MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

A NEW CALIBRATION METHOD OF CONSTRUCTING EMPIRICAL LIKELIHOOD-BASED CONFIDENCE INTERVALS FOR THE TAIL INDEX

Empirical likelihood has attracted much attention in the literature as a nonparametric method. A recent paper by Lu & Peng (2002) [Likelihood based confidence intervals for the tail index. Extremes 5, 337–352] applied this method to construct a confidence interval for the tail index of a heavy‐tailed distribution. It turns out that the empirical likelihood method, as well as other likelihood‐based methods, performs better than the normal approximation method in terms of coverage probability. However, when the sample size is small, the confidence interval computed using the χ² approximation has a serious undercoverage problem. Motivated by Tsao (2004) [A new method of calibration for the empirical loglikelihood ratio. Statist. Probab. Lett. 68, 305–314], this paper proposes a new method of calibration, which corrects the undercoverage problem.     

COMPARATIVE CALIBRATION WITH SUBGROUPS

The main object of this paper is to consider structural comparative calibration models under the assumption that the unknown quantity being measured is not identically distributed for all units. We consider the situation where the mean of the unknown quantity being measured is different within subgroups of the population. Method of moments and maximum likelihood estimators are considered for estimating the parameters in the model. Large sample inference is facilitated by the derivation of the asymptotic variances. An application to a data set which indeed motivated the consideration of such general model and was obtained by measuring the heights of a group of trees with five different instruments is considered.     

A New STATISTIC FOR DETECTING INFLUENTIAL OBSERVATIONS IN A SCHEFFE' TYPE CALIBRATION CURVE

A statistic for identifying influential observations in calibration is given. The statistic is easy to interpret, and provides a useful measure of influence for Scheffé type calibration curves.     

THE SIMPLE LINEAR CALIBRATION PROBLEM AS AN OPTIMAL EXPERIMENTAL DESIGN

ABSTRACT

The aim of this paper is to consider the linear calibration problem through an optimal design approach, to evaluate the approximate variance of the calibrating value and provide approximate confidence intervals. The sequential approach through Stochastic Approximation is also applied to obtain a D-optimal design is considered.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

ESTIMATION OF THE PARAMETER OF A POISSON DISTRIBUTION USING A LINEX LOSS FUNCTION

This paper considers estimation of the parameter of a Poisson distribution using Varian's (1975) asymmetric LINEX loss function L (δ) = b{exp(aδ) - aδ - 1}, where δ is the estimation error and b > 0, a 0. It is shown that for a < 0, the sample mean X¯ is admissible whereas for a > 0, X¯ is dominated by c*X¯, where c*= (n/a)log(1+a/n). Practical implications of this result are indicated. More general results, concerning the admissibility of estimators of the form cX¯+ d are also presented.     

A NEW MEASURE OF ROTATABILITY OF RESPONSE SURFACE DESIGNS

This paper proposes a new measure of rotatability of a response surface design. It is based on a comparison between the design moments of a given design and the design moments of a rotatable design closest to the given design in the least squares sense. This measure is easier to calculate than Khur?s measure. An alternative easier non-geometrical way of deriving Khuri's measure is also presented. Both measures are calculated to illustrate some designs considered by Khuri.     

ESTIMATION OF THE NUMBER OF PEOPLE IN A DEMONSTRATION

It is of interest to estimate the size of a crowd in a demonstration. We propose a practical method to obtain an estimate of the size of the crowd and its standard error. This method has been implemented in practice and, compared with other counting methods, is found to be more efficient, more timely and have less scope for bias. The method described in this paper was motivated by the annual 1 July demonstrations in Hong Kong, and data from the 2006 demonstration are used as an example of the proposed method.     

A COMPARISON OF TESTS OF UNIFORMITY FOR SPHERICAL DATA

A comparison is made af three tests of uniformity for spherical vector data, and of the corresponding tests of uniformity for spherical axial data.     

TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS

A previous paper which studied the distribution of the smallest distance between N independent random points on the surface of a sphere is generalised to higher dimensions in order to study the distribution of the largest sample correlation coefficient between a set of independent normally distributed variables. Inclusion-exclusion arguments provide accurate bounds for the tail of this distribution, and by another argument more exact bounds are also found, one of which is an improvement on the result in the previous paper. Bounds are also found for the power of the test against the alternative hypothesis that one only of the population correlation coefficients is non-zero. The test is also shown to be the likelihood ratio test against the latter alternative.     

A FINE-TUNED ESTIMATOR OF A GENERAL CONVERGENCE RATE

A general rate estimation method based on the in‐sample evolution of appropriately chosen diverging/converging statistics has recently been proposed by D.N. Politis [C. R. Acad. Sci. Paris, Ser. I, vol. 335, pp. 279–282, 2002] and T. McElroy & D.N. Politis [Ann. Statist., vol. 35, pp. 1827–1848, 2007]. In this paper, we show how a modification of the original estimators achieves a competitive rate of convergence. The modified estimators require the choice of a tuning parameter; an optimal such choice is generally a non‐trivial problem in practice. Some discussion to that effect is given, as well as a small simulation study in a heavy‐tailed setting.     

A GENEALOGY OF CORRESPONDENCE ANALYSIS

Over the past half a century correspondence analysis has grown from a little known statistical technique designed to graphically depict the association structure of categorical variables that form a contingency table to a very popular tool used in a wide variety of disciplines. Despite this growth, correspondence analysis remains relatively unknown in some parts of the world, including the Australasian statistical community. This paper provides a non‐technical, bibliographic exploration of correspondence analysis. We take a step back to view the development of this statistical technique and provide a brief account of its genealogy with a selection of over 270 key publications that have contributed to its growth. We also look at its maturity over the decades.     

A NOTE ON THE DEMOGRAPHIC IMPACT OF CONTRACEPTION IN A MATHEMATICAL MODEL OF HUMAN REPRODUCTION

The application of mathematical models of human reproduction to the study of reproductive behaviour as a function of contraceptive behaviour was pioneered by Perrin and Sheps (1963). More recent work in this area continues to focus on an examination of the birth rate as the principal dependent variable. This note suggests an alternative procedure for studying the demographic impact of contraception through the analysis of birth intervals. A mathematical model is formulated for the waiting time between successive live births, and a procedure is described for incorporating into the model certain contraceptive parameters. A controlled experiment is then performed to determine the effect of these parameters on expected child spacing patterns.     

A NOTE ON THE NUMBER OF DUST PARTICLES IN A SAMPLE OF AIR-BORNE MINE DUST

A stochastic model is proposed for the distribution of the number of particles in a sample of air-borne mine dust, and it, is shown that in special circumstances the distribution will be that of a Poisson variate.
I am indebted to Professor Bartlett for drawing my attention to his own more general result and to the referee for the suggestion that led to the unconditional result of Section 4.     

A NEW FAMILY OF DIVERGENCE MEASURES FOR TESTS OF FIT

The aim of this work is to investigate a new family of divergence measures based on the recently introduced Basu, Harris, Hjort and Jones (BHHJ) measure of divergence (Biometrika 85 , 549–559). The new family is investigated in connection with hypothesis testing problems, and new test statistics are proposed. Simulations are performed to check the appropriateness of the proposed test statistics.     

A TEST OF THE LOCATION PARAMETER OF THE EXPONENTIAL DISTRIBUTION

In this paper, we propose a procedure for testing the location parameter of the exponential distribution for certain alternative hypotheses, which could result in the early rejection of the null hypothesis. This is a consequence of the monotone property of the test statistic which is based on the extremal quotient. The test being scale-free does not require the scale parameter to be known.     

INFERENCE ABOUT THE MEAN OF A POISSON DISTRIBUTION IN THE PRESENCE OF A NUISANCE PARAMETER

In this note we examine the problem of estimating the mean of a Poisson distribution when a nuisance parameter is present. Using a condition of Cox (1958) about ancillarity in the presence of a nuisance parameter, we justify that inference about the parameter should be carried out using the conditional distribution given the appropriate ancillary statistics. A small simulation study has been done to compare the performance of the conditional likelihood approach and the standard likelihood approach.