Recursive Schemes for Calculating Common Cumulative Distributions

This article gives a method for obtaining accurate (5 decimal places) estimates of nine common cumulative distributions. Starting with a positive series expansion, we use the common ratio of each term to the preceding term and proceed as with a geometric series (the ratio may involve the term number). This avoids calculating terms in the the numerator or denominator which can be large enough to overflow or small enough to underflow the machine. The method is fast because it eliminates the necessity of calculating each term of the series in its entirety.     

Inference about the Ratio of Two Parameters,with Application to Whale Censusing

Inference for the quotient of two parameters estimated separately may be obtained by the delta method. When the distribution of linear transformations involving the numerator and the denominator is available, more exact and elementary methods may be used. Non-Bayesian and Bayesian approaches are developed. The application of the methods to estimating the stock abundance of Northeastern Atlantic minke whales, where the ratio is a raw estimate divided by a measure of observation efficiency, is explained and discussed. The Bayesian approach allows exact inference in quite general situations using only a single, rapidly implemented, one-dimensional numerical integration. A simple analytic approximation is given for the common situation where the joint posterior distribution of the numerator and denominator can be approximated by a normal distribution that gives very little probability to negative values of the denominator. The Bayesian approach also permits the incorporation of model uncertainty (or disagreement) in a natural way, and this was the basis for the conclusions of the International Whaling Commission Scientific Committee at its 1990 meeting.     

Confidence Bounds for Positive Ratios of Normal Random Variables

Some applications of ratios of normal random variables require both the numerator and denominator of the ratio to be positive if the ratio is to have a meaningful interpretation. In these applications, there may also be substantial likelihood that the variables will assume negative values. An example of such an application is when comparisons are made in which treatments may have either efficacious or deleterious effects on different trials. Classical theory on ratios of normal variables has focused on the distribution of the ratio and has not formally incorporated this practical consideration. When this issue has arisen, approximations have been used to address it. In this article, we provide an exact method for determining (1 ? α) confidence bounds for ratios of normal variables under the constraint that the ratio is composed of positive values and connect this theory to classical work in this area. We then illustrate several practical applications of this method.     

Characterizations Based on Cumulative Residual Entropy of First-Order Statistics

Two different distributions may have equal cumulative residual entropy (CRE), thus a distribution cannot be determined by its CRE. In this article, we explore properties of the CRE and study conditions under which the CRE of the first-order statistics can uniquely determines the parent distribution. Weibull family is characterized through ratio of the CRE of the first-order statistics to its expectation. We have also some bounds for the CRE of residual lifetime of a series system.     

Testing the homogeneity of several two‐parameter populations

The authors extend Fisher's method of combining two independent test statistics to test homogeneity of several two‐parameter populations. They explore two procedures combining asymptotically independent test statistics: the first pools two likelihood ratio statistics and the other, score test statistics. They then give specific results to test homogeneity of several normal, negative binomial or beta‐binomial populations. Their simulations provide evidence that in this context, Fisher's method performs generally well, even when the statistics to be combined are only asymptotically independent. They are led to recommend Fisher's test based on score statistics, since the latter have simple forms, are easy to calculate, and have uniformly good level properties.     

New Goodness of Fit Tests Based on Stochastic EDF

A new approach of randomization is proposed to construct goodness of fit tests generally. Some new test statistics are derived, which are based on the stochastic empirical distribution function (EDF). Note that the stochastic EDF for a set of given sample observations is a randomized distribution function. By substituting the stochastic EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling, Berk–Jones, and Einmahl–Mckeague statistics, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. In comparison to existing tests, it is shown, by a simulation study, that the new test statistics are generally more powerful than the corresponding ones based on the classical EDF or modified EDF in most cases.     

Statistical Artifacts in the Ratio of Discrete Quantities

The ratio is a familiar statistic, but it is often misused. One frequently overlooked problem occurs when ratioing two discrete (digitized) variables. Fine structure appears in the histogram of the ratio that can be very subtle, or can sometimes even dominate the histogram. It disappears when the numerator and/or denominator become continuous. This statistical artifact is not a binning error, nor is it removed by taking more data. It is important to be aware of the artifact in order to avoid misinterpretation of ratio data. We provide examples of the statistical artifact (including one from baseball) and discuss ways to avoid or minimize the problems it can cause.     

Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure

For a postulated common odds ratio for several 2 × 2 contingency tables one may, by conditioning on the marginals of the seperate tables, determine the exact expectation and variance of the entry in a particular cell of each table, hence for the total of such cells across all tables. This makes it feasible to determine limiting values, via single-degree-of-freedom, continuity-corrected chi-square tests on the common odds ratio–one determines lower and upper limits corresponding to just barely significant chi-square values. The Mantel-Haenszel approach can be viewed as a special application of this, but directed specifically to the case of unity for the odds ratio, for which the expectation and variance formulas are particularly simple. Computation of exact expectations and variances may be feasible only for 2 × 2 tables of limited size, but asymptotic formulas can be applied in other instances.Illustration is given for a particular set of four 2 × 2 tables in which both exact limits and limits by the proposed method could be applied, the two methods giving reasonably good agreement. Both procedures are directed at the distribution of the total over the designated cells, the proposed method treating that distribution as being asymptotically normal. Especially good agreement of proposed with exact limits could be anticipated in more asymptotic situations (overall, not for individual tables) but in practice this may not be demonstrable as the computation of exact limits is then unfeasible.     

Assessing a hypothesis test for the difference between two quantiles from independent populations

An empirical distribution function estimator for the difference of order statistics from two independent populations can be used for inference between quantiles from these populations. The inferential properties of the approach are evaluated in a simulation study where different sample sizes, theoretical distributions, and quantiles are studied. Small to moderate sample sizes, tail quantiles, and quantiles which do not coincide with the expectation of an order statistic are identified as problematic for appropriate Type I error control.     

Robust minimum distance inference based on combined distances

The minimum disparity estimators proposed by Lindsay (1994) for discrete models form an attractive subclass of minimum distance estimators which achieve their robustness without sacrificing first order efficiency at the model. Similarly, disparity test statistics are useful robust alternatives to the likelihood ratio test for testing of hypotheses in parametric models; they are asymptotically equivalent to the likelihood ratio test statistics under the null hypothesis and contiguous alternatives. Despite their asymptotic optimality properties, the small sample performance of many of the minimum disparity estimators and disparity tests can be considerably worse compared to the maximum likelihood estimator and the likelihood ratio test respectively. In this paper we focus on the class of blended weight Hellinger distances, a general subfamily of disparities, and study the effects of combining two different distances within this class to generate the family of “combined” blended weight Hellinger distances, and identify the members of this family which generally perform well. More generally, we investigate the class of "combined and penal-ized" blended weight Hellinger distances; the penalty is based on reweighting the empty cells, following Harris and Basu (1994). It is shown that some members of the combined and penalized family have rather attractive properties     

Taylor linearization sampling errors and design effects for poverty measures and other complex statistics

A systematic procedure for the derivation of linearized variables for the estimation of sampling errors of complex nonlinear statistics involved in the analysis of poverty and income inequality is developed. The linearized variable extends the use of standard variance estimation formulae, developed for linear statistics such as sample aggregates, to nonlinear statistics. The context is that of cross-sectional samples of complex design and reasonably large size, as typically used in population-based surveys. Results of application of the procedure to a wide range of poverty and inequality measures are presented. A standardized software for the purpose has been developed and can be provided to interested users on request. Procedures are provided for the estimation of the design effect and its decomposition into the contribution of unequal sample weights and of other design complexities such as clustering and stratification. The consequence of treating a complex statistic as a simple ratio in estimating its sampling error is also quantified. The second theme of the paper is to compare the linearization approach with an alternative approach based on the concept of replication, namely the Jackknife repeated replication (JRR) method. The basis and application of the JRR method is described, the exposition paralleling that of the linearization method but in somewhat less detail. Based on data from an actual national survey, estimates of standard errors and design effects from the two methods are analysed and compared. The numerical results confirm that the two alternative approaches generally give very similar results, though notable differences can exist for certain statistics. Relative advantages and limitations of the approaches are identified.     

