Smooth tests of goodness of fit for regular distributions

Smooth tests of goodness of fit based on orthonormal functions for location-scale families were introducedin Rayner and Best (1986).This paper extends this class of tests from location -scale families to ‘regular’ families. The extension preserves the desirable properties of the class, such as weak optimality, accessible components and convenient distribution theory     

Probabilistic tolerance design for a subsystem under burr distribution using taguchi's loss functions

Taguchi (1986) has derived tolerances for subcomponents, subsystems, parts and materials in which the relationship between a higher-level (Y) and a lower-level (X) quality characteristic is assumed to be deterministic and linear, namely, Y=_α+βX, without an error term. Tsai (1990) developed a probabilistic tolerance design for a subsystem in which a bivariate normal distribution between the above two quality characteristics as well as Taguchi's quadratic loss function were considered together to develop a closed form solution of the tolerance design for a subsystem. The Burr family is very rich for fitting sample data, and has positive domain. A bivariate Burr distribution can describe a nonlinear relationship between two quality characteristics, hence, it is adopted instead of a bivariate normal distribution and the simple solutions of three probabilistic tolerance desings for a subsystem are obtained for three cases of “nominal-is-best”, “smaller-is-berrer”, and “larger-is-beter” quality characteristics, by using Taguchi’ los functions, respectively.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.     

Correlation between the numbers of two types of children in a family with the mpsd for the family size

The family size (sibship size) N is regarded as an integer-valued random variable having a Modified Power Series distribution (MPSD) of Gupta (1974). The family produces two types of children, with probabilities p and q (p+q =1) . It is proved that the correlation between the numbers B and C of these children is positive or negative according as the function log f(θ) is convex or concave with respect to the function g(θ), (see Section 2). This condition is a simple and a natural extension of the one given by Rao et al (1973). Several examples are discussed to illustrate the result.     

The asymptotic moment profile and maximum likelihood: Applications to gamma and gamma ratio densities

In previous papers ( Bowman and Shenton, 1998, 1999a ) we have given expressions for the asymptotic skewness and kurtosis for maximum likelihood estimators in the case of several parameters. Skewness is measured by the third standardized central moment, and kurtosis by the fourth standardized central moment. Moments of the basic structure are assumed to exist. The overarching entity is the covariance matrix ( Hessian form ), and elements of its inverse. These entities involve Stieltjes integrals relating to sums of products of multiple derivatives linked to the basic structure. The first paper dealt with skewness and gives a simple expression read¬ily computerized. The second paper is devoted to the forth standardized central moment and although a certain simplification is discovered, the resulting formula is still somewhat complicated, ft is surprising to find that the asymptotic kurtosis in general requires the evaluation of several hundred components. The present paper studies cases involving estimator for two parameter gamma and one, and three pa¬rameter gamma ratio densities, and mentions strategies aimed at avoiding algebraic and numerical errors.     

Dependence structures and asymptotic properties of Baker's distributions with fixed marginals

We investigate the properties of Baker's (2008) bivariate distributions with fixed marginals and their multivariate extensions. The properties include the weak convergence to the Fréchet–Hoeffding upper bound, the product-moment convergence, as well as the dependence structures TP₂ (totally positive of order 2), or MTP₂ (multivariate TP₂). In proving the weak convergence, a generalized local limit theorem for binomial distribution is provided.     

Estimation of misclassification probabilities by bootstrap methods

Several methods have been proposed to estimate the misclassification probabilities when a linear discriminant function is used to classify an observation into one of several populations. We describe the application of bootstrap sampling to the above problem. The proposed method has the advantage of not only furnishing the estimates of misclassification probabilities but also provides an estimate of the standard error of estimate. The method is illustrated by a small simulation experiment. It is then applied to three published, well accessible data sets, which are typical of large, medium and small data sets encountered in practice.     

Properties of Entropies of Record Values in Reliability and Life Testing Context

In this article, the entropies of record value distributions from some continuous probability models are computed and their properties are investigated, both analytically when feasible, and numerically. In an attempt to establish relationships between entropies of the parent and the corresponding record value distributions, the entropies of record value distributions associated with the uniform, exponential, Weibull, classical Pareto, normal, gamma, beta, and Cauchy distributions are considered in this article. The entropy of record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, have tractable closed forms. The entropies of record value distributions associated with the normal, gamma, beta, and Cauchy distributions do not have tractable closed forms and need further investigations. Some general conclusions are drawn in the final section.     

Extensions of multiple linear regression

Many statistical methods are linked together through their connection with weighted least squares and hence regression. This article reviews these connections, emphasising the iteratively weighted least squares algorithm.