract: SUMMARY Assume that a random sample of size n has been taken from a multinomial distribution with N cells. Let 4k be the number of observations in the kth cell and set

Bayesian methods for binomial data

Abstract: SUMMARY A Bayesian procedure is obtained for the simultaneous estimation of the parameters of m binomial distributions. The method uses logistic transformations for the parameters and an exchangeable prior distribution. Information is combined between the binomial distributions to obtain estimates which, under certain circumstances, will be superior to the usual proportions. This paper is intended as a forerunner to a more general theory for the analysis of nonlinear models.

Planning experiments to detect inadequate regression models

Abstract: SUMMARY The adequacy of a regression model can be tested by fitting an extended model which reduces to the given model for particular parameter values. The choice of the extended model for nonlinear regression is discussed. Two experimental design criteria are compared and designs are developed for simultaneously providing information about both the original regression model and the hypothesis that the model is adequate. Designs are tabulated for detecting second-order departures from two k-factor models and an example is given of a nonlinear application.

Least squares estimation in dynamic-disturbance time series models

Abstract: SUMMARY Recent results of the author concerning large-sample properties of the nonlinear least squares estimates of the parameters, in a regression model in which the disturbances follow a stationary mixed autoregressive-moving average time series, are extended to models in which lagged values of the dependent and independent variables occur.

Time-ordered classification

Abstract: SUMMARY Classification for the special situation where observations after a certain time no longer come from the initial population is considered. The discussion focuses on inference about that time, often called the change-point. Some useful approximations are derived for the distributions of change-point statistics, and large-sample results are established for nuisance-parameter situations.

Tests for randomness of directions against equatorial and bimodal alternatives

Abstract: SUMMARY Tests of randomness of directions in three-dimensional space or equivalently tests of uniform distribution of points on the unit sphere are treated. One test is against alternatives which concentrate probability density near an equator, and the other is against alternatives which concentrate probability density near opposite poles; in each case the poles are unspecified. The tests are based on the latent roots of the matrix of sums of squares and cross-products of the co-ordinates of the observed points on the unit sphere. Against equatorial alternatives the null hypothesis is rejected if the smallest root is less than the appropriate significance point, and against bimodal alternatives the null hypothesis is rejected if the largest root is greater than the appropriate significance point. Tables of significance points are given, based on Monte-Carlo studies and the asymptotic distributions which are derived. The two-dimensional problem is also discussed.

A comparison of some methods for estimating mixed normal distributions

Abstract: SUMMARY Fisher's method of maximum likelihood breaks down when applied to the problem of estimating the five parameters of a mixture of two normal densities from a continuous random sample of size n. Two alternative methods, (i) moment estimates and (ii) multinomial maximum likelihood and minimum x2 estimates obtained by grouping the underlying variable, are compared both for bias to n-1, and mean squared error to n-2 for a variety of mixed distributions. The methods do not differ essentially with regard to bias but for the mean squared error, the grouped estimates are shown to be more accurate than the moment estimates for most distributions, though the moment estimates seem preferable for distributions which are particularly difficult to estimate. It is also found that the accuracy levels of the grouped maximum likelihood and minimum x2 estimates do not differ greatly.

On the correlation structure of some two-dimensional stationary processes

Abstract: SUMMARY The correlation structure of the class of stationary, unilateral, linear autoregressions defined on a plane lattice is considered. It is shown that although standard techniques for deriving the correlogram analytically are not immediately appropriate, it is, nevertheless, possible to give a simple solution in a certain region of the plane provided that the number of regressors is small. The remainder of the correlogram tends to be analytically awkward but can be very easily calculated numerically. Three specific examples arising from the literature are discussed in detail.

Un-Bayesian implications of improper Bayes inference in routine statistical problems

Abstract: SUMMARY For two routine statistical problems, inference about the ratio of two exponential means and inference about the coefficient of variation of a normal random variable, a serious pathology of Bayesian inference based on improper priors is uncovered and investigated.

Birth, death and migration processes

Abstract: SUMMARY The effect of migration between a finite number of colonies each of which undergoes a simple birth and death process is studied. The first two moments are obtained for the general process and deterministic solutions are developed for several special models including the finite linear model proposed by Bailey (1968). In a recent paper Bailey (1968) constructs a model for spatially distributed populations by considering a population to be composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis represented by -xo < i < x. Each colony is assumed to be subject to a simple birth and death process with birth and death rates A and Iu respectively, and with migration rates to each of the two neighbouring colonies. He derives the mean number of individuals in each colony at time t together with the variances and covariances, and briefly considers the corresponding models where the population is extended to two and three dimensions. Clearly by assuming the existence of an infinite number of colonies he avoids the problem of 'edge effects' at the boundaries when the number of colonies is finite. We shall later examine this finite analogue of Bailey's model and show how his results for the infinite model follow as a special case. His paper has recently been extended by several authors. Adke (1969) generalizes Bailey's process to include time-dependent birth and death rates whilst Usher & Williamson (1970) use discrete time intervals to analyze the model when the number of colonies is finite. They consider the population to be split into migrants and nonmigrants, each group having different birth and death rates. Davis (1970) presents some results for a general Markov branching-diffusion process and then applies them to Bailey's model. Crump (1970) studies a general age-dependent branching process in which the population is distributed in N colonies with migration between them and he obtains asymptotic expressions for the first two moments in several special cases. The stochastic equations for the general parameter case of the simple birth-death-migration process are considered by Puri (1968).

