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.     

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.     

SOME REMARKS ON THE MEAN, MEDIAN, MODE AND SKEWNESS

Many sufficient conditions for inequalities about the mean, median, mode and skewness have been obtained. The paper presents a theorem that unifies already known results, gives some counter-examples, and considers the cases of Pearson distributions.     

ESTIMATION OF THE CONCENTRATION PARAMETERS OF THE FISHER MATRIX DISTRIBUTION ON 50(3) AND THE BINGHAM DISTRIBUTION ON Sq, q≥ 2

Two procedures are considered for estimating the concentration parameters of the Fisher matrix distribution for rotations or orientations in three dimensions. The first is maximum likelihood. The use of a convenient 1-dimensional integral representation of the normalising constant, which greatly simplifies the computation, is suggested. The second approach exploits the equivalence of the Fisher distribution for rotations in three dimensions, and the Bingham distribution for axes in four dimensions. We describe a pseudo likelihood procedure which works for the Bingham distribution in any dimension. This alternative approach does not require numerical integration. Results on the asymptotic efficiency of the pseudo likelihood estimator relative to the maximum likelihood estimator are given, and the two estimators are compared in the analysis of a well-known vectorcardiography dataset.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

LOSS OF INFORMATION ASSOCIATED WITH THE ORDER STATISTICS AND RELATED ESTIMATORS IN THE DOUBLE EXPONENTIAL DISTRIBUTION CASE

Fisher (1934) derived the loss of information of the maximum likelihood estimator (MLE) of the location parameter in the case of the double exponential distribution. Takeuchi & Akahira (1976) showed that the MLE is not second order asymptotically efficient. This paper extends these results by obtaining the (asymptotic) losses of information of order statistics and related estimators, and by comparing them via their asymptotic distributions up to the second order.     

SOME SIMPLE APPROXIMATIONS FOR THE DOUBLY NONCENTRAL z DISTRIBUTION

We give two simple approximations for evaluating the cumulative probabilities of the doubly noncentral z distribution. These can easily be used for evaluating the cumulative probabilities of the doubly noncentral F distribution as well. We compare our results with those obtained by Tiku (1965) using series expansion. An industrial situation where a quality characteristic of interest follows the doubly noncentral z distribution is also cited. However, in this case the exact probabilities could be calculated using results on the ratio of two normal variables.     

ON A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD RATIO CRITERIA

In this paper the exact distribution of a statistic with general moment function has been derived in a gamma series and also in a beta series. It is shown that the Box expansion can be obtained from the gamma series by collecting terms of the same order.     

ESTIMATION OF PARAMETERS IN THE MARGINAL FISHER DISTRIBUTION1

The Fisher distribution is frequently used as a model for the probability distribution of directional data, which may be specified either in terms of unit vectors or angular co-ordinates (co-latitude and azimuth). If, in practical situations, only the co-latitudes can be observed, the available data must be regarded as a sample from the corresponding marginal distribution. This paper discusses the estimation by Maximum Likelihood (ML) and the Method of Moments of the two parameters of this marginal Fisher distribution. The moment estimators are generally simpler to compute than the ML estimators, and have high asymptotic efficiency.     

A CUSUM SCHEME BASED ON THE EXPONENTIAL DISTRIBUTION FOR SURVEDLLANCE OF RARE CONGENITAL MALFORMATIONS

A method of monitoring the incidence of malformations is described. It is suitable for systems where the number of births between successive malformations is known or can be estimated with reasonable accuracy. The method utilises a cusum technique based on the exponential distribution to detect an increase in the incidence of malformations above a baseline level. Adequate information to enable the implementation of the method is presented. The proposed method compares favourably with others such as the Poisson cusum and the modified sets technique.     

CORRELATIONS AND CHARACTERIZATIONS OF THE UNIFORM DISTRIBUTION

Two characterizations of the uniform distribution on a suitable compact space are proved. These characterizations are applied to a number of particular examples of which the most interesting is the following: if X , Y and Z are independent n-vectors whose components are independent and identically distributed within a vector, then the pairwise independence of the product moment correlation coefficients between X , Y and Z implies that these vectors are normally distributed.     

ON THE CAUSALITY TEST IN TIME SERIES MODELS WITH HEAVY-TAILED DISTRIBUTION

ABSTRACT

In this article, we consider the problem of testing the Granger causality in stationary time series models with non-normal heavy-tailed distributions. We consider a normal mixture model to cover the heavy-tailed distribution, and propose a test statistic based on the partially adaptive estimator proposed by Phillips [1] Phillips, R.F. 1994. Partially Adaptive Estimation via a Normal Mixture. J. Econometics, 64: 123–144. [Crossref], [Web of Science ®] , [Google Scholar]. It is shown that the test statistic asymptotically follows a chi-squared distribution. Simulation results indicate that our test outperforms the conventional test based on the least squares estimator when the observations follow a heavy-tailed distribution.     

GOODNESS OF FIT FOR THE BINOMIAL DISTRIBUTION

Goodness of fit testing for the binomial distribution can be carried out using Pearson's X²_p statistic and its components. Applications of this technique are considered and compared with recently suggested empirical distribution function tests. Diagnostic use of components is discussed.     

A REGRESSION CHARACTERISATION OF THE MEIXNER HYPERGEOMETRIC DISTRIBUTION

The paper describes the Meixner hypergeometric distribution, characterised by properties of the regression of products of linear transformations of random variables with respect to residuals.     

ON QUANTILES OF SUMS

In this paper we investigate the relationship between the quantiles of a sum of independent continuous random variables and those of its components. Results concerning this relationship are given for the special cases of symmetric distributions, gamma distributions, and for the difference of identically distributed random variables.     

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.