Thomas J. DiCiccio

Computing Bayes Factors by Combining Simulation and Asymptotic Approximations

Abstract The Bayes factor is a ratio of two posterior normalizing constants, which may be difficult to compute. We compare several methods of estimating Bayes factors when it is possible to simulate observations from the posterior distributions, via Markov chain Monte Carlo or other techniques. The methods that we study are all easily applied without consideration of special features of the problem, provided that each posterior distribution is well behaved in the sense of having a single dominant mode. We consider a simulated version of Laplaces method, a simulated version of Bartlett correction, importance sampling, and a reciprocal importance sampling technique. We also introduce local volume corrections for each of these. In addition, we apply the bridge sampling method of Meng and Wong. We find that a simulated version of Laplaces method, with local volume correction, furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly. A simple bridge sampli...

Inferential Aspects of the Skew Exponential Power Distribution

When the analysis of real data indicates that normality assumptions are untenable, more flexible models that cope with the most prevalent deviations from normality can be adopted. In this context, the skew exponential power (SEP) distribution warrants special attention, because it encompasses distributions having both heavy tails and skewness, it allows likelihood inference, and it includes the normal model as a special case. This article concerns likelihood inference about the parameters of the SEP family. In particular, the information matrix of the maximum likelihood estimators (MLEs) is obtained and finite-sample properties of the estimators are investigated numerically. Special attention is given to the properties of the MLEs and likelihood ratio statistics when the data are drawn from a normal distribution, because this case is relevant for using the SEP distribution to test for normality. Application of the SEP distribution in robust estimation problems is considered for both independent and dependent data. Under moderate deviations from normality, estimators obtained under the SEP distribution are shown to outperform the normal-based estimators and to compete with robust estimators; furthermore, the SEP distribution offers the benefits of having a specified distribution, such as interpretable location and scale parameters under nonnormality.

Analytical approximations for iterated bootstrap confidence intervals

Standard algorithms for the construction of iterated bootstrap confidence intervals are computationally very demanding, requiring nested levels of bootstrap resampling. We propose an alternative approach to constructing double bootstrap confidence intervals that involves replacing the inner level of resampling by an analytical approximation. This approximation is based on saddlepoint methods and a tail probability approximation of DiCiccio and Martin (1991). Our technique significantly reduces the computational expense of iterated bootstrap calculations. A formal algorithm for the construction of our approximate iterated bootstrap confidence intervals is presented, and some crucial practical issues arising in its implementation are discussed. Our procedure is illustrated in the case of constructing confidence intervals for ratios of means using both real and simulated data. We repeat an experiment of Schenker (1985) involving the construction of bootstrap confidence intervals for a variance and demonstrate that our technique makes feasible the construction of accurate bootstrap confidence intervals in that context. Finally, we investigate the use of our technique in a more complex setting, that of constructing confidence intervals for a correlation coefficient.

On expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods

We consider a general multiparameter set-up, where both the interest and the nuisance parameters are possibly vector valued. We derive an explicit higher order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof. Our general framework also allows us to include highest posterior density regions, with approximate frequentist validity, in the study. The fact that our interest parameter is possibly vector valued complicates the derivation and warrants the development of special tools and techniques.

Robust portfolio estimation under skew-normal return processes

In this paper, we study issues related to the optimal portfolio estimators and the local asymptotic normality (LAN) of the return process under the assumption that the return process has an infinite moving average (MA) (∞) representation with skew-normal innovations. The paper consists of two parts. In the first part, we discuss the influence of the skewness parameter δ of the skew-normal distribution on the optimal portfolio estimators. Based on the asymptotic distribution of the portfolio estimator ĝ for a non-Gaussian dependent return process, we evaluate the influence of δ on the asymptotic variance V(δ) of ĝ. We also investigate the robustness of the estimators of a standard optimal portfolio via numerical computations. In the second part of the paper, we assume that the MA coefficients and the mean vector of the return process depend on a lower-dimensional set of parameters. Based on this assumption, we discuss the LAN property of the returns distribution when the innovations follow a skew-normal law. The influence of δ on the central sequence of LAN is evaluated both theoretically and numerically.

Objective Bayes, conditional inference and the signed root likelihood ratio statistic

Bayesian properties of the signed root likelihood ratio statistic are analysed. Conditions for first-order probability matching are derived by the examination of the Bayesian posterior and frequentist means of this statistic. Second-order matching conditions are shown to arise from matching of the Bayesian posterior and frequentist variances of a mean-adjusted version of the signed root statistic. Conditions for conditional probability matching in ancillary statistic models are derived and discussed. Copyright 2012, Oxford University Press.

Computer-intensive Conditional Inference

Conditional inference is a fundamental part of statistical theory. However, exact conditional inference is often awkward, leading to the desire for methods which offer accurate approximations. Such a methodology is provided by small-sample likelihood asymptotics. We argue in this paper that simple, simulation-based methods also offer accurate approximations to exact conditional inference in multiparameter exponential family and ancillary statistic settings. Bootstrap simulation of the marginal distribution of an appropriate statistic provides a conceptually simple and highly effective alternative to analytic procedures of approximate conditional inference.

Testing for sub-models of the skew t-distribution

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically

A Simple Analysis of the Exact Probability Matching Prior in the Location-Scale Model

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