Discriminating between distributions using feed-forward neural networks

A relevant problem in many applicatory contexts is to test whether some given observations follow one of two possible probability distributions. The vast literature produced over the years on this topic does not identify a tool which can be easily adopted to any situation but only finds solutions to specific comparisons. Recently, an easy to implement procedure for discrimination between two distributions based on feed-forward neural networks has been proposed giving interesting results. In this work this procedure is further investigated in terms of power, neural network architecture and expected statistical properties of the test statistic for small, moderate and large sample sizes, in a wide range of symmetric and skewed alternatives.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

Rank correlation plots for use with correlated input variables

A method for inducing a desired rank correlation matrix on multivariate input vectors for simulation studies has recently been developed by Iman and Conover (1982). The primary intention of this procedure is to produce correlated input variables for use with computer models. Since this procedure is distribution free and allows the exact marginal distributions to remain intact it can be used with any marginal distributions for which it is reasonable to think in terms of correlation. In this paper we present a series of rank correlation plots based on this procedure when the marginal distributions are normal, lognormal, uniform and loguniform. These plots provide a convenient tool both for aiding the modeler in determining the degree of dependence among input variables (rather than guessing) and for communicating with the modeler the effect of different correlation assumptions. In addition this procedure can be used with sample multivariate data by sampling directly from the respective marginal empirical distribution functions.     

Discriminating Between the Log-Normal and Log-Logistic Distributions

Log-normal and log-logistic distributions are often used to analyze lifetime data. For certain ranges of the parameters, the shape of the probability density functions or the hazard functions can be very similar in nature. It might be very difficult to discriminate between the two distribution functions. In this article, we consider the discrimination procedure between the two distribution functions. We use the ratio of maximized likelihood for discrimination purposes. The asymptotic properties of the proposed criterion are investigated. It is observed that the asymptotic distributions are independent of the unknown parameters. The asymptotic distributions are used to determine the minimum sample size needed to discriminate between these two distribution functions for a user specified probability of correct selection. We perform some simulation experiments to see how the asymptotic results work for small sizes. For illustrative purpose, two data sets are analyzed.     

Discriminating between the Weibull and log-normal distributions for Type-II censored data

Log-normal and Weibull distributions are the two most popular distributions for analysing lifetime data. In this paper, we consider the problem of discriminating between the two distribution functions. It is assumed that the data are coming either from log-normal or Weibull distributions and that they are Type-II censored. We use the difference of the maximized log-likelihood functions, in discriminating between the two distribution functions. We obtain the asymptotic distribution of the discrimination statistic. It is used to determine the probability of correct selection in this discrimination process. We perform some simulation studies to observe how the asymptotic results work for different sample sizes and for different censoring proportions. It is observed that the asymptotic results work quite well even for small sizes if the censoring proportions are not very low. We further suggest a modified discrimination procedure. Two real data sets are analysed for illustrative purposes.     

Tests for exponentiality

The idea of measuring the departure of data bu a plot of obeserved observations against their expectation has been expeetations has been exploited in this paper to develop tests for exponentiality the tests are for parameter two parameter exponential distribution with complete sample and one parameter exponential distribution with complete sample and one large sample distributions of the test statistics critical points have been computed for different levels of significance and applications of these have been computed for differents levels of significance and applications of these tests have been discussed in case of three data sets.     

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.     

Bayesian estimation and prediction based on generalized Type-I hybrid censored sample

In this paper, we consider an exponential form for the underlying distributionand a conjugate prior, and develop a procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-I hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

A Numerical Study of Small Parameter Behavior of Some Families of Distributions

Limiting distributions play an important role in approximating the exact distributions, especially when they have a rather cumbersome analytic form, or simply when they do not have a closed from. The question that naturally arises is how good the approximation is. In this article, we propose a procedure for the numerical assessment of the “goodness” of some easy-to-calculate limiting distributions, originally proposed in Bar-Lev and Enis, in various cases of the underlying distributions, some of which are inherently computationally challenging. The details of the procedure are provided in three examples. The first example deals with the gamma distributions; the second deals with Bessel distributions related to a symmetric random walk, and the third example deals with positive stable distributions. The details of two additional variations of these examples are also discussed. These examples illustrate the ease with which the limiting approximations could be applied in the various cases, well-demonstrating their computational simplicity and attractiveness.     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

Multivariate Three-Parameter Log-Elliptical Distributions

The three-parameter log-elliptical distribution class is developed for the general situation in which the hypothesis of independence for the elements in a sample is not assumed. The parameter estimators are theoretically showed to be invariant under all distributions in the class by considering only a change in the constant of the scale parameter estimator. An estimation procedure based on the three-parameter lognormal distribution is proposed for the parameter estimation problem in any three-parameter log-elliptical distribution. Two classical lognormal data sets are analyzed without assuming independence in the sample in order to illustrate the proposed estimation procedure.     

