Limit theorems for dependent Bernoulli variables with statistical inference

We consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

An algorithm for maximum likelihood estimation in incomplete block variance component models

In the recovery of interblock information to improve the treatment differences estimates in incomplete block designs, the parameter p is usually unknown. Many authors have worked on the problem of estimating it and of studying its properties together with the properties of the treatment differences estimates. In this paper a numerically efficient algorithm is developed which yields the maximum likelihood estimates (MLE) of all the parameters in the mixed incomplete block design model (treatment effects, ρ and variance)     

An Extended Single‐index Model with Missing Response at Random

An extended single‐index model is considered when responses are missing at random. A three‐step estimation procedure is developed to define an estimator for the single‐index parameter vector by a joint estimating equation. The proposed estimator is shown to be asymptotically normal. An algorithm for computing this estimator is proposed. This algorithm only involves one‐dimensional nonparametric smoothers, thereby avoiding the data sparsity problem caused by high model dimensionality. Some simulation studies are conducted to investigate the finite sample performances of the proposed estimators.     

Estimation of nonseparable models with censored dependent variables and endogenous regressors

In this article we develop a nonparametric estimator for the local average response of a censored dependent variable to endogenous regressors in a nonseparable model where the unobservable error term is not restricted to be scalar and where the nonseparable function need not be monotone in the unobservables. We formalize the identification argument put forward in Altonji, Ichimura, and Otsu (2012 Altonji, J. G., Ichimura, H., Otsu, T. (2012). Estimating derivatives in nonseparable models with limited dependent variables. Econometrica 80:1701–1719.[Crossref], [Web of Science ®] , [Google Scholar]), construct a nonparametric estimator, characterize its asymptotic property, and conduct a Monte Carlo investigation to study its small sample properties. Identification is constructive and is achieved through a control function approach. We show that the estimator is consistent and asymptotically normally distributed. The Monte Carlo results are encouraging.     

A goodness of fit test for the multilevel proportional odds model

It is essential to test the goodness of fit of the model before making inferences based on it. Multilevel modeling of ordinal categorical responses is not as developed as for continuous responses. Assessing model adequacy in terms of the goodness of fit with ordinal categorical responses is still being developed and no satisfactory tests are available so far. As a consequence of that, this study concentrates on developing such a goodness of fit test for Multilevel Proportional Odds models and to study the properties of the test.     

Computation of an efficient and robust estimator in a semiparametric mixture model

In this article, we propose an efficient and robust estimation for the semiparametric mixture model that is a mixture of unknown location-shifted symmetric distributions. Our estimation is derived by minimizing the profile Hellinger distance (MPHD) between the model and a nonparametric density estimate. We propose a simple and efficient algorithm to find the proposed MPHD estimation. Monte Carlo simulation study is conducted to examine the finite sample performance of the proposed procedure and to compare it with other existing methods. Based on our empirical studies, the newly proposed procedure works very competitively compared to the existing methods for normal component cases and much better for non-normal component cases. More importantly, the proposed procedure is robust when the data are contaminated with outlying observations. A real data application is also provided to illustrate the proposed estimation procedure.     

Correlated random effects regression analysis for a log-normally distributed variable

In regression analysis, it is assumed that the response (or dependent variable) distribution is Normal, and errors are homoscedastic and uncorrelated. However, in practice, these assumptions are rarely satisfied by a real data set. To stabilize the heteroscedastic response variance, generally, log-transformation is suggested. Consequently, the response variable distribution approaches nearer to the Normal distribution. As a result, the model fit of the data is improved. Practically, a proper (seems to be suitable) transformation may not always stabilize the variance, and the response distribution may not reduce to Normal distribution. The present article assumes that the response distribution is log-normal with compound autocorrelated errors. Under these situations, estimation and testing of hypotheses regarding regression parameters have been derived. From a set of reduced data, we have derived the best linear unbiased estimators of all the regression coefficients, except the intercept which is often unimportant in practice. Unknown correlation parameters have been estimated. In this connection, we have derived a test rule for testing any set of linear hypotheses of the unknown regression coefficients. In addition, we have developed the confidence ellipsoids of a set of estimable functions of regression coefficients. For the fitted regression equation, an index of fit has been proposed. A simulated study illustrates the results derived in this report.     

Hazard rate ordering of the second-order statistics from multiple-outlier PHR samples

In this paper, we compare the hazard rate functions of the second-order statistics arising from two sets of independent multiple-outlier proportional hazard rates (PHR) samples. It is proved that the submajorization order between the sample size vectors together with the supermajorization order between the hazard rate vectors imply the hazard rate ordering between the corresponding second-order statistics from multiple-outlier PHR random variables. The results established here provide theoretical guidance both for the winner's price for the bid in the second-price reverse auction in auction theory and fail-safe system design in reliability. Some numerical examples are also provided for illustration.     

Marginal likelihood estimation from the Metropolis output: tips and tricks for efficient implementation in generalized linear latent variable models

The marginal likelihood can be notoriously difficult to compute, and particularly so in high-dimensional problems. Chib and Jeliazkov employed the local reversibility of the Metropolis–Hastings algorithm to construct an estimator in models where full conditional densities are not available analytically. The estimator is free of distributional assumptions and is directly linked to the simulation algorithm. However, it generally requires a sequence of reduced Markov chain Monte Carlo runs which makes the method computationally demanding especially in cases when the parameter space is large. In this article, we study the implementation of this estimator on latent variable models which embed independence of the responses to the observables given the latent variables (conditional or local independence). This property is employed in the construction of a multi-block Metropolis-within-Gibbs algorithm that allows to compute the estimator in a single run, regardless of the dimensionality of the parameter space. The counterpart one-block algorithm is also considered here, by pointing out the difference between the two approaches. The paper closes with the illustration of the estimator in simulated and real-life data sets.