Showing papers in "Australian & New Zealand Journal of Statistics in 1972"

Confidence intervals for power, with special reference to medical trials

Abstract: A technique is given for drawing valid inferences in cases where performance characteristics of statistical procedures (e.g. power for a test, or probability of a correct selection for a selection procedure) depend upon unknown parameters (e.g. an unknown variance). The technique is especially useful in situations where sample sizes are small (e.g. in many medical trials); the “usual” approximate procedures are found to be misleading in such cases.

An asymptotic expansion for the variance of the errors of misclassification of the linear discriminant function1

Abstract: SUMMARY A technique for deriving asymptotic expansions for the variances of the errors of misclassification of the linear discriminant function (Anderson's classification statistic) is developed. These expansions are shown to be in reasonable agreement with the sample values of the variances of the errors obtained from some sampling experiments.

A note on the distribution of the doubly non‐central f distribution1

Abstract: SUMMARY A three-moment central-F approximation is examined for its usefulness in giving accurate values of the probability integral of the doubly non-central F distribution.

Martingales: a case for a place in the statistician's repertoire1

Abstract: It is usually the case that martingale methods do not form a valued part of the statistician’s repertoire of techniques. The reasons are partly historical and partly environmental, and while modern texts on probability theory deal in great detail with sums of independent random variables, the essentially broader subject of martingales is often relegated to a small section or the back pages. The object of this note is to give a brief exposition of what martingales are and what they can do. It is my personal feeUng that the role of martingale theory in probabiUty and statistics will increase rapidly in importance within the next few years.

Moments of a linearly truncated bivariate normal distribution1

Abstract: SUMMARY The moments are obtained for a bivariate normal distribution which is linearly truncated with respect to both variables; the variables may be correlated. From these moments the parameters of the distribution can be estimated.

Estimation of scale and location parameters, using quadratic coefficients

Abstract: A particular linear estimate with three terms approximation using ordered observations is discussed which seems to be simple to calculate. This estimate is applicable for any sample size for some special distributions without considerable loss in efficiency and is, therefore, of practical value.

The limiting form of the non‐central wishart distribution1,2

Abstract: SUMMARY Let S (p×p) have a Wishart distribution -with v degrees of freedom and non-centrality matrix θ= [θjK] (p×p). Define θ0= min {| θjk |}, let θ0∞, and suppose that | θjK | = 0(θo). Then the limiting form of the standardized non-central distribution, as θ while n remains fixed, is a multivariate Gaussian distribution. This result in turn is used to obtain known asymptotic properties of multivariate chi-square and Rayleigh distributions under somewhat weaker conditions.

Exact distributions of a few arbitrary roots of some complex random matrices1

Abstract: : In this paper, the authors derived expressions for the joint probability density of any few unordered roots of a class of noncentral complex random matrices. This class includes A-matrix which is of interest to physicists, Wishart matrix, MANOVA matrix and canonical correlation matrix. (Author)

Household consumption of food in australia

Abstract: This paper is the second (see the first reference) in the series of studies entitled “Household Consumption and Income in Australia.” undertaken by the author to analyse data from an Australia-wide consumption survey (Drane, Edwards and Gates, 1969). Whereas the first study was an investigation into the overall consumption patterns, the present study is devoted to analysing intensively the patterns of household food consumption. Food is especially important since it is the largest single consumption item in the family budget. In this paper the analysis is extended in two main directions. The first is the disaggregation of total food consumption into its components. The degree of disaggregation, however, is limited by the availability of detailed information. Accordingly, only five components of expenditure on food could be effectively considered. The second direction is the investigation of the possible differences in food consumption patterns among various sociological and demographic groups. To that end multivariate statistical techniques have been employed. The first section of this paper gives a brief description of the scope and limitations of the study and sets out different hypotheses that could be tested. The second section discusses the appropriate statistical techniques and the estimation methods that would be useful, while the results and their exposition have been presented in the third section. The final section contains a short comparative study and a few concluding remarks.

Sampling on several successive occasions with equal and unequal probabilities and without replacement1

Abstract: SUMMARY The applicability of the technique of controlled selection with equal probabilities proposed by the authors, Avadhani and Sukhatme (1965, 1968), and that of unequal probability sampling suggested by Rao et al. (1962) to sampling on several successive occasions is investigated in this paper. Appropriate recurrence formulae for the optimum replacement fraction and the corresponding weight are also derived.

On the distributions of order statistics1

Abstract: SUMMARY Approximations to the distributions of order statistics based on x2t are obtained. These are easy to compute and provide reasonably accurate values for the percentage points and probability integrals of the distributions.

Horvitz‐thompson and des raj estimators revisited

Abstract: Summary In this paper it is shown that the generalized πPS sampling strategy consisting of the design with πi, the probability of inclusion of the ith unit in the sample, proportional to the modified size together with the corresponding Horvitz-Thompson estimator (Rao, 1971), is superior to the symmetrized Des Raj strategy under a general super-population set-up for all values of the super-population parameter g, when the samples are of size two.

Interaction models for the latin square1

Abstract: This paper presents a general method for testing if a particular type of interaction is present in latin square models with one observation per cell. Computing instructions are given along with a detailed numerical example. A data analysis approach is explained where an experimenter can use various models to determine whether interaction is present. If there is interaction, then a procedure is given for examining a number of models to decide what factors (row, column, treatment) are interacting. In addition, a number of sets of latin square data from the literature are reanalysed to determine if they contain interaction.

On the behrens–fisher problem1

Abstract: SUMMARY In this article the Behrens-Fisher problem is reformulated in terms of a structural model of inference. For this version of the problem a solution is obtained which is valid for arbitrary absolutely continuous error distributions. These results are further discussed for the standard normal distribution and for some other special cases with not normally distributed populations.

Least squares analysis of freight‐rate anomalies1

Abstract: SUMMARY Regression methods are applied to freight rate data providing a simple method of identifying anomalies, and calculating and adjusting charges. Application to problems of assessing economic consequences of changes are indicated.

A note on sampling over two occasions1

Abstract: SUMMARY Using simple random sampling without replacement (SRSWOR) on both occasions, a new scheme of selecting the unmatched part of the sample on second occasion is presented. The proposed estimate of the population mean has been shown to be more efficient than the estimate of Pathak and Rao (1967) for the same expected cost. Kulldorff's estimate as considered by Rao and Ghangurde (1969), however, remains superior to the proposed estimate.

Confidentiality, privacy and sensitivity in household surveys

Abstract: Editorial Note: The personal nature of questions included in some surveys is a matter of interest and possibly concern to statisticians. At the Editor's request Dr. Ware and Professor Caldwell take up this issue, their comments being largely based on experience gained from a project carried out in 1971.

Likelihoods with rounded data

Abstract: Summary Observed continuous random variates are effectively “discretized” by the limited accuracy with which they can be recorded. The effects of this on the likelihood function and on maximum likelihood estimation are often overlooked. Some published data for the exponential distribution are examined and the results compared with earlier studies. A note on estimation for the normal distribution is also given.

On some product type estimators in varying probability1

Abstract: SUMMARY It is shown that varying probability in product method estimation increases the efficiency of estimate under some usual conditions. Further, a necessary condition, for the product estimate from a sample drawn with pps and without replacement to be more efficient than the product estimate from a sample drawn with pps and replacement, has been obtained; incidentally, it is the same as obtained by Narain (1951) for varying probability without replacement to be more efficient than varying probability with replacement. The use of the R.H.C. sampling scheme has also been suggested, and its variance and bias has been shown to be always less than that of pps with replacement.