A modification of the Fisher-Cornish approximation for the student t percentiles

An asymptotic expansion of the Student t distribution is derived by expanding the standardized Student t distribution in terms of the normal distribution. This expansion is inverted to obtain corresponding asymptotic expansions for the Student t percentiles as functions of the standard normal percentiles0 Using the first two, three or four terms of these expansions, we get approximations of the Student t percentiles which are generally more accurate than the approximations given by Fisher and Cornish(1960) and Koehler (1983).An approximation of the distribution function obtained from this expansion is compared with the approximations discussed by Ling (1978) andfound to be more accurate for moderate degrees of freedom.     

On Stein Identity,Chernoff Inequality,and Orthogonal Polynomials

In this article, we present a general method for deriving Stein-like identity and Chernoff-like inequality based on orthogonal polynomials. In order to illustrate our method, some applications are given with respect to normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution, not only for random variables but also for random vectors, resulting corresponding Stein-like identity and Chernoff-like inequality are obtained consequently. Within our best knowledge, some of our matrix version results are new in the literature. In addition, forward difference formulae of Charlier polynomials, Krawtchouk polynomials and Meixner polynomials, Stein-like identity, and Chernoff-like inequality with respect to Beta distribution, as well as Rodrigues formula of Meixner polynomials are also prepared in the first time within our limited information. Interestingly, as far as normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution are concerned, we found that their Stein-like identity and corresponding Chernoff-like inequality are related closely, by examining their Rodrigues formula.     

Inference in affine shape theory under elliptical models

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.     

Testing sphericity using small samples

It is shown here that with small sample sizes, the null distribution of the ellipticity, or sphericity, likelihood criterion W(n, p) can be obtained very accurately, either by computation using the Meijer function, or by Monte Carlo simulation. Testing in repeated measures design can now be carried out with much more accuracy.     

Empiric bayes estimation of binomial probability

An empirical Bayes estimator of a binomial parameter, based on orthogonal polynomials on (0,1), is introduced. The resulting estimator of the prior density is asymptotically optimal. The method allows one to combine Bayes and empiric Bayes methods with smoothing in a natural way.     

Normality testing for a long-memory sequence using the empirical moment generating function

Moment generating functions and more generally, integral transforms for goodness-of-fit tests have been in use in the last several decades. Given a set of observations, the empirical transforms are easy to compute, being simply a sample mean, and due to uniqueness properties, these functions can be used for goodness-of-fit tests. This paper focuses on time series observations from a stationary process for which the moment generating function exists and the correlations have long-memory. For long-memory processes, the infinite sum of the correlations diverges and the realizations tend to have spurious trend like patterns where there may be none. Our aim is to use the empirical moment generating function to test the null hypothesis that the marginal distribution is Gaussian. We provide a simple proof of a central limit theorem using ideas from Gaussian subordination models (Taqqu, 1975) and derive critical regions for a graphical test of normality, namely the T₃-plot ( Ghosh, 1996). Some simulated and real data examples are used for illustration.     

Regression with fractional polynomials when interactions are erroneously omitted

We propose a new summary tool, so-called average predictive comparison (APC), which summarizes the effect of a particular predictor in a context of regression. Different from the definition in our earlier work (Liu and Gustafson, 2008), the new definition allows a pointwise evaluation of a predictor's effect for any given value of this predictor. We employ this summary tool to examine the consequence of erroneously omitting interactions in regression models. To be able to involve curved relationships between a response variable and predictors, we consider fractional polynomial regression models (Royston and Altman, 1994). We derive the asymptotic properties of the APC estimates under a general setting with p(≥2)$p(\ge 2)$ predictors involved. In particular, when there are only two predictors of interest, we find out that the APC estimator is robust to the model misspecification under some certain conditions. We illustrate the application of the proposed summary tool via a real data example. We also conduct simulation experiments to further check the performance of the APC estimates.     

On the boundary properties of Bernstein polynomial estimators of density and distribution functions

For density and distribution functions supported on [0,1], Bernstein polynomial estimators are known to have optimal mean integrated squared error (MISE) properties under the usual smoothness conditions on the function to be estimated. These estimators are also known to be well-behaved in terms of bias: they have uniform bias over the entire unit interval. What is less known, however, is that some of these estimators do experience a boundary effect, but of a different nature than what is seen with the usual kernel estimators.     

A flexible Bayesian non-parametric approach for fitting the odds to case II interval-censored data

Interval-censored survival data arise often in medical applications and clinical trials [Wang L, Sun J, Tong X. Regression analyis of case II interval-censored failure time data with the additive hazards model. Statistica Sinica. 2010;20:1709–1723]. However, most of existing interval-censored survival analysis techniques suffer from challenges such as heavy computational cost or non-proportionality of hazard rates due to complicated data structure [Wang L, Lin X. A Bayesian approach for analyzing case 2 interval-censored data under the semiparametric proportional odds model. Statistics & Probability Letters. 2011;81:876–883; Banerjee T, Chen M-H, Dey DK, et al. Bayesian analysis of generalized odds-rate hazards models for survival data. Lifetime Data Analysis. 2007;13:241–260]. To address these challenges, in this paper, we introduce a flexible Bayesian non-parametric procedure for the estimation of the odds under interval censoring, case II. We use Bernstein polynomials to introduce a prior for modeling the odds and propose a novel and easy-to-implement sampling manner based on the Markov chain Monte Carlo algorithms to study the posterior distributions. We also give general results on asymptotic properties of the posterior distributions. The simulated examples show that the proposed approach is quite satisfactory in the cases considered. The use of the proposed method is further illustrated by analyzing the hemophilia study data [McMahan CS, Wang L. A package for semiparametric regression analysis of interval-censored data; 2015. http://CRAN.R-project.org/package=ICsurv.