An R2criterion based on optimal predictors

Abstract: The predictor that minimizes mean-squared prediction error is used to derive a goodness-of-fit measure that offers an asymptotically valid model selection criterion for a wide variety of regression models. In particular, a new goodness-of-fit criterion (cr²) is proposed for censored or otherwise limited dependent variables. The new goodness-of-fit measure is then applied to the analysis of duration.     

A New Test of Randomness for Lehmer Generators Based on the Manhattan Distance Between Pairs of Consecutive Random Numbers

This article considers the Marsaglia effect by proposing a new test of randomness for Lehmer random number generators. Our test is based on the Manhattan distance criterion between consecutive pairs of random numbers rather than the usually adopted Euclidian distance. We derive the theoretical distribution functions for the Manhattan distance for both overlapping (two dimensional) as well as non-overlapping cases. Extensive goodness-of-fit testing as well as empirical experimentation provides ample proof of the merits of the proposed criterion.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

Tests of fit for a lognormal distribution

The problem of goodness of fit of a lognormal distribution is usually reduced to testing goodness of fit of the logarithmic data to a normal distribution. In this paper, new goodness-of-fit tests for a lognormal distribution are proposed. The new procedures make use of a characterization property of the lognormal distribution which states that the Kullback–Leibler measure of divergence between a probability density function (p.d.f) and its r-size weighted p.d.f is symmetric only for the lognormal distribution [Tzavelas G, Economou P. Characterization properties of the log-normal distribution obtained with the help of divergence measures. Stat Probab Lett. 2012;82(10):1837–1840]. A simulation study examines the performance of the new procedures in comparison with existing goodness-of-fit tests for the lognormal distribution. Finally, two well-known data sets are used to illustrate the methods developed.     

Probability Generating Function Based Jeffrey's Divergence for Statistical Inference

Statistical inference procedures based on transforms such as characteristic function and probability generating function have been examined by many researchers because they are much simpler than probability density functions. Here, a probability generating function based Jeffrey's divergence measure is proposed for parameter estimation and goodness-of-fit test. Being a member of the M-estimators, the proposed estimator is consistent. Also, the proposed goodness-of-fit test has good statistical power. The proposed divergence measure shows improved performance over existing probability generating function based measures. Real data examples are given to illustrate the proposed parameter estimation method and goodness-of-fit test.     

Parameter changes in GARCH model

A new method for detecting the parameter changes in generalized autoregressive heteroskedasticity GARCH (1,1) model is proposed. In the proposed method, time series observations are divided into several segments and a GARCH (1,1) model is fitted to each segment. The goodness-of-fit of the global model composed of these local GARCH (1,1) models is evaluated using the corresponding information criterion (IC). The division that minimizes IC defines the best model. Furthermore, since the simultaneous estimation of all possible models requires huge computational time, a new time-saving algorithm is proposed. Simulation results and empirical results both indicate that the proposed method is useful in analysing financial data.     

A New Goodness-of-Fit Test Based on the Empirical Characteristic Function

In this article, we present a goodness-of-fit test for a distribution based on some comparisons between the empirical characteristic function c_n(t) and the characteristic function of a random variable under the simple null hypothesis, c₀(t). We do this by introducing a suitable distance measure. Empirical critical values for the new test statistic for testing normality are computed. In addition, the new test is compared via simulation to other omnibus tests for normality and it is shown that this new test is more powerful than others.     

A new variational Bayesian algorithm with application to human mobility pattern modeling

A new variational Bayesian (VB) algorithm, split and eliminate VB (SEVB), for modeling data via a Gaussian mixture model (GMM) is developed. This new algorithm makes use of component splitting in a way that is more appropriate for analyzing a large number of highly heterogeneous spiky spatial patterns with weak prior information than existing VB-based approaches. SEVB is a highly computationally efficient approach to Bayesian inference and like any VB-based algorithm it can perform model selection and parameter value estimation simultaneously. A significant feature of our algorithm is that the fitted number of components is not limited by the initial proposal giving increased modeling flexibility. We introduce two types of split operation in addition to proposing a new goodness-of-fit measure for evaluating mixture models. We evaluate their usefulness through empirical studies. In addition, we illustrate the utility of our new approach in an application on modeling human mobility patterns. This application involves large volumes of highly heterogeneous spiky data; it is difficult to model this type of data well using the standard VB approach as it is too restrictive and lacking in the flexibility required. Empirical results suggest that our algorithm has also improved upon the goodness-of-fit that would have been achieved using the standard VB method, and that it is also more robust to various initialization settings.     

Minimum aberration criterion for screening experiments at two levels from an entropy-based perspective

This paper presents a new criterion for selecting a two-level fractional factorial design. The theoretical underpinning for the criterion is the Shannon entropy. The criterion, which is referred to as the entropy-based minimum aberration criterion, has several advantages. The advantage of the entropy-based criterion over the classical minimum aberration criterion is that it utilizes a measure of uncertainty on the skewness of the distribution of word length patterns in the selection of the “best” design in a family of two-level fractional factorial plans. The criterion evades the trauma associated with the lack of prior knowledge on the important effects.     

