Chernoff distance for doubly truncated distributions

In a recent paper, Nair et al. [Stat Pap 52:893–909, 2011] proposed Chernoff distance measure for left/right-truncated random variables and studied their properties in the context of reliability analysis. Here we extend the definition of Chernoff distance for doubly truncated distributions. This measure may help the information theorists and reliability analysts to study the various characteristics of a system/component when it fails between two time points. We study some properties of this measure and obtain its upper and lower bounds. We also study the interval Chernoff distance between the original and weighted distributions. These results generalize and enhance the related existing results that are developed based on Chernoff distance for one-sided truncated random variables.     

Improved testing for the binomial distribution using chi-squared components with data-dependent cells

A power study suggests that a good test of fit analysis for the binomial distribution is provided by a data-dependent Chernoff–Lehmann X ² test with class expectations greater than unity, and its components. These data-dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions and provide a comprehensive assessment of how well the data agree with a binomial distribution. We suggest that a well-performed single test of fit statistic is the Anderson–Darling statistic.     

General characterization theorems based on versions of the Chernoff inequality and the Cox representation

In this short communication, we extend characterization theorems for distributions based on versions of the Chernoff inequality to the case where the distributions are not necessarily purely discrete or absolutely continuous (in the usual sense) and relate these to Cox's representation for a survivor function in terms of the hazard measure, as presented by Kotz and Shanbhag (1980). (The original version of the representation referred to had appeared in Cox, 1972). Some corollaries of the results giving characteristic properties of certain well-known distributions explicitly are also presented.     

Penalized likelihood-ratio test for finite mixture models with multinomial observations

Due to the irregularity of finite mixture models, the commonly used likelihood-ratio statistics often have complicated limiting distributions. We propose to add a particular type of penalty function to the log-likelihood function. The resulting penalized likelihood-ratio statistics have simple limiting distributions when applied to finite mixture models with multinomial observations. The method is especially effective in addressing the problems discussed by Chernoff and Lander (1995). The theory developed and simulations conducted show that the penalized likelihood method can give very good results, better than the well-known C(α) procedure, for example. The paper does not, however, fully explore the choice of penalty function and weight. The full potential of the new procedure is to be explored in the future.     

General characterization theorems via the mean absolute deviation

The Chernoff–Borovkov–Utev inequality resulted owing to earlier inequalities established by Chernoff (1981) and Borovkov and Utev (1983), respectively, giving bounds for the variance of functions of normal r.v.’s and leading to characterizations of normality. Subsequently, several analytic properties of variance bounds and other relevant results were established by others. Defining the mean absolute deviation (about a median) as E|X−med(X)| where med(X) is a median of the distribution of the random variable X, Freimer and Mudholkar (1991) gave a bound for the mean absolute deviation of a certain real-valued function of an absolutely continuous random variable (w.r.t. Lebesgue measure) and Korwar (1991) presented an analogue of this in the discrete case; these authors, also, characterized the Laplace and a mixture of two Waring distributions via the respective bounds.We extend these latter results theorems to the case where the distributions are not necessarily purely discrete or absolutely continuous, via the approach of Alharbi and Shanbhag (1996). The results in Freimer and Mudholkar (1991) and Korwar (1991) are now corollaries to our findings. Also, following Alharbi and Shanbhag (1996), we relate these results to the modified version of Cox’s representation for a survival function in terms of the hazard measure, given in Kotz and Shanbhag (1980). (The original version of the representation mentioned had appeared in Cox (1972).)     

Estimation for u-shaped beta distributions: minimum hellinger distance and related methods

We compare minimum Hellinger distance and minimum Heiiinger disparity estimates for U-shaped beta distributions. Given suitable density estimates, both methods are known to be asymptotically efficient when the data come from the assumed model family, and robust to small perturbations from the model family. Most implementations use kernel density estimates, which may not be appropriate for U-shaped distributions. We compare fixed binwidth histograms, percentile mesh histograms, and averaged shifted histograms. Minimum disparity estimates are less sensitive to the choice of density estimate than are minimum distance estimates, and the percentile mesh histogram gives the best results for both minimum distance and minimum disparity estimates. Minimum distance estimates are biased and a bias-corrected method is proposed. Minimum disparity estimates and bias-corrected minimum distance estimates are comparable to maximum likelihood estimates when the model holds, and give better results than either method of moments or maximum likelihood when the data are discretized or contaminated, Although our re¬sults are for the beta density, the implementations are easily modified for other U-shaped distributions such as the Dirkhlet or normal generated distribution.     

Calculation of the Prokhorov distance by optimal quantization and maximum flow

Calculating exact values of the Prokhorov metric for the set of probability distributions on a metric space is a challenging problem. In this paper probability distributions are approximated by finite-support distributions through optimal or quasi-optimal quantization, in such a way that exact calculation of the Prokhorov distance between a distribution and a quantizer can be performed. The exact value of the Prokhorov distance between two quantizers is obtained by solving an optimization problem through the Simplex method. This last value is used to approximate the Prokhorov distance between the two initial distributions, and the accuracy of the approximation is measured. We illustrate the method on various univariate and bivariate probability distributions. Approximation of bivariate standard normal distributions by quasi-optimal quantizers is also considered.     

Large deviations for the L1-distance in kernel density estimation

We establish a large deviation limit theorem of Chernoff type for the L₁-distance between the nonparametric kernel density estimator and the underlying density. The estimation is based on a sequence of independent and identically distributed random variables. The rate function is well identified, distribution-free and independent of the choice of the kernel.     

