Mixtures of regression models with incomplete and noisy data

The estimation of the mixtures of regression models is usually based on the normal assumption of components and maximum likelihood estimation of the normal components is sensitive to noise, outliers, or high-leverage points. Missing values are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this article, we propose the mixtures of regression models for contaminated incomplete heterogeneous data. The proposed models provide robust estimates of regression coefficients varying across latent subgroups even under the presence of missing values. The methodology is illustrated through simulation studies and a real data analysis.     

An update on ‘a comparative study of tests for homogeneity of variance’

Tests for equality of variances using independent samples are widely used in data analysis. Conover et al. [A comparative study of tests for homogeneity of variance, with applications to the outer continental shelf bidding data. Technometrics. 1981;23:351–361], won the Youden Prize by comparing 56 variations of popular tests for variance on the basis of robustness and power in 60 different scenarios. None of the tests they compared were robust and powerful for the skewed distributions they considered. This study looks at 12 variations they did not consider, and shows that 10 are robust for the skewed distributions they considered plus the lognormal distribution, which they did not study. Three of these 12 have clearly superior power for skewed distributions, and are competitive in terms of robustness and power for all of the distributions considered. They are recommended for general use based on robustness, power, and ease of application.     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.     

Another look at the robustness of colton's model for comparing two medical treatments

The robustness of an extended version of Colton's decision theoretic model is considered. The extended version includes the losses due to the patients who are not entered in the experiment, but require treatment while the experiment is in progress. Among the topics considered are the effects of risk of using a sample size considerably less than the optimum, use of an incorrect patient horizon, application of a modified loss function, and use of a two point prior distribution. It is shown that the investigated model is robust with respect to all these changes with the exception of the use of the modified prior density.     

A robust test for autocorrelation in the presence of a structural break in variance

It has been known that when there is a break in the variance (unconditional heteroskedasticity) of the error term in linear regression models, a routine application of the Lagrange multiplier (LM) test for autocorrelation can cause potentially significant size distortions. We propose a new test for autocorrelation that is robust in the presence of a break in variance. The proposed test is a modified LM test based on a generalized least squares regression. Monte Carlo simulations show that the new test performs well in finite samples and it is especially comparable to other existing heteroskedasticity-robust tests in terms of size, and much better in terms of power.     

Optimal quantization applied to sliced inverse regression

In this paper we consider a semiparametric regression model involving a d-dimensional quantitative explanatory variable X and including a dimension reduction of X via an index β′X. In this model, the main goal is to estimate the Euclidean parameter β and to predict the real response variable Y conditionally to X. Our approach is based on sliced inverse regression (SIR) method and optimal quantization in L^p-norm. We obtain the convergence of the proposed estimators of β and of the conditional distribution. Simulation studies show the good numerical behavior of the proposed estimators for finite sample size.     

An m-estimation-based model selection criterion with a data-oriented penalty

In Wu and Zen (1999), a linear model selection procedure based on M-estimation is proposed, which includes many classical model selection criteria as its special cases, and it is shown that the selection procedure is strongly consistent for a variety of penalty functions. In this paper, we will investigate its small sample performances for some choices of fixed penalty functions. It can be seen that the performance varies with the choice of the penalty. Hence, a randomized penalty based on observed data is proposed, which preserves the consistency property and provides improved performance over a fixed choice of penalty functions.     

BUSDM – an algorithm for the bottom-up search of departures from a model

Searching for regions of the input space where a statistical model is inappropriate is useful in many applications. The study proposes an algorithm for finding local departures from a regression-type prediction model. The algorithm returns low-dimensional hypercubes where the average prediction error clearly departs from zero. The study describes the developed algorithm, and shows successful applications on the simulated and real data from the steel plate production. The algorithms that have been originally developed for searching regions of the high-response value from the input space are reviewed and considered as alternative methods for locating model departures. The proposed algorithm succeeds in locating the model departure regions better than the compared alternatives. The algorithm can be utilized in sequential follow-up of a model as time goes along and new data are observed.     

Interval-valued data regression using partial linear model

Semi-parametric modelling of interval-valued data is of great practical importance, as exampled by applications in economic and financial data analysis. We propose a flexible semi-parametric modelling of interval-valued data by integrating the partial linear regression model based on the Center & Range method, and investigate its estimation procedure. Furthermore, we introduce a test statistic that allows one to decide between a parametric linear model and a semi-parametric model, and approximate its null asymptotic distribution based on wild Bootstrap method to obtain the critical values. Extensive simulation studies are carried out to evaluate the performance of the proposed methodology and the new test. Moreover, several empirical data sets are analysed to document its practical applications.     

A general class of test statistics for Van Valen's Red Queen hypothesis

Van Valen's Red Queen hypothesis states that within a homogeneous taxonomic group the age is statistically independent of the rate of extinction. The case of the Red Queen hypothesis being addressed here is when the homogeneous taxonomic group is a group of similar species. Since Van Valen's work, various statistical approaches have been used to address the relationship between taxon age and the rate of extinction. We propose a general class of test statistics that can be used to test for the effect of age on the rate of extinction. These test statistics allow for a varying background rate of extinction and attempt to remove the effects of other covariates when assessing the effect of age on extinction. No model is assumed for the covariate effects. Instead we control for covariate effects by pairing or grouping together similar species. Simulations are used to compare the power of the statistics. We apply the test statistics to data on Foram extinctions and find that age has a positive effect on the rate of extinction. A derivation of the null distribution of one of the test statistics is provided in the supplementary material.