Bayesian estimation and prediction based on generalized Type-II hybrid censored sample

ABSTRACT

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies

The generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.     

A Convergent Iterative Procedure for Constructing Bivariate Distributions

For many years there has been interest in families of bivariate distributions with the marginals as parameters. Questions of this kind arise if one is to build a stochastic model in a situation where one has some idea about the dependence structure and marginal distributions. In this article, among all bivariate distributions which satisfy the constraints imposed by the known marginals and/or dependence structure, one that has the maximum entropy is obtained by using iterative procedure, and its convergence is proved.     

Effect of a new selection procedure on adaptive estimation

Adaptive estimation of parameters of some failure time distributionsis considered. A new procedure named the F-procedure has beendeveloped for selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimatorsof parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptiveestimators by simulation of samples using the Monte Carlo method.Adaptive estimation of parameters of some failure time distributions is considered. A new procedure named the F-procedure has been developedfor selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimators of parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptive estimators by simulation of samples using the Monte Carlo method.     

Adaptive m estimation of symmetric distribution location

An adaptive M esitmation procedure for using a random sample to estimate the location parameter of an unknown symmetric distribution is developed. The procedure may be applied to samples from distributions with tail lenghts at least as heavy as normal distribution tails. Simulation studies demonstrate the potential of the new estimator for producing good location estimates.     

A likelihood ratio test to discriminate exponential–Poisson and gamma distributions

The exponential–Poisson (EP) distribution with scale and shape parameters β>0 and λ∈?, respectively, is a lifetime distribution obtained by mixing exponential and zero-truncated Poisson models. The EP distribution has been a good alternative to the gamma distribution for modelling lifetime, reliability and time intervals of successive natural disasters. Both EP and gamma distributions have some similarities and properties in common, for example, their densities may be strictly decreasing or unimodal, and their hazard rate functions may be decreasing, increasing or constant depending on their shape parameters. On the other hand, the EP distribution has several interesting applications based on stochastic representations involving maximum and minimum of iid exponential variables (with random sample size) which make it of distinguishable scientific importance from the gamma distribution. Given the similarities and different scientific relevance between these models, one question of interest is how to discriminate them. With this in mind, we propose a likelihood ratio test based on Cox's statistic to discriminate the EP and gamma distributions. The asymptotic distribution of the normalized logarithm of the ratio of the maximized likelihoods under two null hypotheses – data come from EP or gamma distributions – is provided. With this, we obtain the probabilities of correct selection. Hence, we propose to choose the model that maximizes the probability of correct selection (PCS). We also determinate the minimum sample size required to discriminate the EP and gamma distributions when the PCS and a given tolerance level based on some distance are before stated. A simulation study to evaluate the accuracy of the asymptotic probabilities of correct selection is also presented. The paper is motivated by two applications to real data sets.     

Goodness of fit for the poisson distribution based on the probability generating function

For testing the fit of a discrete distribution, use of the probability generating function and its empirical counterpart has been suggested in Koeherlakota and Kocherlakota (1986). In the present paper, a particular functional of the corresponding empirical probability generating function process is proposed as a measure to test the discrepancy between the evidence and the hypothesis. The asymptotic behavior of the empirical probability generating function when a parameter is estimated is obtained, The study is exemplified for the Poisson case only but the procedure can be extended to other discrete distributions.     

Stein's two sample test of a general linear hypothesis and a noncentral f–type distribution

Stein's two–sample procedure for a general linear model is studied and derived in terms of matrices in which the error tems are distributed as multivatriate student t–error terms. Tests and confidence regions are constructed in a similar way to classical linear models which involves percentage points of student t and F distributions. The advantages of taking two samples are: the variance of the error terms is known, and the power of tests are size of confidence regions are controllable. A new distribution called noncentral F–type distribution different from the nencentral F is found when considerinf the power of the test of general linear hypothesis.     

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.