The shape of the hazard function: Does the generalized gamma have the last word?

This paper compares the five-parameter beta generalized gamma (BGG) distribution to the three-parameter generalized gamma (GG). Both distributions include the four standard hazard shapes that we believe is an important property for any parametric family. For several BGG distributions, we select matching GGs and compute the Kullback-Liebler distance, observing remarkable agreement. We explore the beta parameters' influence on the matched GG parameters, detecting a strong connection between the distributions. Lastly, we compare the distributions using two real-data examples. We conclude from these comparisons that the BGG is not likely to be more useful for analytical purposes than the simpler GG.     

Certain nonparametric classification rules and their asymptotic efficiencies

Two nonparametric classification rules for e-univariace populations are proposed, one in which the probability of correct classification is a specified number and the other in which one has to evaluate the probability of correct classification. In each case the classification is with respect to the Chernoff and Savage (1958) class of statistics, with possible specialization to populations having different location shifts and different changes of scale. An optimum property, namely the consistency of the classification procedure, is established for the second rule, when the distributions are either fixed or “near” in the Pitman sense and are tending to a common distribution at a specified rate. A measure of asymptotic efficiency is defined for the second rule and its asymptotic efficiency based on the Chernoff-Savage class of statistics relative to the parametric competitors ie the case of location shifts and scale changes is shown to be equal to the analogous Pitman efficiency.     

Some theoretical results regarding the polygonal distribution

Polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. We demonstrate that the densities of polygonal distributions are dense in the class of continuous and concave densities with bounded second derivatives. Furthermore, we prove that polygonal density functions provide O(g^{? 2}) approximations (where g is the number of triangular distribution components), in the supremum distance, to any density function from the hypothesized class. Parametric consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A result regarding model selection via penalized ML estimation is proved.     

Discriminating among Weibull,log-normal,and log-logistic distributions

In this article, we consider the problem of the model selection/discrimination among three different positively skewed lifetime distributions. All these three distributions, namely; the Weibull, log-normal, and log-logistic, have been used quite effectively to analyze positively skewed lifetime data. In this article, we have used three different methods to discriminate among these three distributions. We have used the maximized likelihood method to choose the correct model and computed the asymptotic probability of correct selection. We have further obtained the Fisher information matrices of these three different distributions and compare them for complete and censored observations. These measures can be used to discriminate among these three distributions. We have also proposed to use the Kolmogorov–Smirnov distance to choose the correct model. Extensive simulations have been performed to compare the performances of the three different methods. It is observed that each method performs better than the other two for some distributions and for certain range of parameters. Further, the loss of information due to censoring are compared for these three distributions. The analysis of a real dataset has been performed for illustrative purposes.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

The chernoff efficiency of the wilcoxon rank test

Exact limiting Chernoff efficiencies of the Wilcoxon rank test are derived using Hoadley's results. Efficiency curves are derived for the two-sample Wilcoxon rank test relative to the two-sample t test for normal shift alternatives when the null hypothesis is that of common normality. The comparisons with Bahadur efficiency and small sample Hodges-Lehmann efficiency-are also made.     

Forecasting Accuracy of Alternative Techniques: A Comparison of U.S. Macroeconomic Forecasts

This article introduces and discusses a new measure of the relative economic affluence (REA) between income distributions with different means. The REA measure D is applied to the U.S. white and black household income distributions of 1967 and 1979. The measure D shows that the REA of the white households with respect to the black households decreased from 1967 to 1979. This conclusion contrasts with those obtained by applications of distance or quasi-distance functions. It is shown in this study that REA measures and distance functions address different and relevant issues. An REA measure deals with the relation “more affluent than” and defines a partial strict ordering over the set of pairs of income distributions—that is, the relation is asymmetric and transitive—whereas a distance function accounts for the dissimilarity between distributions without imposing an ordering relation and hence fulfills the symmetry property.     

Compromise Pitman estimators

In this paper, we investigate the construction of compromise estimators of location and scale, by averaging over several models selected among a specified large set of possible models. The weight given to each distribution is based on the profile likelihood, which leads to a notion of distance between distributions as we study the asymptotic behaviour of such estimators. The selection of the models is made in a minimax way, in order to choose distributions that are close to any possible distribution. We also present simulation results of such compromise estimators based on contaminated Gaussian and Student's t distributions.     

Signed binomial approximation of binomial mixtures via differential calculus for linear operators

We obtain sharp estimates in signed binomial approximation of binomial mixtures with respect to the total variation distance. We provide closed form expressions for the leading terms, and show that the corresponding leading coefficients depend on the zeros of appropriate Krawtchouk polynomials. The special case of Pólya–Eggenberger distributions is discussed in detail. Our approach is based on a differential calculus for linear operators represented by stochastic processes, which allows us to give unified proofs.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

A fast robust method for fitting gamma distributions

The art of fitting gamma distributions robustly is described. In particular we compare methods of fitting via minimizing a Cramér Von Mises distance, an L ₂ minimum distance estimator, and fitting a B-optimal M-estimator. After a brief prelude on robust estimation explaining the merits in terms of weak continuity and Fréchet differentiability of all the aforesaid estimators from an asymptotic point of view, a comparison is drawn with classical estimation and fitting. In summary, we give a practical example where minimizing a Cramér Von Mises distance is both efficacious in terms of efficiency and robustness as well as being easily implemented. Here gamma distributions arise naturally for “in control” representation indicators from measurements of spectra when using fourier transform infrared (FTIR) spectroscopy. However, estimating the in-control parameters for these distributions is often difficult, due to the occasional occurrence of outliers.