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 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.     

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.     

ON THE DISTRIBUTION OF THE REGRESSION COEFFICIENT MATRIX OF A NORMAL DISTRIBUTION

In this paper we generalize a result of Kshirsagar's (1960, pp. 83–84) on the distribution of the regression coefficient matrix for a multivariate normal population.     

A ROBUST TEST FOR MULTIPLE COMPARISONS OF CORRELATION COEFFICIENTS1

A limiting expression is derived for the tail of the distribution of the maximum of a set of product moment correlation coefficients. The technique used is quite general and may be applied to non-normal observations as well as to rank correlation coefficients. The result obtained for the latter leads to a test procedure for multiple comparisons of these non-parametric measures of dependence.     

A LIMIT THEOREM FOR THE GALTON-WATSON PROCESS WITH IMMIGRATION

It is difficult, in general, to optain an explicit expression for the limiting-stationary distribution, when such a distribution exists, of the process in which teh individuals reproduce as in a Galton-Wastson process, but are also subject to an independent immigration component at each generation. The main result of this paper is a limit theorem which suggests a means of approximating this distribution by a gamma density, when the mean of the offspring distribution is less than, but close to, unity. Following along the same lines, it is easy to show that a similar limit theorem holds for the asymptotic conditional limit distribution of an ordinary subcritical Galton-Watson process, whereby this distribution approaches the exponential as the offspring mean approaches unity.     

ON ASSESSING THE POTENTIAL SEVERITY OF AN OUTBREAK OF A RARE INFECTIOUS DISEASE: A BAYESIAN APPROACH

In this paper an infective population, during the early stages of the outbreak of a disease, is approximated by a Galton-Watson process. Attention is focused on the threshold theorem which assesses an epidemic as major if the associated Galton-Watson process is supercritical (i.e. the mean μ of the offspring distribution is greater than one). A Bayesian formulation is adopted together with the assumption of a power series offspring distribution and an approximate form is found for P(μ >1) computed from the posterior distribution of μ. Exact results are given for the case of a Poisson offspring distribution. The results are illustrated with applications to three sets of data on smallpox outbreaks. The Bayesian approach has a number of advantages over classical methods and in particular allows the cases μ < 1, μ > 1 to be treated without distinction.     

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.     

TESTING THE CONCORDANCE OF EVENTS WITH THEIR PROBABILITIES

A test due to D. R. Cox (1958a, 1958b) for the agreement between an observed set of independent events and independently prescribed probabilities is considered. Although the exact distribution can be found for small numbers of events this is often not possible. A method of modifying the scoring system slightly is shown to result in much simpler calculation with very little loss in power. The exact distribution is desirable because of slow convergence to normality. The logic of using a simulation test is considered.     

DISTRIBUTION OF THE LEAST SQUARES ESTIMATOR IN A FIRST-ORDER AUTOREGRESSIVE MODEL

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. A uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exist a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.     

A Simple Simulational Algorithm for Generating Progressive Type-II Censored Samples

We establish an independence result concerning a progressive Type-II censored sample from the uniform distribution. This result is used to present a simple and efficient simulational algorithm for generating a progressive Type-II censored sample from any continuous distribution.     

SOME PROPERTIES OF A RANDOM LINEAR DIFFERENCE EQUATION1

Some simple conditions are given for the absolute continuity of the limiting distribution of a random linear difference equation. These results are applied to the super-critical Bellman-Harris branching process with immigration. When the coefficients of the difference equation are non-negative and there is no limiting distribution, it is shown that the asymptotic behaviour of the solutions is the same as that of the partial sums of a divergent random power series. A number of limit theorems are given for the latter situation.     

A characterization of a bimodal distribution

As a sequel to Khlnchirie's definition of unimodality a bimodal distribution function is defined, A characterization for such a distribution is given using the well-known result of Khinchine on unimodality and a characterization theorem for a U-shaped probability density function by Ghosh and Shanbhag(1972).     

A characteristic function-based approach to approximate maximum likelihood estimation

The choice of the summary statistics in approximate maximum likelihood is often a crucial issue. We develop a criterion for choosing the most effective summary statistic and then focus on the empirical characteristic function. In the iid setting, the approximating posterior distribution converges to the approximate distribution of the parameters conditional upon the empirical characteristic function. Simulation experiments suggest that the method is often preferable to numerical maximum likelihood. In a time-series framework, no optimality result can be proved, but the simulations indicate that the method is effective in small samples.     

A flexible approach for multivariate mixed-effects models with non-ignorable missing values

We propose a flexible model approach for the distribution of random effects when both response variables and covariates have non-ignorable missing values in a longitudinal study. A Bayesian approach is developed with a choice of nonparametric prior for the distribution of random effects. We apply the proposed method to a real data example from a national long-term survey by Statistics Canada. We also design simulation studies to further check the performance of the proposed approach. The result of simulation studies indicates that the proposed approach outperforms the conventional approach with normality assumption when the heterogeneity in random effects distribution is salient.     

A pivot function and its limiting distribution: applications in goodness of fit and testing hypothesis

ABSTRACT

In this paper, a pivot function which is in terms of the sample and the underlying population distribution is introduced. It is assumed that the population distribution is continuous and strictly increasing on its support. Then, the martingale central limit theorem is applied to prove that limiting distribution of the pivot function is the standard normal. Interestingly, this result provides a unified procedure that can be applied for the goodness of fit, and for the purpose of parametric and nonparametric inferences, for the populations having distribution functions that are continuous and strictly increasing on their supports. The method is fairly simple and can be easily applied.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

INCOMPLETE SAMPLING FOR SELECTION OF A FIXED PROPORTION

Testing only a sample of a population from which a fixed proportion are to be chosen for further use is considered in relation to the balance between testing costs and extra value achieved by culling the poorest members of the population. It is shown that in some circumstances the greatest return is obtained by testing some, but not all, members of the population. Optimum sampling fractions are deducted for selection on a quantitative chaaracter having a normal distribution, and approximate results are given for selection on a qualitative character.     

THE SOJOURN TIME DISTRIBUTION OF AN ALTERNATING RENEWAL PROCESS

Ordinary, modified, and equilibrium alternating renewal processes are defined in such a way that the distribution of the sojourn time in one state of a two-state system may be considered conditional on the starting state. The double Laplace transform of a non-negative stochastic process is also defined, and used to express a result of Takacs (1957) on sojourn time distributions.     

ESTIMATION OF QUANTILES FOR A FRÉCHET-TYPE DISTRIBUTION

Estimation of high quantiles of a distribution in the domain of attraction of the Fréchet distribution is based on the extremal distribution of the k largest order statistics. The problem is treated by a local maximum likelihood method on a three parameter model. The estimators are shown to be asymptotically consistent for the whole range of the tail index parameter.