Best bounds for order statistics and their expectations in range and mean units with applications

Best bounds for the order statistics are obtained in terras of the sample range and, for non-negative samples, in terms of the sample mean. Best bounds for the differences of two order statistics are found in this case also. Corresponding bounds on the expectation of order statistics and their differences are also found. Several applications of these bounds are considered     

Two notions of positive ageing based on stochastic dominance

The negative effects of age on the life length of a device or positive ageing are commonly used criteria for classifying life distributions. In this paper two notions of positive ageing are considered. These are the new better (worse) than used in average conditional survival probability and harmonic new better (worse) than used in upper tail. Closure of these notions under mixture and convolution are studied. The survivals of a device subject to discrete shocks of these notions of ageing which occur according to homogeneous Poisson processes are studied. A cumulative damage model is considered. Two test statistics are proposed for the two notions of ageing.     

Efficient method for identification of cyclic trends in incidence

Various methods for estimating the parameters of the simple harmonic curve and corresponding statistics for testing the significance of the sinusoidal trend are investigated. The locally reasonable method is almost fully efficient when the size of the trend is very small; however, the maximum likelihood method is preferred generally, especially when the trend is not very small. The log likelihood ratio test is more powerful than the R test which is based on locally reasonable estimates. The efficient method and the log likelihood ratio or equivalent tests are the best statistical techniques for identifying the cyclical trend. Thus they are the methods of choice when adequate computing facilities are available.     

New tests based on Rubin's empirical distribution function

In this paper we derive some new tests for goodness-of-fit based on Rubin's empirical distribution function (EDF). Substituting Rubin's EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling statistics, since Rubin's EDF for a given sample is a randomized distribution function, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. We show that the new tests are consistent under simple hypothesis. Several power comparisons are also performed to show that the new tests are generally more powerful than the classical ones.     

Estimation of the linear EV model with censored data

In this paper, a censored linear errors-in-variables model is investigated. The asymptotic normality of the unknown parameter's estimator is obtained. Two empirical log-likelihood ratio statistics for the unknown parameter in the model are suggested. It is proved that the proposed statistics are asymptotically chi-squared under some mild conditions, and hence can be used to construct the confidence regions of the parameter of interest. Finite sample performance of the proposed method is illustrated in a simulation study.     

Characterizations of exponentiality within the HNBUE class and related tests

A harmonic new better than used in expectation (HNBUE) variable is a random variable which is dominated by an exponential distribution in the convex stochastic order. We use a recently obtained condition on stochastic equality under convex domination to derive characterizations of the exponential distribution and bounds for HNBUE variables based on the mean values of the order statistics of the variable. We apply the results to generate discrepancy measures to test if a random variable is exponential against the alternative that is HNBUE, but not exponential.     

The quantile-based flattened logistic distribution: Some properties and applications

In this paper, the quantile-based flattened logistic distribution has been studied. Some classical and quantile-based properties of the distribution have been obtained. Closed form expressions of L-moments, L-moment ratios and expectation of order statistics of the distribution have been obtained. A quantile-based analysis concerning the method of matching L-moments estimation is employed to estimate the parameters of the proposed model. We further derive the asymptotic variance–covariance matrix of the matching L-Moments estimators of the proposed model. Finally, we apply the proposed model to simulated as well as two real life datasets and compare the fit with the logistic distribution.     

Goodness-of-fit tests for one-shot device accelerated life testing data

In this article, we develop a formal goodness-of-fit testing procedure for one-shot device testing data, in which each observation in the sample is either left censored or right censored. Such data are also called current status data. We provide an algorithm for calculating the nonparametric maximum likelihood estimate (NPMLE) of the unknown lifetime distribution based on such data. Then, we consider four different test statistics that can be used for testing the goodness-of-fit of accelerated failure time (AFT) model by the use of samples of residuals: a chi-square-type statistic based on the difference between the empirical and expected numbers of failures at each inspection time; two other statistics based on the difference between the NPMLE of the lifetime distribution obtained from one-shot device testing data and the distribution specified under the null hypothesis; as a final statistic, we use White's idea of comparing two estimators of the Fisher Information (FI) to propose a test statistic. We then compare these tests in terms of power, and draw some conclusions. Finally, we present an example to illustrate the proposed tests.     

Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.