A matching prior for the product of normal means based on the modified profile likelihood

In this paper, we develop a matching prior for the product of means in several normal distributions with unrestricted means and unknown variances. For this problem, properly assigning priors for the product of normal means has been issued because of the presence of nuisance parameters. Matching priors, which are priors matching the posterior probabilities of certain regions with their frequentist coverage probabilities, are commonly used but difficult to derive in this problem. We developed the first order probability matching priors for this problem; however, the developed matching priors are unproper. Thus, we apply an alternative method and derive a matching prior based on a modification of the profile likelihood. Simulation studies show that the derived matching prior performs better than the uniform prior and Jeffreys’ prior in meeting the target coverage probabilities, and meets well the target coverage probabilities even for the small sample sizes. In addition, to evaluate the validity of the proposed matching prior, Bayesian credible interval for the product of normal means using the matching prior is compared to Bayesian credible intervals using the uniform prior and Jeffrey’s prior, and the confidence interval using the method of Yfantis and Flatman.     

Large deviations of multiclass M/G/1 queues

Consider a multiclass M/G/1 queue where queued customers are served in their order of arrival at a rate which depends on the customer class. We model this system using a chain with states represented by a tree. Since the service time distribution depends on the customer class, the stationary distribution is not of product form so there is no simple expression for the stationary distribution. Nevertheless, we can find a harmonic function on this chain which provides information about the asymptotics of this stationary distribution. The associated h‐transformation produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. The Canadian Journal of Statistics 37: 327–346; 2009 © 2009 Statistical Society of Canada     

A comparison of efficiencies between quasi-likelihood and pseudo-likelihood estimates in non-separable spatial–temporal binary data

The goal of this paper is to compare the performance of two estimation approaches, the quasi-likelihood estimating equation and the pseudo-likelihood equation, against model mis-specification for non-separable binary data. This comparison, to the authors’ knowledge, has not been done yet. In this paper, we first extend the quasi-likelihood work on spatial data to non-separable binary data. Some asymptotic properties of the quasi-likelihood estimate are also briefly discussed. We then use the techniques of a truncated Gaussian random field with a quasi-likelihood type model and a Gibbs sampler with a conditional model in the Markov random field to generate spatial–temporal binary data, respectively. For each simulated data set, both of the estimation methods are used to estimate parameters. Some discussion about the simulation results are also included.     

Error variance estimation via least squares for small sample nonparametric regression

In this paper we explore statistical properties of some difference-based approaches to estimate an error variance for small sample based on nonparametric regression which satisfies Lipschitz condition. Our study is motivated by Tong and Wang (2005), who estimated error variance using a least squares approach. They considered the error variance as the intercept in a simple linear regression which was obtained from the expectation of their lag-k Rice estimator. Their variance estimators are highly dependent on the setting of a regressor and weight of their simple linear regression. Although this regressor and weight can be varied based on the characteristic of an unknown nonparametric mean function, Tong and Wang (2005) have used a fixed regressor and weight in a large sample and gave no indication of how to determine the regressor and the weight. In this paper, we propose a new approach via local quadratic approximation to determine this regressor and weight. Using our proposed regressor and weight, we estimate the error variance as the intercept of simple linear regression using both ordinary least squares and weighted least squares. Our approach applies to both small and large samples, while most existing difference-based methods are appropriate solely for large samples. We compare the performance of our approach with other existing approaches using extensive simulation study. The advantage of our approach is demonstrated using a real data set.     

Order test for high-dimensional two-sample means

We propose a new method to test the order between two high-dimensional mean curves. The new statistic extends the approach of Follmann (1996) to high-dimensional data by adapting the strategy of Bai and Saranadasa (1996). The proposed procedure is an alternative to the non-negative basis matrix factorization (NBMF) based test of Lee et al. (2008) for the same hypothesis, but it is much easier to implement. We derive the asymptotic mean and variance of the proposed test statistic under the null hypothesis of equal mean curves. Based on theoretical results, we put forward a permutation procedure to approximate the null distribution of the new test statistic. We compare the power of the proposed test with that of the NBMF-based test via simulations. We illustrate the approach by an application to tidal volume traces.     

Inference for comparing a multinomial distribution with a known standard

We propose a new type of stochastic ordering which imposes a monotone tendency in differences between one multinomial probability and a known standard one. An estimation procedure is proposed for the constrained maximum likelihood estimate, and then the asymptotic null distribution is derived for the likelihood ratio test statistic for testing equality of two multinomial distributions against the new stochastic ordering. An alternative test is also discussed based on Neyman modified minimum chi-square estimator. These tests are illustrated with a set of heart disease data.     

Modifications of REML algorithm for HGLMs

Hierarchical generalized linear models (HGLMs) have become popular in data analysis. However, their maximum likelihood (ML) and restricted maximum likelihood (REML) estimators are often difficult to compute, especially when the random effects are correlated; this is because obtaining the likelihood function involves high-dimensional integration. Recently, an h-likelihood method that does not involve numerical integration has been proposed. In this study, we show how an h-likelihood method can be implemented by modifying the existing ML and REML procedures. A small simulation study is carried out to investigate the performances of the proposed methods for HGLMs with correlated random effects.     

A note on statistical inference for differences of covariances

This article investigates statistical inferences about differences of covariances matrices when the response has more than two values. The subspace constructed by differences of covariance matrices is related to the sufficient dimension subspace and the central space. The asymptotic distribution of test statistic for structural dimension is outlined.     

Another look at linear programming for feature selection via methods of regularization

We consider statistical procedures for feature selection defined by a family of regularization problems with convex piecewise linear loss functions and penalties of l ₁ nature. Many known statistical procedures (e.g. quantile regression and support vector machines with l ₁-norm penalty) are subsumed under this category. Computationally, the regularization problems are linear programming (LP) problems indexed by a single parameter, which are known as ‘parametric cost LP’ or ‘parametric right-hand-side LP’ in the optimization theory. Exploiting the connection with the LP theory, we lay out general algorithms, namely, the simplex algorithm and its variant for generating regularized solution paths for the feature selection problems. The significance of such algorithms is that they allow a complete exploration of the model space along the paths and provide a broad view of persistent features in the data. The implications of the general path-finding algorithms are outlined for several statistical procedures, and they are illustrated with numerical examples.