Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Goodness-of-fit measures in binary choice models

In this paper, a review is given of various goodness-of-fit measures that have been proposed for the binary choice model in the last two decades. The relative behaviour of several pseudo-R² measures is analysed in a series of misspecified binary choice models, the misspecification being omitted variables or an included irrelevant variable. A comparison is made with the OLS-R² of the underlying latent variable model and with the squared sample correlation coefficient of the true and predicted probabilities. Further, it is investigated how the values of the measures change with a changing frequency rate of successes.     

Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions

Summary. We propose a goodness-of-fit statistic Q _n based on the Hoeffding maximum correlation for testing uniformity and we show its relationship to Gini's mean difference. We compute exact and asymptotic critical values and study the power of the test proposed against a representative set of alternatives.     

An Objective Graphical Method for Testing Normal Distributional Assumptions using Probability Plots

Formulas for plotting probability and techniques for subjectively drawing lines on probability plots are reviewed. A method is presented for plotting data and drawing an objective line on the probability plot to obtain a test of the distributional assumption.     

Estimation of the Generalized Lambda Distribution Parameters for Grouped Data

Abstract

In this article we consider the problem of fitting a five-parameter generalization of the lambda distribution to data given in the form of a grouped frequency table. The estimation of parameters is done by six different procedures: percentiles, moments, probability-weighted moments, minimum Cramér-Von Mises, maximum likelihood, and pseudo least squares. These methods are evaluated and compared using a Monte Carlo study where the parent populations were generalized lambda distribution (GLD) approximations of Normal, Beta, Gamma random variables, and for nine combinations of sample sizes and number of classes. Of the estimators analyzed it is concluded that, although the method of pseudo least squares suffers from a number of limitations, it appears to be the candidate procedure to estimate the parameters of a GLD from grouped data.     

The Siable-Law Model of Stock Returns

This study investigates the tail shapes of empirical distributions of returns on an extensive group of common stocks. The tails of the return distributions are found to be thinner than those of infinite variance stable distributions. Therefore, although homogeneity is evident in general, economic and statistical inferences drawn from stable-law parameters estimated from samples of stock returns may be misleading. This is in spite of the apparent overall similarity (in shape) between empirical and stable distributions.     

Bayesian Model Selection: Measuring the χ2 Discrepancy with the Uniform Distribution

In the last years, many articles have been written about Bayesian model selection. In this article, a different and easier method is proposed and analyzed. The key idea of this method is based on the well-known property that, under the true model, the cumulative distribution function is distributed as a uniform distribution over the interval (0, 1). The method is first introduced for the continuous case and then for the discrete case by smoothing the cumulative distribution function. Some asymptotical properties of the method are obtained by developing an alternative to Helly's theorems. Finally, the performance of the method is evaluated by simulation, showing a good behavior.     

Nearly Gaussian Distributions and Application

We consider the random variable X that is not Gaussian but for which X ^c , where c = (2k + 1)/(2j + 1) with k, j ? {0, 1,…}, is approximately Gaussian. A variable of this type is used to model the errors made by meteorologists when forecasting temperatures.     

Lack-of-fit Tests Based On Partial Sums of Residuals

To evaluate the validity of the mean function in generalized linear models, Su and Wei (1991 Su, J.Q., Wei, L.T. (1991). A lack-of-fit test for the mean function in a generalized linear model. J. Amer. Statist. Assoc. 86:420–426.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed a lack-of-fit test based on partial sums of residuals. They compute P values using an unusual bootstrapping simulation. However, the simulations can hardly be performed with more than a few predictor variables because it is prohibitively time consuming. We modify their test for linear models and propose another lack of fit test based on partial sums of residuals. We find the non normal limiting distributions for both tests thus enabling more direct calculation of P values. Finally, we examine how the nature of the simulation reduces the power of Su-Wei’s test.     

An Efficient Method to Generate Data and Compute Exact P-Values in Goodness-of-Fit Testing

In this article, we use a characterization of the set of sample counts that do not match with the null hypothesis of the test of goodness of fit. Two direct applications arise: first, to instantaneously generate data sets whose corresponding asymptotic P-values belong to a certain pre-defined range; and second, to compute exact P-values for this test in an efficient way. We present both issues before illustrating them by analyzing a couple of data sets. Method's efficiency is also assessed by means of simulations. We focus on Pearson's X ² statistic but the case of likelihood-ratio statistic is also discussed.