Dempster's combination rule and sufficient statistics

In his articles (1966-1968) concerning statistical inference based on lower and upper probabilities, Dempster refers to the connection between Fisher's fiducial argument and his own ideas of statistical inference. Dempster's main concern however focuses on the “Bayesian” aspects of his theory and not on an elaboration of the relation between Fisher's and his ideas. This article attempts to work out the connection between those two approaches and focuses primarily on the question, whether Dempster's combination rule, his upper and lower probabilty based on sufficient statistics and inference based on sufficient statistics in Fisher's sense are consistent. To be adequate to Fisher's reasoning, we deal with absolutely continuous, one parametric families of distributions.This is certainly not the usual assumption in context with Dempster's theory and implies a normative but straightforward definition concerning the underlying conditional distribution; this definition however is done in Dempster's spirit as can be seen from his articles, (1966, 1968,a,b). Under those assumptions it can be shown that - similar to Lindley's results concerning consistency in fiducial reasoning (1958) - the combination rule, Dempster's procedure based on sufficient statistics and fiducial inference by sufficient statistics agree iff the parametric family under consideration can be transformed to location parameter form.     

Applications of schur theory to reliability inference for k of n systems

The results of Hoeffding (1956), Pledger and Proschan (1971), Gleser (1975) and Boland and Proschan (1983) are used to obtain Buehler (1957) 1-α lower confidence limits for the reliability of k of n systems of independent components when the subsystem data have equal sample sizes and the observed failures satisfy certain conditions. To the best of our knowledge, for k ≠ 1 or n, this is the first time the exact optimal lower confidence limits for system reliability have been given. The observed failure vectors are a generalization of key test results for k of n systems, k ≠ n (Soms (1984) and Winterbottom (1974)). Two examples applying the above theory are also given.     

Distribution theory and inference for polynomial-normal densities

This paper considers a class of densities formed by taking the product of nonnegative polynomials and normal densities. These densities provide a rich class of distributions that can be used in modelling when faced with non-normal characteristics such as skewness and multimodality. In this paper we address inferential and computational issues arising in the practical implementation of this parametric family in the context of the linear model. Exact results are recorded for the conditional analysis of location-scale models and an importance sampling algorithm is developed for the implementation of a conditional analysis for the general linear model when using polynomial-normal distributions for the error.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Testing Covariance Stationarity

In this paper, we show that the widely used stationarity tests such as the Kwiatkowski Phillips, Schmidt, and Shin (KPSS) test have power close to size in the presence of time-varying unconditional variance. We propose a new test as a complement of the existing tests. Monte Carlo experiments show that the proposed test possesses the following characteristics: (i) in the presence of unit root or a structural change in the mean, the proposed test is as powerful as the KPSS and other tests; (ii) in the presence of a changing variance, the traditional tests perform badly whereas the proposed test has high power comparing to the existing tests; (iii) the proposed test has the same size as traditional stationarity tests under the null hypothesis of stationarity. An application to daily observations of return on U.S. Dollar/Euro exchange rate reveals the existence of instability in the unconditional variance when the entire sample is considered, but stability is found in subsamples.     

Nonparametric Entropy-Based Tests of Independence Between Stochastic Processes

This article develops nonparametric tests of independence between two stochastic processes satisfying β-mixing conditions. The testing strategy boils down to gauging the closeness between the joint and the product of the marginal stationary densities. For that purpose, we take advantage of a generalized entropic measure so as to build a whole family of nonparametric tests of independence. We derive asymptotic normality and local power using the functional delta method for kernels. As a corollary, we also develop a class of entropy-based tests for serial independence. The latter are nuisance parameter free, and hence also qualify for dynamic misspecification analyses. We then investigate the finite-sample properties of our serial independence tests through Monte Carlo simulations. They perform quite well, entailing more power against some nonlinear AR alternatives than two popular nonparametric serial-independence tests.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Using a mixture model for multiple imputation in the presence of outliers: the 'Healthy for life' project

Summary.  We consider the problem of obtaining population-based inference in the presence of missing data and outliers in the context of estimating the prevalence of obesity and body mass index measures from the 'Healthy for life' study. Identifying multiple outliers in a multivariate setting is problematic because of problems such as masking, in which groups of outliers inflate the covariance matrix in a fashion that prevents their identification when included, and swamping, in which outliers skew covariances in a fashion that makes non-outlying observations appear to be outliers. We develop a latent class model that assumes that each observation belongs to one of K unobserved latent classes, with each latent class having a distinct covariance matrix. We consider the latent class covariance matrix with the largest determinant to form an 'outlier class'. By separating the covariance matrix for the outliers from the covariance matrices for the remainder of the data, we avoid the problems of masking and swamping. As did Ghosh-Dastidar and Schafer, we use a multiple-imputation approach, which allows us simultaneously to conduct inference after removing cases that appear to be outliers and to promulgate uncertainty in the outlier status through the model inference. We extend the work of Ghosh-Dastidar and Schafer by embedding the outlier class in a larger mixture model, consider penalized likelihood and posterior predictive distributions to assess model choice and model fit, and develop the model in a fashion to account for the complex sample design. We also consider the repeated sampling properties of the multiple imputation removal of outliers.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.