The First Exit Time Theory Applied to Life Table Data: The Health State Function of a Population and Other Characteristics

In this article, we summarize the main parts of the first exit time theory developed in connection to the life table data and the resulting theoretical and applied issues. New tools arise from the development of this theory and especially the Health State Function and some important characteristics of this function.

We provide both simple and complex models and propose a methodology for reconstructing the health state function from the provided first exit time density distribution (for the appropriate computer program, see http://www.cmsim.net). In the simpler case, this theory is applied for the reconstruction of the so-called Inverse-Gaussian function.     

Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case

Abstract. In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high‐dimensional vector after standardizing marginals. Hence, it is of importance to model and estimate tail dependence functions. Even for moderate dimension, non‐parametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper, we propose a semi‐parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Under this model assumption, we propose a novel estimator for the tail dependence function, which proves favourable compared to the empirical tail dependence function estimator, both theoretically and empirically.     

A moment-generating function with application to the birnbaum-saunders distribution

The Birnbaum-Saunders distribution is a fatigue life distribution that was derived from a model assuming that failure is due to the development and growth of a dominant crack. This distribution has been shown to be applicable not only for fatigue analysis but also in other areas of engineering science. Because of its increasing use, it would be desirable to obtain expressions for the expected value of different powers of this distribution.

In this article, the moment-generating function for the sinh-normal distribution is derived. It is shown that this moment-generating function can be used to obtain both integer and fractional moments for the Birnbaum-Saunders distribution. Thus it is now possible to obtain an expression for the expected value of the square root of a Birnbaum-Saunders random variable. A general expression for integer noncentral moments for the Birnbaum-Saunders distribution is derived using the moment-generating function of the sinh-normal distribution. Also included is an approximation of the moment-generating function that can be used fcx small values of the shape parameter.     

Enhancement for two commonly-used approximations for the inverse cumulative function of the normal distribution

Two commonly used approximations for the inverse distribution function of the normal distribution are Schmeiser's and Shore's. Both approximations are based on a power transformation of either the cumulative density function (CDF) or a simple function of it. In this note we demonstrate, that if these approximations are presented in the form of the classical one-parameter Box-Cox transformation, and the exponent of the transformation is expressed as a simple function of the CDF, then the accuracy of both approximations may be considerably enhanced, without losing much in algebraic simplicity. Since both approximations are special cases of more general four-parameter systems of distributions, the results presented here indicate that the accuracy of the latter, when used to represent non-normal density functions, may also be considerably enhanced.     

Local quadratic estimation of the curvature in a functional single index model

The nonlinear responses of species to environmental variability can play an important role in the maintenance of ecological diversity. Nonetheless, many models use parametric nonlinear terms which pre-determine the ecological conclusions. Motivated by this concern, we study the estimate of the second derivative (curvature) of the link function in a functional single index model. Since the coefficient function and the link function are both unknown, the estimate is expressed as a nested optimization. We first estimate the coefficient function by minimizing squared error where the link function is estimated with a Nadaraya-Watson estimator for each candidate coefficient function. The first and second derivatives of the link function are then estimated via local-quadratic regression using the estimated coefficient function. In this paper, we derive a convergence rate for the curvature of the nonlinear response. In addition, we prove that the argument of the linear predictor can be estimated root-n consistently. However, practical implementation of the method requires solving a nonlinear optimization problem, and our results show that the estimates of the link function and the coefficient function are quite sensitive to the choices of starting values.     

Some results concerning off-training-set and IID error for the Gibbs and the Bayes optimal generalizers

In this paper we analyse the average behaviour of the Bayes-optimal and Gibbs learning algorithms. We do this both for off-training-set error and conventional IID (independent identically distributed) error (for which test sets overlap with training sets). For the IID case we provide a major extension to one of the better known results. We also show that expected IID test set error is a non-increasing function of training set size for either algorithm. On the other hand, as we show, the expected off-training-set error for both learning algorithms can increase with training set size, for non-uniform sampling distributions. We characterize the relationship the sampling distribution must have with the prior for such an increase. We show in particular that for uniform sampling distributions and either algorithm, the expected off-training-set error is a non-increasing function of training set size. For uniform sampling distributions, we also characterize the priors for which the expected error of the Bayes-optimal algorithm stays constant. In addition we show that for the Bayes-optimal algorithm, expected off-training-set error can increase with training set size when the target function is fixed, but if and only if the expected error averaged over all targets decreases with training set size. Our results hold for arbitrary noise and arbitrary loss functions.     

