Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

On generalized geometric distributions and improved estimation of batting average in cricket

Batting average is the most popular way of measuring a batsman's performance in cricket. However, in light of scores from not-out innings, the conventional way of computing the batting average is unsatisfactory from theoretical statistical perspective, as well as from intuitive and practitioner's point of view. We adopt alternative methods of calculating batting average, treating not-outs as right-censored data and using generalized class of geometric distributions (GGD) as models for the runs scored. In the proposed family of GGD, the generalization lies in the hazard of getting out possibly changing from one score to another. Each postulated structure of the hazards leads to a different member of the GGD family. Selection of appropriate member from the GGD family and maximum likelihood estimation of the hazard parameters in the model are discussed theoretically with illustrations. The proposed method subsumes the traditional average and product limit (Kaplan-Meier) estimate as the two extreme scenarios within this structure. We also discuss two alternative methods of estimating the true mean under the proposed framework and deliberate on issues while adopting these practices in practice.     

THE GENERALIZED SECANT HYPERBOLIC DISTRIBUTION AND ITS PROPERTIES

ABSTRACT

The properties of a family of distributions generalizing the secant hyperbolic are developed. This family consists of symmetric distributions, with kurtosis ranging from 1.8 to infinity, and includes the logistic as a special case, the uniform as a limiting case, and closely approximates the normal and Student's t-distributions with corresponding kurtosis. A significant difference between this family and Student's t is that for any member of the generalized secant hyperbolic family, all moments are finite. Further, technical difficulties associated with evaluating moments of Student's t (especially for fractional degrees of freedom) are not present with this family. The properties of the maximum likelihood and modified maximum likelihood estimates of the location and scale parameters for complete samples are considered. Examples illustrate the methods developed in this work.     

An approximated distribution of the gini's rank associatio coefficient

An approximate distribution is proposed for the Gini's rank association coefficient g which is, like Kendall's and Spearman's rank correlation coefficient, a statistic to test independence between two random variables. The purposed distribution can be simply transformed into a Student's T distribution; so, hypothesis testing is made much easier.     

On efficient inferences in familial-longitudinal binary models with two variance components

When a generalized linear mixed model (GLMM) with multiple (two or more) sources of random effects is considered, the inferences may vary depending on the nature of the random effects. For example, the inference in GLMMs with two independent random effects with two distinct components of dispersion will be different from the inference in GLMMs with two random effects in a two factor factorial design set-up. In this paper, we consider a familial-longitudinal model for repeated binary data where the binary response of an individual member of a family at a given time point is assumed to be influenced by the past responses of the member as well as two but independent sources of random family effects. For the estimation of the parameters of the proposed model, we discuss the well-known maximum-likelihood (ML) method as well as a generalized quasi-likelihood (GQL) approach. The main objective of the paper is to examine the relative asymptotic efficiency performance of the ML and GQL estimators for the regression effects, dynamic (longitudinal) dependence and variance parameters of the random family effects from two sources.     

Comparison of Efficiency of Stratified and Unequal Probability Sampling

In this article, a choice of the optimum sampling design to study a finite population is studied. Three sampling schemes are compared, viz., Sunter's procedure of unequal probability sampling, stratified sampling under optimum stratification, and simple random sampling without replacement. The comparison is made against a background of various correlation between stratification and survey variables and various variability in the variables. Under weak correlation and large variability, stratification appeared to be more efficient than Sunter's procedure. Under strong correlation and/or low variability in the variables, the latter procedure was the most efficient. Simple random sampling was usually the least efficient.     

Libby and Novick's generalized beta exponential distribution

A new family of skewed distributions is presented. Some properties and estimation procedures for Libby and Novick's generalized beta exponential distribution, a particular member of the family, are derived. Real applications using two original data sets are described to show superior performance versus at least six known models.     

Comparison of experiments for a class of positively dependent random variables

Using Blackwell's definition for comparison of experiments, it is shown that some sets of positively dependent random variables are less informative than similar sets of independent random variables. It is also shown that the information content of symmetric multivariate normal random vectors with a common known variance increases as the common correlation coefficient decreases. Some results which compare members of two-parameter exponential families are also included.     

A method to generate autocorrelated uniform random numbers

We present a simple, fast method to generate autocorrelated uniform random numbers. The “sum of uniforms” method adds a pair of U(0,1) random numbers, transforms the sum to a third U(0,1) random number, and uses this third random number as one member of the next pair. The method produces any desired level of positive or negative correlation between successive random numbers.     