A negative binomial model with varying element sizes

Abstract: SUMMARY Compound Poisson distributions are generally associated with counts of discrete events on sample elements of uniform size. The case of variable element sizes, applicable to many compound distributions, is considered here for the particular case of the negative binomial model. Maximum likelihood and weighted least squares estimators for its parameters are presented, with examples.

Pearson chi-squared test of fit with random intervals

Abstract: SUMMARY A modified form of the Pearson x2 test statistic is considered where estimators based on the ungrouped sample are employed in the test statistic as well as in determining the class interval end points. In the case of continuous distributions with location and scale parameters, it is shown here that the null distribution of such a modified test statistic does not depend on the parameters if these are estimated by sample mean and variance. By utilizing known results concerning the distribution of a weighted sum of independent chi-squared variates, a table of certain percentage points of the asymptotic distribution of this modified test statistic is developed in order to facilitate its use for testing normality. case of continuous distributions. In the first place, how should the class intervals be formed and how many should there be? Secondly, if there are unknown parameters, how should they be estimated and what is their effect on the test? Aside from the complexities arising in deriving estimators ofthe parameters, the resulting distribution ofX2, the Pearson x2 statistic, can be quite different from that of x2, depending upon the method of estimation, as was observed by Chernoff & Lehmann (1954). Thirdly, the estimation of class probabilities when parameters are unknown is also usually complicated in the case of continuous dis- tributions. In considering the problem of how the class intervals should be formed in using the X2 statistic, A.R. Roy in an unpublished report, and Watson (1957, 1958, 1959) relaxed the requirement of fixed class boundaries and utilized estimators of the parameters to determine the end points of the intervals. The estimators employed for this purpose as well as for substitution in the X2 statistic were based on the ungrouped sample. In the present paper we extend a result of Roy to show, in the case of unknown location and scale parameters, that the asymptotic null distribution of the test statistic does not involve the unknown parameters when they are estimated by the sample mean and sample variance respectively. As an application of the technique developed here, we provide a table of some percentage points of the asymptotic distribution of the Pearson statistic modified in the above manner, based on equal probability classes, and used to test for normality. The number of class intervals to be employed in the test should depend on the alternative distributions. This requires knowledge of the corresponding asymptotic nonnull distribution of the test statistic and will be given for various alternatives in a subsequent

The power function of range and studentized range tests in normal samples

Abstract: where X = EXi/n. Without loss of generality, we take EEi = 0 and 61 ... 6 ,T. Note that unlike the distribution of x'2, which is a noncentral x2 with n 1 degrees of freedom depending only on E62 the cumulative distribution function (1) depends on n 1 differences of the Ei, for example 7 61 (h = 2, ..., n). For a few selected cases (1) was evaluated by David (1953) in the course of an investigation of the power of short-cut tests, based on the range, of the hypothesis, Ho, of equality of the Ei. It was seen that, for a configuration of the 6 favourable to the range, W' could give a more powerful test than x'2. It may also be lnoted that, in an empirical study of various tests of homogeneity of variance in normal populations, Pearson (1966) found Smax/Smin and Wmax/Wmin to be as good as the standard 111 test for certain patterns of the population variances. These results are closely related to the behaviour of the range in nonnormal samples since, for example,

On the error of prediction of a time series

Abstract: : Parametric and nonparametric procedures for the prediction of a time series are discussed. In each case the increase in the mean squared error of prediction over its minimum level due to the use of estimated spectra is assessed. The fitting of simple parametric models as approximations is also discussed. (Author)

Some results on the distribution of Wilks's likelihood-ratio criterion

Abstract: SUMMARY The null distribution of Wilks's likelihood ratio criterion A is examined in its exact and asymptotic forms. The case whenp, the number of variates, and q, the number of hypothesis degrees of freedom, are both odd is considered and in this case the distribution is given in simple integral form. The results are used to develop techniques for computing the distribution. Tables of the chi-squared correction factors for the percentage points are extended to cover, in conjunction with existing tables, values for all p < q < 20 and pq < 144 with omission when p or q is odd and greater than 10.