Goodness-of-fit tests and minimum distance estimation via optimal transformation to uniformity

A unified approach of parameter-estimation and goodness-of-fit testing is proposed. The new procedures may be applied to arbitrary laws with continuous distribution function. Specifically, both the method of estimation and the goodness-of-fit test are based on the idea of optimally transforming the original data to the uniform distribution, the criterion of optimality being an L2-type distance between the empirical characteristic function of the transformed data, and the characteristic function of the uniform (0,1)$(0,1)$ distribution. Theoretical properties of the new estimators and tests are studied and some connections with classical statistics, moment-based procedures and non-parametric methods are investigated. Comparison with standard procedures via Monte Carlo is also included, along with a real-data application.     

The Vanishing 2×2 Table: Linking the Hypergeometric,Binomial and Poisson

A simple procedure for establishing minimum sample size in X ² goodness-of-fit tests is presented. Samples of this size will automatically satisfy Yarnold's criterion.     

Local Power Analyses of Goodness-of-fit Tests for Copulas

Abstract.  The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramér–von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramér–von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods.     

Log-ratios and the logarithmic mean

Logarithmically transformed ratios have an important role to play in some branches of statistics and economics. Focus of this paper is on an unfamiliar interpretation of these logratios, which are in fact absolute differences being divided by the logarithmic mean. It is demonstrated that familiarity with the logarithmic mean and its relation to more common mean values leads to view a problem from a new angle whenever logratios are involved. Examples investigated are Fisher’s z-transformation, the likelihood-ratio test of goodness-of-fit, Theil’s measure of inequality, and the log-ratio itself as one important device for measuring relative change within timeseries.     

Homogeneity testing in a Weibull mixture model

The mixed Weibull distribution provides a flexible model to analyze random durations in a possibly heterogeneous population. To test for homogeneity against unobserved heterogeneity in a Weibull mixture model, a dispersion score test and a goodness-of-fit test are investigated. The empirical power of these tests is assessed and compared on a broad range of alternatives. It comes out that the dispersion score test, as it is based on a Weibull-to-exponential transformation, often breaks down. A simple new procedure is introduced for Weibull mixtures in scale, which combines the dispersion score test and the goodness-of-fit test. The new test is compared with several known procedures and shown to have a good overall power. To detect mixtures in shape and scale, a goodness-of-fit test is recommended. This research has been partially sponsored by a grant of the Deutsche Forschungsgemeinschaft. We thank Lars Haferkamp for computational assistance and Wilfried Seidel and a referee for their remarks on alternative test procedures.     

A new hybrid estimation method for the generalized pareto distribution

ABSTRACT

The generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

A model selection criterion based on the BHHJ measure of divergence

The aim of this work is the discussion and investigation of measures of divergence and model selection criteria. A recently introduced measure of divergence, the so-called BHHJ measure (Basu, A., Harris, I.R., Hjort, N.L., Jones, M.C., 1998. Robust and efficient estimation by minimising a density power divergence. Biometrika 85, 549–559) is investigated and a new model selection criterion the divergence information criterion (DIC) based on this measure is proposed. Simulations are performed to check the appropriateness of the proposed criterion.     

Use of tail areas for testing goodness-of-fit

Characterizing a set of data as a random sample from a specified distribution is often a precursor to statistical inference or hypothesis testing involving the extremes of the distribution -precisely the regions of greatest uncertainty. It seems reasonable then to exploit as best we can our limited knowledge of this region. Toward this end we investigate here the areas in the tails of the distribution as determined by the extreme order statistics as a criterion for testing goodness-of-fit.     

A mixture of the manova and gmanova models

A mixture of the MANOVA and GMANOVA models is presented. The expected value of the response matrix in this model is the sum of two matrix components. The first component represents the GMANOVA portion and the second component represents the MANOVA portion. Maximum likelihood estimators are derived for the parameters in this model, and goodness-of-fit tests are constructed for fuller models via the likelihood ration criterion. Finally, likelihood ration tests for general liinear hypotheses are developed and a numerical example is presented.     

Inference after separated hypotheses testing: an empirical investigation for linear models

Model selection problems arise while constructing unbiased or asymptotically unbiased estimators of measures known as discrepancies to find the best model. Most of the usual criteria are based on goodness-of-fit and parsimony. They aim to maximize a transformed version of likelihood. For linear regression models with normally distributed error, the situation is less clear when two models are equivalent: are they close to or far from the unknown true model? In this work, based on stochastic simulation and parametric simulation, we study the results of Vuong's test, Cox's test, Akaike's information criterion, Bayesian information criterion, Kullback information criterion and bias corrected Kullback information criterion and the ability of these tests to discriminate between non-nested linear models.