Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

Simultaneous estimation of parameters in different linear models and applications to biometric problems.

Empirical Bayes procedure is employed in simultaneous estimation of vector parameters from a number of Gauss-Markoff linear models. It is shown that with respect to quadratic loss function, empirical Bayes estimators are better than least squares estimators. While estimating the parameter for a particular linear model, a suggestion has been made for distinguishing between the loss due to decision maker and the loss due to individual. A method has been proposed but not fully studied to achieve balance between the two losses. Finally the problem of predicting future observations in a linear model has been considered.     

Parameter Estimation for Partially Complete Time and Type of Failure Data

The theory of competing risks has been developed to asses a specific risk in presence of other risk factors. In this paper we consider the parametric estimation of different failure modes under partially complete time and type of failure data using latent failure times and cause specific hazard functions models. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are obtained when latent failure times and cause specific hazard functions are exponentially distributed. We also consider the case when they follow Weibull distributions. One data set is used to illustrate the proposed techniques. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence functions and robust Bayes and empirical Bayes small area estimation

We introduce new robust small area estimation procedures basedon area-level models. We first find influence functions correspondingto each individual area-level observation by measuring the divergencebetween the posterior density functions of regression coefficientswith and without that observation. Next, based on these influencefunctions, properly standardized, we propose some new robustBayes and empirical Bayes small area estimators. The mean squarederrors and estimated mean squared errors of these estimatorsare also found. A small simulation study compares the performanceof the robust and the regular empirical Bayes estimators. Whenthe model variance is larger than the sample variance, the proposedrobust empirical Bayes estimators are superior.     

Bayesian Estimation in a Generalized Negative Binomial Distribution

A generalized negative binomial (GNB) distribution was introduced by JAIN and CONSUL (1971) and was modified by NELSON (1975). The probability function of the distribution is defined by the function p(x; m, β, θ)= θ^x (1 - θ)^m+βx—x for x=0, 1, …, and zero otherwise, where m>0, 0<θ<1 and β=0 or 1≦β<θ^?1. The Bayes estimators for a number of parametric functions of θ when m and β are known are derived. The prior information on θ may be given by a beta distribution, B(a, b), to which no subjective significance is attached. It has been illustrated that the parameters in the prior distribution can be assigned by a computer. Comparisons are made of the Bayes estimate of P(X=k) to the corresponding ML estimate and the MVU estimate for any given sample to the order n^?1 for different values of k..     

A 3-Component Mixture of Rayleigh Distributions: Properties and Estimation in Bayesian Framework

To study lifetimes of certain engineering processes, a lifetime model which can accommodate the nature of such processes is desired. The mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process as compared to simple models. This paper is about studying a 3-component mixture of the Rayleigh distributionsin Bayesian perspective. The censored sampling environment is considered due to its popularity in reliability theory and survival analysis. The expressions for the Bayes estimators and their posterior risks are derived under different scenarios. In case the case that no or little prior information is available, elicitation of hyperparameters is given. To examine, numerically, the performance of the Bayes estimators using non-informative and informative priors under different loss functions, we have simulated their statistical properties for different sample sizes and test termination times. In addition, to highlight the practical significance, an illustrative example based on a real-life engineering data is also given.     

Estimating the null distribution to adjust observed confidence levels for genome-scale screening

In a novel approach to the multiple testing problem, Efron (2004, Journal of the American Statistical Association 99, 96-104; 2007a Journal of the American Statistical Association 102, 93-103; 2007b, Annals of Statistics 35, 1351-1377) formulated estimators of the distribution of test statistics or nominal p-values under a null distribution suitable for modeling the data of thousands of unaffected genes, nonassociated single-nucleotide polymorphisms, or other biological features. Estimators of the null distribution can improve not only the empirical Bayes procedure for which it was originally intended, but also many other multiple-comparison procedures. Such estimators in some cases improve the proposed multiple-comparison procedure (MCP) based on a recent non-Bayesian framework of minimizing expected loss with respect to a confidence posterior, a probability distribution of confidence levels. The flexibility of that MCP is illustrated with a nonadditive loss function designed for genomic screening rather than for validation. The merit of estimating the null distribution is examined from the vantage point of the confidence-posterior MCP (CPMCP). In a generic simulation study of genome-scale multiple testing, conditioning the observed confidence level on the estimated null distribution as an approximate ancillary statistic markedly improved conditional inference. Specifically simulating gene expression data, however, indicates that estimation of the null distribution tends to exacerbate the conservative bias that results from modeling heavy-tailed data distributions with the normal family. To enable researchers to determine whether to rely on a particular estimated null distribution for inference or decision making, an information-theoretic score is provided. As the sum of the degree of ancillarity and the degree of inferential relevance, the score reflects the balance conditioning would strike between the two conflicting terms. The CPMCP and other methods introduced are applied to gene expression microarray data.     

