THE MAXIMUM RESISTANCE OF TESTS

The breakdown point of an estimator is the smallest fraction of contamination that can force the value of the estimator beyond the boundary of the parameter space. It is well known that the highest possible breakdown point, under equivariance restrictions, is 50% of the sample. However, this upper bound is not always attainable. We give an example of an estimation problem in which the highest possible attainable breakdown point is much less than 50% of the sample. For hypothesis testing, we discuss the resistance of a test and propose new definitions of resistance. The maximum resistance to rejection (acceptance) is the smallest fraction of contamination necessary to force a test to reject (fail to reject) regardless of the original sample. We derive the maximum resistances of the t-test and sign test in the one-sample problem and of the t-test and Mood test in the two-sample problem. We briefly discuss another measure known as the expected resistance.     

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.     

ESTIMATING THE NUMBER OF VISITS TO THE DOCTOR

The frequency of doctor consultations has direct consequences for health care budgets, yet little statistical analysis of the determinants of doctor visits has been reported. We consider the distribution of the number of visits to the doctor and, in particular, we model its dependence on a number of demographic factors. Examination of the Australian 1995 National Health Survey data reveals that generalized linear Poisson or negative binomial models are inadequate for modelling the mean as a function of covariates, because of excessive zero counts, and a mean‐variance relationship that varies enormously over covariate values. A negative binomial model is used, with parameter values estimated in subgroups according to the discrete combinations of the covariate values. Smoothing splines are then used to smooth and interpolate the parameter values. In effect the mean and the shape parameters are each modelled as (different) functions of gender, age and geographical factors. The estimated regressions for the mean have simple and intuitive interpretations. However, the dependence of the (negative binomial) shape parameter on the covariates is more difficult to interpret and is subject to influence by extreme observations. We illustrate the use of the model by estimating the distribution of the number of doctor consultations in the Statistical Local Area of Ryde, based on population numbers from the 1996 census.     

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS OF THE TRUNCATED CAUCHY DISTRIBUTION

Truncated Cauchy distribution with four unknown parameters is considered and derivation and existence of the maximum likelihood estimates is investigated here. We provide a sufficient condition for the maximum likelihood estimate of the scale parameter to be finite, and also show that the condition is necessary for sufficiently large samples. Note that all the moments of the truncated Cauchy distribution exist which makes it much more attractive as a model when compared to the regular Cauchy. We also study, using simulations, the small sample properties of the maximum likelihood estimates.     

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.     

TEST OF FIT FOR THE INVERSE GAUSSIAN AND GAMMA DISTRIBUTIONS UNDER CENSORING

The problem of testing the fit of the inverse Gaussian and the gamma distribution when the sample is censored and some of the parameters are unknown, is studied. Empirical Distribution Function (EDF) statistics, namely Cramér-von Mises' W ² and the Anderson-Darling's A ², are used. The limiting covariance functions of the corresponding empirical processes are derived. Asymptotic percentage points are given for some parameter values and censoring proportions. Moreover, a numerical routine is made available upon request, to obtain p-values for both test statistics, thus eliminating the need of tables and interpolation. Finally, a simple Monte Carlo study is presented to evaluate first, the approximation when using the asymptotic distributions in finite samples and second, to support the use of estimated parameter values instead of the unknown parameters needed in the limiting covariance function.     

THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.     

ON THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER VARIABLE SELECTION

This paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ= a^Tβ where a is specified. When, as a first step, a data‐based variable selection (e.g. minimum Akaike information criterion) is used to select a model, it is common statistical practice to then carry out inference about θ, using the same data, based on the (false) assumption that the selected model had been provided a priori. The paper considers a confidence interval for θ with nominal coverage 1 ‐ α constructed on this (false) assumption, and calls this the naive 1 ‐ α confidence interval. The minimum coverage probability of this confidence interval can be calculated for simple variable selection procedures involving only a single variable. However, the kinds of variable selection procedures used in practice are typically much more complicated. For the real‐life data presented in this paper, there are 20 variables each of which is to be either included or not, leading to 2²⁰ different models. The coverage probability at any given value of the parameters provides an upper bound on the minimum coverage probability of the naive confidence interval. This paper derives a new Monte Carlo simulation estimator of the coverage probability, which uses conditioning for variance reduction. For these real‐life data, the gain in efficiency of this Monte Carlo simulation due to conditioning ranged from 2 to 6. The paper also presents a simple one‐dimensional search strategy for parameter values at which the coverage probability is relatively small. For these real‐life data, this search leads to parameter values for which the coverage probability of the naive 0.95 confidence interval is 0.79 for variable selection using the Akaike information criterion and 0.70 for variable selection using Bayes information criterion, showing that these confidence intervals are completely inadequate.     

ASYMPTOTIC STABILITY OF THE OSCV SMOOTHING PARAMETER SELECTION

The smoothing parameter selection by the one-sided cross-validation (OSCV) method is completely automatic in that it does not require extra parameters estimation. Also it reduces the variability comparable to that of plug-in rules. In this paper we derive analytically the asymptotic variance of the smoothing parameter selected by OSCV. It shows the dependency of the stability on the one-sided kerenl and tells the possibility of the optimal one-sided kernel which minimizes the asymptotic variability.     

