Rényi entropy of m-generalized order statistics

Abstract

Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.     

Inference and Decision Making for 21st-Century Drug Development and Approval

ABSTRACT

The cost and time of pharmaceutical drug development continue to grow at rates that many say are unsustainable. These trends have enormous impact on what treatments get to patients, when they get them and how they are used. The statistical framework for supporting decisions in regulated clinical development of new medicines has followed a traditional path of frequentist methodology. Trials using hypothesis tests of “no treatment effect” are done routinely, and the p-value < 0.05 is often the determinant of what constitutes a “successful” trial. Many drugs fail in clinical development, adding to the cost of new medicines, and some evidence points blame at the deficiencies of the frequentist paradigm. An unknown number effective medicines may have been abandoned because trials were declared “unsuccessful” due to a p-value exceeding 0.05. Recently, the Bayesian paradigm has shown utility in the clinical drug development process for its probability-based inference. We argue for a Bayesian approach that employs data from other trials as a “prior” for Phase 3 trials so that synthesized evidence across trials can be utilized to compute probability statements that are valuable for understanding the magnitude of treatment effect. Such a Bayesian paradigm provides a promising framework for improving statistical inference and regulatory decision making.     

Beyond tail median and conditional tail expectation: Extreme risk estimation using tail Lp-optimization

The conditional tail expectation (CTE) is an indicator of tail behavior that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A valuable alternative is the median shortfall (MS), although it only gives information about the frequency of a tail event. We construct a class of tail L^p-medians encompassing the MS and CTE. For p in (1,2), a tail L^p-median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on real fire insurance data.     

Modelling daily multivariate pollutant data at multiple sites

Summary. This paper considers the spatiotemporal modelling of four pollutants measured daily at eight monitoring sites in London over a 4-year period. Such multiple-pollutant data sets measured over time at multiple sites within a region of interest are typical. Here, the modelling was carried out to provide the exposure for a study investigating the health effects of air pollution. Alternative objectives include the design problem of the positioning of a new monitoring site, or for regulatory purposes to determine whether environmental standards are being met. In general, analyses are hampered by missing data due, for example, to a particular pollutant not being measured at a site, a monitor being inactive by design (e.g. a 6-day monitoring schedule) or because of an unreliable or faulty monitor. Data of this type are modelled here within a dynamic linear modelling framework, in which the dependences across time, space and pollutants are exploited. Throughout the approach is Bayesian, with implementation via Markov chain Monte Carlo sampling.     

Computing maximum likelihood estimates from type II doubly censored exponential data

It is well-known that, under Type II double censoring, the maximum likelihood (ML) estimators of the location and scale parameters, θ and δ, of a twoparameter exponential distribution are linear functions of the order statistics. In contrast, when θ is known, theML estimator of δ does not admit a closed form expression. It is shown, however, that theML estimator of the scale parameter exists and is unique. Moreover, it has good large-sample properties. In addition, sharp lower and upper bounds for this estimator are provided, which can serve as starting points for iterative interpolation methods such as regula falsi. Explicit expressions for the expected Fisher information and Cramér-Rao lower bound are also derived. In the Bayesian context, assuming an inverted gamma prior on δ, the uniqueness, boundedness and asymptotics of the highest posterior density estimator of δ can be deduced in a similar way. Finally, an illustrative example is included.     

非统计专业统计学教育刍议

针对非统计专业统计学教育目前存在的问题,阐述了提高对统计学重要性的认识是保证统计教学效果的前提;统计学教学环节的改革是提高教学效果的重要保证;统计教师的业务素质是提高统计教学效果的关键。     

Two-stage Huber estimation

In this paper we propose a new robust estimator in the context of two-stage estimation methods directed towards the correction of endogeneity problems in linear models. Our estimator is a combination of Huber estimators for each of the two stages, with scale corrections implemented using preliminary median absolute deviation estimators. In this way we obtain a two-stage estimation procedure that is an interesting compromise between concerns of simplicity of calculation, robustness and efficiency. This method compares well with other possible estimators such as two-stage least-squares (2SLS) and two-stage least-absolute-deviations (2SLAD), asymptotically and in finite samples. It is notably interesting to deal with contamination affecting more heavily the distribution tails than a few outliers and not losing as much efficiency as other popular estimators in that case, e.g. under normality. An additional originality resides in the fact that we deal with random regressors and asymmetric errors, which is not often the case in the literature on robust estimators.     

Generalized Laplacian approximations in Bayesian inference

This paper presents a new Laplacian approximation to the posterior density of η = g(θ). It has a simpler analytical form than that described by Leonard et al. (1989). The approximation derived by Leonard et al. requires a conditional information matrix R_η to be positive definite for every fixed η. However, in many cases, not all R_η are positive definite. In such cases, the computations of their approximations fail, since the approximation cannot be normalized. However, the new approximation may be modified so that the corresponding conditional information matrix can be made positive definite for every fixed η. In addition, a Bayesian procedure for contingency-table model checking is provided. An example of cross-classification between the educational level of a wife and fertility-planning status of couples is used for explanation. Various Laplacian approximations are computed and compared in this example and in an example of public school expenditures in the context of Bayesian analysis of the multiparameter Fisher-Behrens problem.     

Divergence-based estimation and testing with misclassified data

The well-known chi-squared goodness-of-fit test for a multinomial distribution is generally biased when the observations are subject to misclassification. In Pardo and Zografos (2000) the problem was considered using a double sampling scheme and ø-divergence test statistics. A new problem appears if the null hypothesis is not simple because it is necessary to give estimators for the unknown parameters. In this paper the minimum ø-divergence estimators are considered and some of their properties are established. The proposed ø-divergence test statistics are obtained by calculating ø-divergences between probability density functions and by replacing parameters by their minimum ø-divergence estimators in the derived expressions. Asymptotic distributions of the new test statistics are also obtained. The testing procedure is illustrated with an example.     

Test for homogeneity in Hardy–Weinberg normal mixture model

The phenotype of a quantitative trait locus (QTL) is often modeled by a finite mixture of normal distributions. If the QTL effect depends on the number of copies of a specific allele one carries, then the mixture model has three components. In this case, the mixing proportions have a binomial structure according to the Hardy–Weinberg equilibrium. In the search for QTL, a significance test of homogeneity against the Hardy–Weinberg normal mixture model alternative is an important first step. The LOD score method, a likelihood ratio test used in genetics, is a favored choice. However, there is not yet a general theory for the limiting distribution of the likelihood ratio statistic in the presence of unknown variance. This paper derives the limiting distribution of the likelihood ratio statistic, which can be described by the supremum of a quadratic form of a Gaussian process. Further, the result implies that the distribution of the modified likelihood ratio statistic is well approximated by a chi-squared distribution. Simulation results show that the approximation has satisfactory precision for the cases considered. We also give a real-data example.