Random coefficient regressions: parametric goodness of fit tests

Random coefficient regression models have been applied in different fields during recent years and they are a unifying frame for many statistical models. Recently, Beran and Hall (1992) opened the question of the nonparametric study of the distribution of the coefficients. Nonparametric goodness of fit tests were considered in Delicado and Romo (1994.). In this paper we propose statistics for parametric goodness of fit tests and we obtain their asymptotic distributions. Moreover, we construct bootstrap approximations to these distributions, proving their validity. Finally, a simulation study illustrates our results

Goodness of Fit: an axiomatic approach

An axiomatic approach is used to develop a one-parameter family of measures of divergence between distributions. These measures can be used to perform goodness-of-fit tests with good statistical properties. Asymptotic theory shows that the test statistics have well-defined limiting distributions which are however analytically intractable. A parametric bootstrap procedure is proposed for implementation of the tests. The procedure is shown to work very well in a set of simulation experiments, and to compare favourably with other commonly used goodness-of-fit tests. By varying the parameter of the statistic, one can obtain information on how the distribution that generated a sample diverges from the target family of distributions when the true distribution does not belong to that family. An empirical application analyses a UK income data set.Goodness of fit; axiomatic approach; measures of divergence; parametric bootstrap

The art of fitting financial time series with Levy stable distributions

This paper illustrates a procedure for fitting financial data with alpha-stable distributions. After using all the available methods to evaluate the distribution parameters, one can qualitatively select the best estimate and run some goodness-of-fit tests on this estimate, in order to quantitatively assess its quality. It turns out that, for the two investigated data sets (MIB30 and DJIA from 2000 to present), an alpha-stable fit of log-returns is reasonably good.finance; statistical methods; stable distributions

Some Extended Classes of Distributions: Characterizations and Properties

Based on a simple relationship between two truncated moments and certain functions of the th order statistic, we characterize some extended classes of distributions recently proposed in the statistical literature, videlicet Beta-G, Gamma-G, Kumaraswamy-G and McDonald-G. Several properties of these extended classes and some special cases are discussed. We compare these classes in terms of goodness-of-fit criteria using some baseline distributions by means of two real data sets

Multimodel Response Assessment for Monthly Rainfall Distribution in Some Selected Indian Cities Using Best Fit Probability as a Tool

We carry out a study of the statistical distribution of rainfall precipitation data for 20 cites in India. We have determined the best-fit probability distribution for these cities from the monthly precipitation data spanning 100 years of observations from 1901 to 2002. To fit the observed data, we considered 10 different distributions. The efficacy of the fits for these distributions was evaluated using four empirical non-parametric goodness-of-fit tests namely Kolmogorov-Smirnov, Anderson-Darling, Chi-Square, Akaike information criterion, and Bayesian Information criterion. Finally, the best-fit distribution using each of these tests were reported, by combining the results from the model comparison tests. We then find that for most of the cities, Generalized Extreme-Value Distribution or Inverse Gaussian Distribution most adequately fits the observed data.Comment: 14 pages, 5 figure

Maximum Fidelity

The most fundamental problem in statistics is the inference of an unknown probability distribution from a finite number of samples. For a specific observed data set, answers to the following questions would be desirable: (1) Estimation: Which candidate distribution provides the best fit to the observed data?, (2) Goodness-of-fit: How concordant is this distribution with the observed data?, and (3) Uncertainty: How concordant are other candidate distributions with the observed data? A simple unified approach for univariate data that addresses these traditionally distinct statistical notions is presented called "maximum fidelity". Maximum fidelity is a strict frequentist approach that is fundamentally based on model concordance with the observed data. The fidelity statistic is a general information measure based on the coordinate-independent cumulative distribution and critical yet previously neglected symmetry considerations. An approximation for the null distribution of the fidelity allows its direct conversion to absolute model concordance (p value). Fidelity maximization allows identification of the most concordant model distribution, generating a method for parameter estimation, with neighboring, less concordant distributions providing the "uncertainty" in this estimate. Maximum fidelity provides an optimal approach for parameter estimation (superior to maximum likelihood) and a generally optimal approach for goodness-of-fit assessment of arbitrary models applied to univariate data. Extensions to binary data, binned data, multidimensional data, and classical parametric and nonparametric statistical tests are described. Maximum fidelity provides a philosophically consistent, robust, and seemingly optimal foundation for statistical inference. All findings are presented in an elementary way to be immediately accessible to all researchers utilizing statistical analysis.Comment: 66 pages, 32 figures, 7 tables, submitte

Goodness of fit tests in random coefficient regression models

Random coefficient regressions have been applied in a wide range of fields, from biology to economics, and constitute a common frame for several important statistical models. A nonparametric approach to inference in random coefficient models was initiated by Beran and Hall. In this paper we introduce and study goodness of fit tests for the coefficient distributions; their asymptotic behaviour under the null hypothesis is obtained. We also propose bootstrap resampling strategies to approach these distributions and prove their asymptotic validity using results by Gine and Zinn on bootstrap empirical processes. A simulation study illustrates the properties of these tests

Goodness of Fit: an axiomatic approach

An axiomatic approach is used to develop a one-parameter family of measures of divergence between distributions. These measures can be used to perform goodness-of-fit tests with good statistical properties. Asymptotic theory shows that the test statistics have well-defined limiting distributions which are however analytically intractable. A parametric bootstrap procedure is proposed for implementation of the tests. The procedure is shown to work very well in a set of simulation experiments, and to compare favourably with other commonly used goodness-of-fit tests. By varying the parameter of the statistic, one can obtain information on how the distribution that generated a sample diverges from the target family of distributions when the true distribution does not belong to that family. An empirical application analyses a UK income data set