A Neural Network-Based Approach for Statistical Probability Distribution Recognition

The statistical probability distribution of data should be known in advance, so that we can make some statistical inference based on the data and realize what information the data provides. Until now, a nonparametric goodness of fit test has been widely used in probability distribution recognition. However, such a procedure cannot guarantee a precise distribution recognition when only small data samples are available. In addition, the number of the divided groups will influence the results. This study proposes a neural network-based approach for probability distribution recognition. Two types of neural networks, backpropagation and learning vector quantization, are used in classifying normal, exponential, Weibull, Uniform, Chi-square, t, F, and Lognormal distributions. Implementation results demonstrate that the proposed approach outperforms the traditional statistical approach.     

Improved estimation of hydrologic data using the chi‐square goodness‐of‐fit test

Abstract

This study considers the complete Chi‐Square goodness‐of‐fit test procedures and numerical analysis to improve the intermediate value method for an approximate solution of the cumulative distribution function. The goodness‐of‐fit test is applied for evaluating the appropriate distribution of frequency analysis for estimating annual maximum flows of hydrologic data, from four measurement stations (Lounung, Yuemei, Santimem, and Kaoping stations) located in the Kaoping River Basin. Analytical results indicate that the error percentage of the Chi‐Square statistical values is improved by from 0.3 to 38.3%. Thus, the improved estimation of the Chi‐Square goodness‐of‐fit test procedure increases the accuracy of some frequency distributions.     

Quality Control Methods for Several Related Variables

The methods described in an earlier article devoted to control methods for two related variables are extended to the case of more than two related variables. The concept of matrix notation is introduced because of the resultant simplification in multivariate analysis and the original two-variable problem is restated in matrix form. The method of principal components is introduced both as a method of charscterizing a multivariate process and as a control tool associated with control procedures. These methods are illustrated with a numerical example from the field of ballistic missiles. Approximate multivariate techniques, designed to simplify the administration of these control programs, are also discussed.     

A new approach to MLE of Weibull distribution with interval data

We study the two-parameter maximum likelihood estimation (MLE) problem for the Weibull distribution with consideration of interval data. Without interval data, the problem can be solved easily by regular MLE methods because the restricted MLE of the scale parameter β for a given shape parameter α has an analytical form, thus α can be efficiently solved from its profile score function by traditional numerical methods. In the presence of interval data, however, the analytical form for the restricted MLE of β does not exist and directly applying regular MLE methods could be less efficient and effective. To improve efficiency and effectiveness in handling interval data in the MLE problem, a new approach is developed in this paper. The new approach combines the Weibull-to-exponential transformation technique and the equivalent failure and lifetime technique. The concept of equivalence is developed to estimate exponential failure rates from uncertain data including interval data. Since the definition of equivalent failures and lifetimes follows EM algorithms, convergence of failure rate estimation by applying equivalent failures and lifetimes is mathematically proved. The new approach is demonstrated and validated through two published examples, and its performance in different conditions is studied by Monte Carlo simulations. It indicates that the profile score function for α has only one maximum in most cases. Such good characteristic enables efficient search for the optimal value of α.     

Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions

We introduce two families of statistics, functionals of the empirical moment generating function process of the logarithmically transformed data, for testing goodness of fit to the two-parameter Weibull distribution or, equivalently, to the type I extreme value model. We show that when affine invariant estimators are used for the parameters of the extreme value distribution, the distributions of these statistics to not depend on the underlying parameters and one of them has a limiting chi-squared distribution. We estimate, via simulations, some finite sample quantiles for the statistics introduced and evaluate their power against a varied set of alternatives.     

Data Conformance Testing by Digital Analysis - A Critical Review and an Approach to More Appropriate Testing

The use of Benford's digital law for data conformance testing has recently received growing interest, particularly in business auditing. A closer examination shows that the currently propagated approaches are questionable: i) The theoretical foundations consist in asymptotic laws and approximation theorems which give no constructive guidelines for industrial application. ii) The statistical methods used are not well-matched to the asymptotic and approximative character of Benford's law. Starting from a close analysis of these aspects, the paper suggests an approach to more appropriate statistical methodology for digital analysis.     

The posterior predictive p-value for the problem of goodness of fit

The aim of this paper is to explore some features and possible uses of the posterior predictivep-value for the problem of goodness of fit. First, the behaviour of the posterior predictivep-value is compared with the behaviour of the classicalp-value through some interesting examples. Then, we consider a decision problem for simultaneously deciding to accept/reject a modelM and to accept/reject a null hypothesis (if we have accepted the modelM); the posterior predictivep-value is used for estimating the posterior probability of the model. Research partially supported by DGESIC (Spain) under grant number PB97-0021.     

A program for identifying the form of the distribution law of random quantities and their modeling

A program for identifying the form of the distribution law of a random quantity is described. The dependence of the optimum number of intervals when constructing histograms from a volume of the sample (10–100,000) and of the kurtosis (in the limits 2–9.65) is determined by computer modeling. Using 50 different functions, the program can also model random quantities with unimodal and bimodal distribution laws. __________ Translated from Izmeritel’naya Tekhnika, No. 5, pp. 9–14, May, 2007.     

A Robust Criterion for Checking the Uniformity of Two Non-Gaussian Samples with Respect to Variances

A robust F-criterion for checking the uniformity with respect to the variances of two non-Gaussian samples is obtained and investigated. Theoretical algorithms are developed and considered. The effectiveness and efficiency of the proposed criterion for known distribution kurtoses are illustrated using specific examples.__________Translated from Izmeritelnaya Tekhnika, No. 2, pp. 9–12, February, 2005.