On testing hypotheses with divergence statistics

Using divergence measures based on entropy functions, a procedure to test statistical hypotheses is proposed. Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained. Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators are considered, so they can be used to construct confidence intervals and to test statistical hypotheses based on one or more samples. These results can also be applied to multinomial populations. Tests of goodness of fit and tests of homogeneity can be constructed.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Exponentiality test based on alpha-divergence and gamma-divergence

In the present paper, we use the already defined alpha-divergence and gamma-divergence for constructing some goodness of fit tests for exponentiality. These divergence measures are very robust with respect to outliers. Since the existence of outliers among statistical data can be lead to misleading results, therefore utilizing these divergence measures can be of importance. In order to construct test statistics, two estimators are used for alpha-divergence and gamma-divergence. In the first one, we consider the alpha-divergence and gamma-divergence of the equilibrium distribution function, which is well defined on the empirical distribution function (EDF) and is proposed as an EDF-based goodness of fit test statistic. The second one is an estimator in manner of Vasicek entropy estimator. Simulation results indicate that in comparison with the other tests statistics, our mentioned test statistics almost in most of the cases have higher power. Finally, two examples containing outliers illustrate the importance and use of the proposed tests.     

Primal and Dual Capacity Utilization: An Application to Productivity Measurement in the U.S. Automobile Industry

Capacity utilization measures have traditionally been constructed as indexes of actual, as compared to “potential,” output. This potential or capacity output (Y*) can be represented within an economic model of the firm as the tangency between the short- and long-run average cost curves. Economic theoretical measures of capacity utilization (CU) can then be characterized as Y/Y* where Y is the realized level of output. These quantity or primal CU measures allow for economic interpretation; they provide explicit inference as to how changes in exogenous variables affect CU. Additional information for analyzing deviations from capacity production can be obtained by assessing the “dual” cost of the gap.

In this article the definitions and representations of primal-output and dual-cost CU measures are formalized within a dynamic model of a monopolistic firm. As an illustration of this approach to characterizing CU measures, a model is estimated for the U.S. automobile industry, 1959–1980, and primal and dual CU indexes are constructed. Application of these indexes to adjustment-of-productivity measures for “disequilibrium” is then carried out, using the dual-cost measure.     

Tail-weighted measures of dependence

Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall's tau and Spearman's rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures – the portfolio value-at-risk and conditional tail expectation.     

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.     

A “Reverse Order” Elementary Statistics Course

A family of coefficients for measuring monotone association is presented. These include measures of association of ordinal or interval variables such as gamma of Goodman and Kruskal, Somers's d_yx , Kendall's tau, or Spearman's rho as special cases. The article shows how a large number of measures of association can be put into a single general form. These coefficients are used as a basis for defining a variety of data analysis techniques.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.     

Adaptive Inference for Multi-Stage Survey Data

Multi-level models can be used to account for clustering in data from multi-stage surveys. In some cases, the intraclass correlation may be close to zero, so that it may seem reasonable to ignore clustering and fit a single-level model. This article proposes several adaptive strategies for allowing for clustering in regression analysis of multi-stage survey data. The approach is based on testing whether the PSU-level variance component is zero. If this hypothesis is retained, then variance estimates are calculated ignoring clustering; otherwise, clustering is reflected in variance estimation. A simple simulation study is used to evaluate the various procedures.     

An improvement of a monte carlo technique using asymptotic moments with an application to the likelihood ratio statistic

The use of asymptotic moments to increase the precision of the control variate technique for Monte Carlo estimation is dis­cussed. An application is made to the estimation of the mean and variance of the likelihood ratio goodness–of–fit statistic with the Pearson statistic used as a control variate. Estimates of the variance reductions are given.     

Approximate distribution and test of fit for the clustering effect in the dirichlet multinomial model

The Dirichlet-multinomial model is considered as a model for cluster sampling. The model assumes that the design's covariance matrix is a constant times the covariance under multinomial sampling. The use of this model requires estimating a parameter C, that measures the clustering effect. In this paper, a regression estimate for C is obtained. An approximate distribution of this estimator is obtained through the use of asymptotic techniques. A goodness of fit statistic for testing the fit of the Dirichlet Multinomial model is also obtained, based on those asymptotic techniques. These statistics provide a means of knowing when the data satisfy the model assumption. These results are used to analyze data concerning the authorship of Greek prose.     

Splice plots for generalized linear models

ABSTRACT

We develop splice plots as a diagnostic tool for parametric generalized linear models. Splice plots use the independence of the outcome and explanatory measures given the regression function. Plotting differences between the estimated parametric regression function and non-parametric estimates of the regression function computed in small neighborhoods of the fitted values from the parametric model can be used to assess model fit.     

The effect of different data structures,sample sizes on model fit measures

In this study, our aim was to investigate the changes of different data structures and different sample sizes on the structural equation modeling and the influence of these factors on the model fit measures. Examining the created structural equation modeling under different data structures and sample sizes, the evaluation of model fit measures were performed with a simulation study. As a result of the simulation study, optimization and negative variance estimation problems have been encountered depending on the sample size and changing correlations. It was observed that these problems disappeared either by increasing the sample size or the correlations between the variables in factor. For upcoming studies, the choice of RMSEA and IFI model fit measures can be suggested in all sample sizes and the correlation values for data sets are ensured the multivariate normal distribution assumption.     