Maximum entropy estimation of income share function from generalized Gini index

Following Sir Anthony and Atkinson who started thinking about the insensitivity of the Gini index to income shares of the lower and the upper income groups, a generalization of the classical Gini index was introduced by Kakwani, Donaldson, Weymark and Yitzhaki which is sensitive to both high and low incomes. In this paper, the maximum entropy method is used to estimate the underlying true income share function based on the limited information of the generalized Gini index about the income shares of a population's percentiles. The income share function is estimated through maximizing both the Shannon entropy and the second-order entropy. In the end, through parametric bootstrap and analyzing a real dataset, the results are compared with the estimator of the share function, which is obtained based on the total information. In contrast to the classic Gini index, the derived share function based on the generalized Gini index provides more accurate approximations for income shares of the lower and the upper percentiles.     

Application of a predictive distribution formula to Bayesian computation for incomplete data models

We consider exact and approximate Bayesian computation in the presence of latent variables or missing data. Specifically we explore the application of a posterior predictive distribution formula derived in Sweeting And Kharroubi (2003), which is a particular form of Laplace approximation, both as an importance function and a proposal distribution. We show that this formula provides a stable importance function for use within poor man’s data augmentation schemes and that it can also be used as a proposal distribution within a Metropolis-Hastings algorithm for models that are not analytically tractable. We illustrate both uses in the case of a censored regression model and a normal hierarchical model, with both normal and Student t distributed random effects. Although the predictive distribution formula is motivated by regular asymptotic theory, it is not necessary that the likelihood has a closed form or that it possesses a local maximum.     

A distribution estimation method based on level crossings

This paper presents a method of nonparametric distribution estimation based on a sample level-crossing function, which leads to the construction of a level-crossing empirical distribution function (LCEDF). An efficiency function for this LCEDF relative to the classical empirical distribution function (e.d.f.) is derived. The LCEDF gives more efficient estimates than the e.d.f. in the tails of any underlying continuous distribution, for both small and large sample sizes. Simulation experiments, which apply the LCEDF to a smoothing technique for various distributions, confirm the theoretical results.     

Bayesian estimation and prediction intervals for a Rayleigh distribution under a conjugate prior

This paper is an effort to obtain Bayes estimators of Rayleigh parameter and its associated risk based on a conjugate prior (square root inverted gamma prior) with respect to both symmetric loss function (squared error loss), and asymmetric loss function (precautionary loss function). We also derive the highest posterior density (HPD) interval for the Rayleigh parameter as well as the HPD prediction intervals for a future observation from this distribution. An illustrative example to test how the Rayleigh distribution fits a real data set is presented. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different conditions.     

A Composite Likelihood Cross-validation Approach in Selecting Bandwidth for the Estimation of the Pair Correlation Function

Abstract.  A useful tool while analysing spatial point patterns is the pair correlation function (e.g. Fractals, Random Shapes and Point Fields, Wiley, New York, 1994). In practice, this function is often estimated by some nonparametric procedure such as kernel smoothing, where the smoothing parameter (i.e. bandwidth) is often determined arbitrarily. In this article, a data-driven method for the selection of the bandwidth is proposed. The efficacy of the proposed approach is studied through both simulations and an application to a forest data example.     

A Unified Approach to Prior and Loss Robustness

ABSTRACT

We introduce a new statistical framework in order to study Bayesian loss robustness under classes of priors distributions, thus unifying both concepts of robustness. We propose measures that capture variation with respect to both prior selection and selection of loss function and explore general properties of these measures. We illustrate the approach for the continuous exponential family. Robustness in this context is studied first with respect to prior selection where we consider several classes of priors for the parameter of interest, including unimodal and symmetric and unimodal with positive support. After prior variation has been measured we investigate robustness to loss function, using Hellinger and Linex (Linear Exponential) classes of loss functions. The methods are applied to standard examples.     