Serial correlation or random subject effects

In longitudinal data analysis with random subject effects, there is often within subject serial correlation and possibly unequally spaced observations. This serial correlation can be partially confounded with the random between subject effects. In real data, it is often not clear whether there is serial correlation, random subject effects or both. Using inference based on the likelihood function, it is not always possible to identify the correct model, especially in small samples. However, it is important that some effort be made to attempt to find a good model rather than just making assumptions. This often means trying models with random coefficients, with serial correlation, and with both. Model selection criteria such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used. The problem of modelling serial correlation with unequally spaced observations is addressed. A real data example is presented where there is an apparent heterogeneity of variances, possible serial correlation and between subject random effects. In this example, it turns out that the random subject effects explains both the serial correlation and the variance heterogeneity.     

A complete solution to the problem of robustness of Grubbs's test

Srivastava (1980) showed that Grubbs's test for detecting a univariate outlier is robust against the effect of intraclass correlation structure. Young, Pavur, and Marco (1989) extended this result by proving that both the significance level and the power of Grubbs's test remain unchanged within a wider family of dispersion matrices, introduced by Baldessari (1966) in a different context. In this note, we derive a complete solution of the problem by establishing that the characteristics of Grubbs's test are invariant with respect to a given dispersion matrix if and only if it has Baldessari's structure.     

THE WRAPPED t FAMILY OF CIRCULAR DISTRIBUTIONS

This paper considers the three‐parameter family of symmetric unimodal distributions obtained by wrapping the location‐scale extension of Student's t distribution onto the unit circle. The family contains the wrapped normal and wrapped Cauchy distributions as special cases, and can be used to closely approximate the von Mises distribution. In general, the density of the family can only be represented in terms of an infinite summation, but its trigonometric moments are relatively simple expressions involving modified Bessel functions. Point estimation of the parameters is considered, and likelihood‐based methods are used to fit the family of distributions in an illustrative analysis of cross‐bed measurements. The use of the family as a means of approximating the von Mises distribution is investigated in detail, and new efficient algorithms are proposed for the generation of approximate pseudo‐random von Mises variates.     

Inferences Based on a Skipped Correlation Coefficient

The most popular method for trying to detect an association between two random variables is to test H ₀ ?:?ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H ₀ ?:?ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

A new extension of the FGM copula for negative association

In this paper we introduced a single parameter, absolutely continuous and radially symmetric bivariate extension of the Farlie-Gumbel-Morgenstern (FGM) family of copulas. Specifically, this extension measures the higher negative dependencies than most FGM extensions available in literature. Closed-form formulas for distribution, quantile, density, conditional distribution, regression, Spearman's rho, Kendall's tau, and Gini's gamma are obtained. In addition, a formula for random variate generations is presented in closed-form to facilitate simulation studies. We conduct both paired and multiple comparisons with Frank, Gaussian, and Plackett copulas to investigate the performance based on Vuong's test. Furthermore, the new copula is compared with Frank, Gaussian, and Plackett copulas using both Kolmogorov-Smirnov and Cramér-von Mises type test statistics. Finally, a bivariate dataset is analyzed to compare and illustrate the flexibility of the new copula for negative dependence.     

On the distribution of linear combinations of the components of a Dirichlet random vector

On making use of a result of Imhof, an integral representation of the distribution function of linear combinations of the components of a Dirichlet random vector is obtained. In fact, the distributions of several statistics such as Moran and Geary's indices, the Cliff‐Ord statistic for spatial correlation, the sample coefficient of determination, F‐ratios and the sample autocorrelation coefficient can be similarly determined. Linear combinations of the components of Dirichlet random vectors also turn out to be a key component in a decomposition of quadratic forms in spherically symmetric random vectors. An application involving the sample spectrum associated with series generated by ARMA processes is discussed.     

On blest's measure of rank correlation

Blest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.     

A note on the accuracy of fisher's approximation to the large sample variance of an intraclass correlationa

The adequacy of Fisher's approximation to the large sample variance of an intraclass correlation is investigated in the context of family studies. It is found that the approximation is highly accurate in samples of moderately large size (≧ 30 families), and can also be used for significance-testing under a broad range of circumstances. The exact sampling of distribution of the intraclass correlation coefficient is also derived.     

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.     

Monitoring correlation change in a sequence of random variables

We propose a monitoring procedure to test for the constancy of the correlation coefficient of a sequence of random variables. The idea of the method is that a historical sample is available and the goal is to monitor for changes in the correlation as new data become available. We introduce a detector which is based on the first hitting time of a CUSUM-type statistic over a suitably constructed threshold function. We derive the asymptotic distribution of the detector and show that the procedure detects a change with probability approaching unity as the length of the historical period increases. The method is illustrated by Monte Carlo experiments and the analysis of a real application with the log-returns of the Standard & Poor's 500 (S&P 500) and IBM stock assets.