A generalized normal distribution

Undoubtedly, the normal distribution is the most popular distribution in statistics. In this paper, we introduce a natural generalization of the normal distribution and provide a comprehensive treatment of its mathematical properties. We derive expressions for the nth moment, the nth central moment, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, and the asymptotic distribution of the extreme order statistics. We also discuss estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix.     

Variable selection in quantile regression when the models have autoregressive errors

This paper considers a problem of variable selection in quantile regression with autoregressive errors. Recently, Wu and Liu (2009) investigated the oracle properties of the SCAD and adaptive-LASSO penalized quantile regressions under non identical but independent error assumption. We further relax the error assumptions so that the regression model can hold autoregressive errors, and then investigate theoretical properties for our proposed penalized quantile estimators under the relaxed assumption. Optimizing the objective function is often challenging because both quantile loss and penalty functions may be non-differentiable and/or non-concave. We adopt the concept of pseudo data by Oh et al. (2007) to implement a practical algorithm for the quantile estimate. In addition, we discuss the convergence property of the proposed algorithm. The performance of the proposed method is compared with those of the majorization-minimization algorithm (Hunter and Li, 2005) and the difference convex algorithm (Wu and Liu, 2009) through numerical and real examples.     

Bootstrap tests for multivariate directional alternatives

Tests on multivariate means that are hypothesized to be in a specified direction have received attention from both theoretical and applied points of view. One of the most common procedures used to test this cone alternative is the likelihood ratio test (LRT) assuming a multivariate normal model for the data. However, the resulting test for an ordered alternative is biased in that the only usable critical values are bounds on the null distribution. The present paper provides empirical evidence that bootstrapping the null distribution of the likelihood ratio statistic results in a bootstrap test (BT) with comparable power properties without the additional burden of assuming multivariate normality. Additionally, the tests based on the LRT statistic can reject the null hypothesis in favor of the alternative even though the true means are far from the alternative region. The BT also has similar properties for normal and nonnormal data. This anomalous behavior is due to the formulation of the null hypothesis and a possible remedy is to reformulate the null to be the complement of the alternative hypothesis. We discuss properties of a BT for the modified set of hypotheses (MBT) based on a simulation study. The resulting test is conservative in general and in some specific cases has power estimates comparable to those for existing methods. The BT has higher sensitivity but relatively lower specificity, whereas the MBT has higher specificity but relatively lower sensitivity.     

The existence of the strong combined-optimal design

The technique of fold-over is useful for conducting follow-up experiments. Based on the minimum aberration criterion, Li and Lin (2003) developed an algorithm and used computer to search the corresponding optimal foldover designs for 16 and 32 runs in the 2 ^k-p design. In their study, they found that the 2¹⁰⁻⁶ design is the only one that is not a strong combined-optimal design among all the designs. However, they did not interpret the reason causing the phenomenon. This article will explore under what kind of conditions, that the strong combined-optimal design will exist, and the solutions of the related problems.     

Some challenges for statistics

The paper gives a highly personal sketch of some current trends in statistical inference. After an account of the challenges that new forms of data bring, there is a brief overview of some topics in stochastic modelling. The paper then turns to sparsity, illustrated using Bayesian wavelet analysis based on a mixture model and metabolite profiling. Modern likelihood methods including higher order approximation and composite likelihood inference are then discussed, followed by some thoughts on statistical education.     

Change Points with Linear Trend Followed by Abrupt Change for the Exponential Distribution

We discuss testing procedures to detect if a random sequence of exponentially distributed random variables has been subjected to a linear trend change followed by an abrupt change. We propose three statistics and explore their distribution theories. As an illustration, we applied these tests to Stanford heart transplant data and airport inter arrival data.     

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS OF THE TRUNCATED CAUCHY DISTRIBUTION

Truncated Cauchy distribution with four unknown parameters is considered and derivation and existence of the maximum likelihood estimates is investigated here. We provide a sufficient condition for the maximum likelihood estimate of the scale parameter to be finite, and also show that the condition is necessary for sufficiently large samples. Note that all the moments of the truncated Cauchy distribution exist which makes it much more attractive as a model when compared to the regular Cauchy. We also study, using simulations, the small sample properties of the maximum likelihood estimates.     

Testing linear inequality constraints in the standard linear model

In this paper we are concerned with the problem of testing whether the â-parameters of the standard linear model satisfy the linear equality constraints R = r when they are known to satisfy the corresponding linear inequality constraints Râ ? r. In particular we will show that the exact finite sample null distributions of the Likelihood Ratio, Wald and Kuhn-Tucker

statistics are known when R is of full row rank but not known when R has less than full row rank. The less than full row rank problem has not been discussed previously but it is of considerable potential importance.

This paper contains several simple numerical examples which illustrate the computational details of the tests     

Information and the Likelihood Function in Exponential Families

A very strong relationship between the directed Kullback divergence and the likelihood function emerges simply and naturally in the context of exponential families. The directed Kullback divergence is an information-theoretic measure of the distance from one distribution to another.     

A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.