AN EVALUATION OF NON-ITERATIVE METHODS FOR ESTIMATING THE LINEAR-BY-LINEAR PARAMETER OF ORDINAL LOG-LINEAR MODELS

Parameter estimation for association and log-linear models is an important aspect of the analysis of cross-classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear-by-linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non-iterative procedures for estimating the linear-by-linear parameter for ordinal log-linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non-iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered.     

A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

THE ESTIMATION OF PLUTONIUM CONCENTRATIONS IN SOIL

In this paper we examine the consequences, for statistical analysis and interpretation, of the particulate nature of radioactive contamination of a nuclear weapons test site. We propose a probabilistic model which incorporates the particulate nature of the contamination and which is simple enough to be statistically fitted to the data. Parameter estimation involves the reconciliation and combination of measurements of (a) 59.5 ke V gamma rays from americium-241, a decay product of plutonium-241, using a portable medium resolution NaI detector, on a regular survey grid at a test site and (b) 59.5 ke V radiation from soil samples obtained at grid points. The implications of the model for measurement of levels of contamination are considered.     

A BOOTSTRAP TEST OF THE GOODNESS-OF-FIT OF THE MULTIVARIATE TOLERANCE MODEL

The multivariate tolerance (MT) model is used to model multi-agent bioassay survival responses. When the MT model is appropriate, it can aid in identifying the biological mechanism of the survival response or motivate tolerance-based error reduction. A bootstrap test for the goodness-of-fit of the MT model is proposed. The proposed test has little level error, useful power, and makes a novel adjustment for the multiple testing character of the question of MT model goodness-of-fit. This adjustment may be more generally applicable in bootstrap testing.     

LINEAR MINIMAX ESTIMATORS FOR ESTIMATING A FUNCTION OF THE PARAMETER

In this note we provide sufficient conditions for the minimaxity of linear estimators of the form aX+b in the one-parameter exponential family for estimating a differentiable function g(θ) with normalized quadratic loss. We provide some examples which show that the natural estimator X is minimax in estimating a function of the parameter (different from the mean).     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

NONLINEAR TUBE-FITTING FOR THE ANALYSIS OF ANATOMICAL AND FUNCTIONAL STRUCTURES

We are concerned with the estimation of the exterior surface and interior summaries of tube-shaped anatomical structures. This interest is motivated by two distinct scientific goals, one dealing with the distribution of HIV microbicide in the colon and the other with measuring degradation in white-matter tracts in the brain. Our problem is posed as the estimation of the support of a distribution in three dimensions from a sample from that distribution, possibly measured with error. We propose a novel tube-fitting algorithm to construct such estimators. Further, we conduct a simulation study to aid in the choice of a key parameter of the algorithm, and we test our algorithm with validation study tailored to the motivating data sets. Finally, we apply the tube-fitting algorithm to a colon image produced by single photon emission computed tomography (SPECT) and to a white-matter tract image produced using diffusion tensor imaging (DTI).     

ESTIMATION OF THE CONDITIONAL PROBABILITY OF TUBERCULOSIS GIVEN THE RESPONSE OF A MANTOUX TEST

For the planning of community tuberculosis, prophylaxis one must know the prevalence of tuberculosis (TB) infection as a function of response to the Mantoux intradermal tuberculin test; the standard test for indicating Mycobacterium TB infection. The skin induration size used to select individuals for prophylaxis must be chosen carefully, in view of the costs associated with carrying out and supervising such prophylaxis. The Mantoux test was used to obtain measurements on adolescents in metropolitan Victoria and on a small sample of adolescents with clinical TB. These data are employed to obtain estimates and to construct upper confidence bounds for the conditional probability of TB infection, given the level of Mantoux response. Two conservative methods are presented; one is ‘nonparametric’, the other ‘semiparametric’. The analyses indicate that for responses up to and including 13 mm, the probability of TB infection is less than.07 with ninety-five percent confidence.     

THE EXACT DISTRIBUTION OF THE BELL-DOKSUM TEST OF INDEPENDENCE

An expression for the exact distribution of the Bell-Doksum test of independence based upon the use of order statistics from normally distributed random samples is derived and a table of critical values is provided. Two extensions of the theory are considered and possible applications to Poisson processes and queueing theory are noted.     

TESTING HYPOTHESES ABOUT THE COMMON MEAN OF TWO NORMAL POPULATIONS

Three tests are considered concerning the common mean of two normal populations: (1) an F test based on a sample from one population, (2) a test based on the addition of the F statistics from independent samples from two popultions (proposed), and (3) a test based on the maximum of the F statistics from two independent samples from two populations. A condition under which test (2) is locally more powerful than test (1) is given. As the test statistic in test (2) does not follow a standard distribution, a formula for approximating the observed significance level is provided. A simulation study is used to compare the power of these tests.