Marginal analysis of correlated failure time data with informative cluster sizes

We consider modeling correlated survival data when cluster sizes may be informative to the outcome of interest based on a within-cluster resampling (WCR) approach and a weighted score function (WSF) method. We derive the large sample properties for the WCR estimators under the Cox proportional hazards model. We establish consistency and asymptotic normality of the regression coefficient estimators, and the weak convergence property of the estimated baseline cumulative hazard function. The WSF method is to incorporate the inverse of cluster sizes as weights in the score function. We conduct simulation studies to assess and compare the finite-sample behaviors of the estimators and apply the proposed methods to a dental study as an illustration.     

An estimator of number of species from quadrat sampling

We consider the problem of estimating the number of distinct species S in a study area from the recorded presence or absence of species in each of a sample of quadrats. A generalized jackknife estimator of S is derived, along with an estimate of its variance. It is compared with the jackknife estimator for S proposed by Heltshe and Forrester and the empirical Bayes estimator of Mingoti and Meeden. We show that the empirical Bayes estimator has the form of a generalized jackknife estimator under a specific model for species distribution. We compare the new estimators of S to the empirical Bayes estimator via simulation. We characterize circumstances under which each is superior.     

The Estimation of the Segregation Parameter when Ascertainment is not Complete

Procedures to estimate the genetic segregation parameter when ascertainment of families is incomplete, have previously relied on iterative computer algorithms since estimators with closed form are lacking. We now present the Minimum Variance Unbiased Estimator for the segregation parameter under any ascertainment probability. This estimator assumes a simple form when ascertainment is complete. We also present a simple estimator, akin to Li and Mantel's (1968) estimator, but without the restriction that ascertainment be complete. The performance of these estimators is compared with respect to asymptotic efficiency. We also provide tables that define the required number of families of a given size that need to be sampled to achieve a specific power for testing simple hypothesis on the segregation parameter.     

On Bayesian inference for the K function

The K function is a summary of spatial dependence in spatial point processes. In practice one observes a realization of the spatial point process, called a spatial point pattern. Although the K function of a spatial point process is typically unknown, several estimators of the process K function have been put forth. These estimators, however, are based upon empirical averages; the complicated distributional properties of the estimators unfortunately complicates interval estimation. In this paper, we propose a Bayesian inferential framework, allowing inference for the K function of the spatial point process (including interval estimation). Of particular interest is the unique use of the posterior predictive distribution to (efficiently) enable such inferences. To demonstrate our technique, the well known Swedish pine sapling data (Strand, 1972) is analyzed, including a discussion on evaluating model fit.     

Estimation of distribution function in bivariate competing risk models

We consider lifetime data involving pairs of study individuals with more than one possible cause of failure for each individual. Non-parametric estimation of cause-specific distribution functions is considered under independent censoring. Properties of the estimators are discussed and an illustration of their application is given.     

Bayesian analysis for the meiosis I non-disjunction fraction in numerical chromosomal anomalies

The main causes of numerical chromosomal anomalies, including trisomies, arise from an error in the chromosomal segregation during the meiotic process, named a non-disjunction. One of the most used techniques to analyze chromosomal anomalies nowadays is the polymerase chain reaction (PCR), which counts the number of peaks or alleles in a polymorphic microsatellite locus. It was shown in previous works that the number of peaks has a multinomial distribution whose probabilities depend on the non-disjunction fraction F. In this work, we propose a Bayesian approach for estimating the meiosis I non-disjunction fraction F. in the absence of the parental information. Since samples of trisomic patients are, in general, small, the Bayesian approach can be a good alternative for solving this problem. We consider the sampling/importance resampling technique and the Simpson rule to extract information from the posterior distribution of F. Bayes and maximum likelihood estimators are compared through a Monte Carlo simulation, focusing on the influence of different sample sizes and prior specifications in the estimates. We apply the proposed method to estimate F. for patients with trisomy of chromosome 21 providing a sensitivity analysis for the method. The results obtained show that Bayes estimators are better in almost all situations.     

Protecting against nonrandomly missing data in longitudinal studies

Nonrandomly missing data can pose serious problems in longitudinal studies. We generally have little knowledge about how missingness is related to the data values, and longitudinal studies are often far from complete. Two approaches that have been used to handle missing data--use of maximum likelihood with an ignorable mechanism and direct modeling of the missing data mechanism--have the disadvantage of not giving consistent estimates under important classes of nonrandom mechanisms. We introduce two protective estimators, that is, estimators that retain their consistency over a wide range of nonrandom mechanisms. We compare these protective estimators using longitudinal data from a mental health panel study. We also investigate their robustness to certain departures from normality.     

Bayes estimates as an approximation to maximum likelihood estimates

Ronald A. Fisher, who is the founder of maximum likelihood estimation (ML estimation), criticized the Bayes estimation of using a uniform prior distribution, because we can create estimates arbitrarily if we use Bayes estimation by changing the transformation used before the analysis. Thus, the Bayes estimates lack the scientific objectivity, especially when the amount of data is small. However, we can use the Bayes estimates as an approximation to the objective ML estimates if we use an appropriate transformation that makes the posterior distribution close to a normal distribution. One-to-one correspondence exists between a uniform prior distribution under a transformed scale and a non-uniform prior distribution under the original scale. For this reason, the Bayes estimation of ML estimates is essentially identical to the estimation using Jeffreys prior.     