Chi–square tests for comparing vectors of proportions for several cluster samples

Test statistics are developed for comparing vectors of proportions obtained from several independent two–stage cluster samples. It is assumed that clusters are selected with probability proportional to size for each sample. Wald's general method of constructing quadratic forms is used to obtain a large sample chi–square test. More easily evaluted chi–square tests are derived from the Dirichlet–multinnomial model. Corresponding goodness–of–fit test for the Dirichlet–multinomial model are also derived.     

Predictor relative importance and matching regression parameters

Predictor importance in applied regression modeling gives the main operational tools for managers and decision-makers. The paper considers estimation of predictors' importance in regression using measures introduced in works by Gibson and R. Johnson (GJ), then modified by Green, Carroll, and DeSarbo, and developed further by J. Johnson (JJ). These indices of importance are based on the orthonormal decomposition of the data matrix, and the work shows how to improve this approximation. Using predictor importance, the regression coefficients can also be adjusted to reach the best data fit and to be meaningful and interpretable. The results are compared with the robust to multicollinearity, but computationally difficult, Shapley value regression (SVR). They show that the JJ index is good for importance estimation, but the GJ index outperforms it if both predictor importance and coefficients of regression are needed; hence, this index (GJ) can be used in place of the more computationally intensive estimation by SVR. The results can be easily estimated by the considered approach that is very useful in practical regression modeling and analysis, especially for big data.     

A simple test procedure in standardizing the power of Hosmer–Lemeshow test in large data sets

The Hosmer–Lemeshow (H–L) test is a widely used method when assessing the goodness-of-fit of a logistic regression model. However, the H–L test is sensitive to the sample sizes and the number of groups in H–L test. Cautions need to be taken for interpreting an H–L test with a large sample size. In this paper, we propose a simple test procedure to evaluate the model fit of logistic regression model with a large sample size, in which a bootstrap method is used and the test result is determined by the power of H–L test at the target sample size. Simulation studies show that the proposed method can effectively standardize the power of the H–L test under the pre-specified level of type I error. Application to the two datasets illustrates the usefulness of the proposed model.     

Variations of Q–Q Plots: The Power of Our Eyes!

In statistical modeling, we strive to specify models that resemble data collected in studies or observed from processes. Consequently, distributional specification and parameter estimation are central to parametric models. Graphical procedures, such as the quantile–quantile (Q–Q) plot, are arguably the most widely used method of distributional assessment, though critics find their interpretation to be overly subjective. Formal goodness of fit tests are available and are quite powerful, but only indicate whether there is a lack of fit, not why there is lack of fit. In this article, we explore the use of the lineup protocol to inject rigor into graphical distributional assessment and compare its power to that of formal distributional tests. We find that lineup tests are considerably more powerful than traditional tests of normality. A further investigation into the design of Q–Q plots shows that de-trended Q–Q plots are more powerful than the standard approach as long as the plot preserves distances in x and y to be the same. While we focus on diagnosing nonnormality, our approach is general and can be directly extended to the assessment of other distributions.     

Prediction of disease status: A regressive model approach for repeated measures

In this paper, regressive models are proposed for modeling a sequence of transitions in longitudinal data. These models are employed to predict the future status of the outcome variable of the individuals on the basis of their underlying background characteristics or risk factors. The estimation of parameters and also estimates of conditional and unconditional probabilities are shown for repeated measures. The goodness of fit tests are extended in this paper on the basis of the deviance and the Hosmer–Lemeshow procedures and generalized to repeated measures. In addition, to measure the suitability of the proposed models for predicting the disease status, we have extended the ROC curve approach to repeated measures. The procedure is shown for the conditional models for any order as well as for the unconditional model, to predict the outcome at the end of the study. The test procedures are also suggested. For testing the differences between areas under the ROC curves in subsequent follow-ups, two different test procedures are employed, one of which is based on permutation test. In this paper, an unconditional model is proposed on the basis of conditional models for the disease progression of depression among the elderly population in the USA on the basis of the Health and Retirement Survey data collected longitudinally. The illustration shows that the disease progression observed conditionally can be employed to predict the outcome and the role of selected variables and the previous outcomes can be utilized for predictive purposes. The results show that the percentage of correct predictions of a disease is quite high and the measures of sensitivity and specificity are also reasonably impressive. The extended measures of area under the ROC curve show that the models provide a reasonably good fit in terms of predicting the disease status during a long period of time. This procedure will have extensive applications in the field of longitudinal data analysis where the objective is to obtain estimates of unconditional probabilities on the basis of series of conditional transitional models.     

A Goodness of Fit Test for the Multilevel Logistic Model

It is crucial to test the goodness of fit of a model before it is used to make statistical inferences. However, no satisfactory goodness of fit test is available for the case of categorical multilevel data which occur when categorical data are clustered or hierarchical in nature. Hence the aim of this paper is to develop a new goodness of fit test for multilevel binary data based on Hosmer and Lemeshow and Lipsitz et.al. In order to identify the properties of the developed test, simulation studies were carried out to assess the Type I error and the power.     

A new generalized Balakrishnan type skewed–normal distribution: properties and associated inference

It is also shown that our proposed skew-normal model subsumes many other well-known skew-normal model that exists in the literature. Recent work on a new two-parameter generalized skew-normal model has received a lot of attention. This paper presents a new generalized Balakrishnan type skew–normal distribution by introducing two shape parameters. We also provide some useful results for this new generalization. It is also shown that our proposed skew–normal model subsumes the original Balakrishnan skew–normal model (2002) as well as other well–known skew–normal models as special cases. The resulting flexible model can be expected to fit a wider variety of data structures than either of the models involving a single skewing mechanism. For illustrative purposes, a famed data set on IQ scores has been used to exhibit the efficacy of the proposed model.