Partial derivatives and confidence intervals of bivariate tail dependence functions

Bivariate extreme value theory was used to estimate a rare event (see de Haan and de Ronde [1998. Sea and wind: multivariate extremes at work. Extremes 1, 7–45]). This procedure involves estimating a tail dependence function. There are several estimators for the tail dependence function in the literature, but their limiting distributions depend on partial derivatives of the tail dependence function. In this paper smooth estimators are proposed for estimating partial derivatives of bivariate tail dependence functions and their asymptotic distributions are derived as well. A simulation study is conducted to compare different estimators of partial derivatives in terms of both mean squared errors and coverage accuracy of confidence intervals of the bivariate tail dependence function based on these different estimators of partial derivatives.     

Minimum-area confidence sets for a normal distribution

When making inference on a normal distribution, one often seeks either a joint confidence region for the two parameters or a confidence band for the cumulative distribution function. A number of methods for constructing such confidence sets are available, but none of these methods guarantees a minimum-area confidence set. In this paper, we derive both a minimum-area joint confidence region for the two parameters and a minimum-area confidence band for the cumulative distribution function. The minimum-area joint confidence region is asymptotically equivalent to other confidence regions in the literature, but the minimum-area confidence band improves on existing confidence bands even asymptotically.     

Nonparametric modeling of the gap time in recurrent event data

Recurrent event data arise in many biomedical and engineering studies when failure events can occur repeatedly over time for each study subject. In this article, we are interested in nonparametric estimation of the hazard function for gap time. A penalized likelihood model is proposed to estimate the hazard as a function of both gap time and covariate. Method for smoothing parameter selection is developed from subject-wise cross-validation. Confidence intervals for the hazard function are derived using the Bayes model of the penalized likelihood. An eigenvalue analysis establishes the asymptotic convergence rates of the relevant estimates. Empirical studies are performed to evaluate various aspects of the method. The proposed technique is demonstrated through an application to the well-known bladder tumor cancer data.     

Estimation of Survival Function in Cox Regression Model Under Random Censoring From Both Sides

In this article we present three types of parametric–non parametric estimators for conditional survival function in Cox proportional hazards regression model when the lifetime of interest is subjected to random censorship from both sides. We prove consistency and asymptotic normality of estimators.     

On Spiring's normal loss function

Loss functions express the loss to society, incurred through the use of a product, in monetary units. Underlying this concept is the notion that any deviation from target of any product characteristic implies a degradation in the product performance and hence a loss. Spiring (1993), in response to criticisms of the quadratic loss function, developed the reflected normal loss function, which is based on the normal density function. We give some modifications of these loss functions to simplify their application and provide a framework for the reflected normal loss function that accomodates a broader class of symmetric loss situations. These modifications also facilitate the unification of both of these loss functions and their comparison through expected loss. Finally, we give a simple method for determing the parameters of the modified reflected normal loss function based on loss information for multiple values of the product characteristic, and an example to illustrate the flexibility of the proposed model and the determination of its parameters.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

Promotion Time Cure Rate Model with Bivariate Random Effects

In this article, we consider the inclusion of random effects in both the survival function for at-risk subjects and the cure probability assuming a bivariate normal distribution for those effects in each cluster. For parameter estimation, we implemented the restricted maximum likelihood (REML) approach. We consider Weibull and Piecewise Exponential distributions to model the survival function for non-cured individuals. Simulation studies are performed, and based on a real database we evaluate the performance of our proposed model. Effect of different follow-up times and the effect of considering independent random effects instead of bivariate random effects are also studied.     

Modelling growth and decline in lung function in Duchenne's muscular dystrophy with an augmented linear mixed effects model

Summary.  Longitudinal modelling of lung function in Duchenne's muscular dystrophy is complicated by a mixture of both growth and decline in lung function within each subject, an unknown point of separation between these phases and significant heterogeneity between individual trajectories. Linear mixed effects models can be used, assuming a single changepoint for all cases; however, this assumption may be incorrect. The paper describes an extension of linear mixed effects modelling in which random changepoints are integrated into the model as parameters and estimated by using a stochastic EM algorithm. We find that use of this 'mixture modelling' approach improves the fit significantly.