Bayes Estimator of Survival Probability from Randomly Censored Observations with Weibull Distribution of Time to Death

This paper deals with Bayes estimation of survival probability when the data are randomly censored. Such a situation arises in case of a clinical trial which extends for a limited period T. A fixed number of patients (n) are observed whose times to death have identical Weibull distribution with parameters β and θ. The maximum times of observation for different patients are also independent uniform variables as the patients arrive randomly throughout the trial. For the joint prior distribution of (β, θ) as suggested by Sinha and Kale (1980, page 137) Bayes estimator of survival probability at time t (0<t<T) has been obtained. Considering squared error loss function it is the mean of the survival probability with respect to the posterior distribution of (β, θ). This estimator is then compared with the maximum likelihood estimator, by simulation, for various values of β, θ and censoring percentage. The proposed estimator is found to be better under certain conditions.     

Empirical Bayesian estimation of the disease transmission probability in multiple-vector-transfer designs

Plant disease is responsible for major losses in agriculture throughout the world. Diseases are often spread by insect organisms that transmit a bacterium, virus, or other pathogen. To assess disease epidemics, plant pathologists often use multiple-vector-transfers. In such contexts, groups of insect vectors are moved from an infected source to each of n test plants that will then be observed for developing symptoms of infection. The purpose of this paper is to present new estimators for p, the probability of pathogen transmission for an individual vector, motivated from an empirical Bayesian approach. We specifically investigate four such estimators, characterize their small-sample properties, and propose new credible intervals for p. These estimators remove the need to specify hyperparameters a priori and are shown to be easier to compute than the classical Bayes estimators proposed by Chaubey and Li (1995, Journal of Official Statistics 11, 1035-1046) and Chick (1996, Biometrics 52, 1055-1062). Furthermore, some of these estimators are shown to have better frequentist properties than the commonly used maximum likelihood estimator and to provide a smaller Bayes risk than the estimator proposed by Burrows (1987, Phytopathology 77, 363-365).     

Bayes factor between Student t and Gaussian mixed models within an animal breeding context

The implementation of Student t mixed models in animal breeding has been suggested as a useful statistical tool to effectively mute the impact of preferential treatment or other sources of outliers in field data. Nevertheless, these additional sources of variation are undeclared and we do not know whether a Student t mixed model is required or if a standard, and less parameterized, Gaussian mixed model would be sufficient to serve the intended purpose. Within this context, our aim was to develop the Bayes factor between two nested models that only differed in a bounded variable in order to easily compare a Student t and a Gaussian mixed model. It is important to highlight that the Student t density converges to a Gaussian process when degrees of freedom tend to infinity. The twomodels can then be viewed as nested models that differ in terms of degrees of freedom. The Bayes factor can be easily calculated from the output of a Markov chain Monte Carlo sampling of the complex model (Student t mixed model). The performance of this Bayes factor was tested under simulation and on a real dataset, using the deviation information criterion (DIC) as the standard reference criterion. The two statistical tools showed similar trends along the parameter space, although the Bayes factor appeared to be the more conservative. There was considerable evidence favoring the Student t mixed model for data sets simulated under Student t processes with limited degrees of freedom, and moderate advantages associated with using the Gaussian mixed model when working with datasets simulated with 50 or more degrees of freedom. For the analysis of real data (weight of Pietrain pigs at six months), both the Bayes factor and DIC slightly favored the Student t mixed model, with there being a reduced incidence of outlier individuals in this population.     

Nonparametric association analysis of bivariate left‐truncated competing risks data

We develop time‐varying association analyses for onset ages of two lung infections to address the statistical challenges in utilizing registry data where onset ages are left‐truncated by ages of entry and competing‐risk censored by deaths. Two types of association estimators are proposed based on conditional cause‐specific hazard function and cumulative incidence function that are adapted from unconditional quantities to handle left truncation. Asymptotic properties of the estimators are established by using the empirical process techniques. Our simulation study shows that the estimators perform well with moderate sample sizes. We apply our methods to the Cystic Fibrosis Foundation Registry data to study the relationship between onset ages of Pseudomonas aeruginosa and Staphylococcus aureus infections.     

The Autocovariance Function for Marked Point Processes: A Comparison between Two Different Approaches

Exploratory analysis of marked point patterns has previously been conducted using two disjoint techniques, namely the mark correlation function and spectral analysis. Our purpose here is to present two alternative autocovariance estimators to the mark correlation function which not only apply in both planar and lattice situations, but which in the lattice case can also be considered in terms of the inverse Fourier transform of the spectrum. Moreover, they can be applied to isotropic or anisotropic marked point patterns. Various examples are presented to show how these estimators perform when applied to data sets possessing different kinds of mark structure, and a rank test procedure is proposed to enable the construction of empirical tests of hypothesis.