Random weighting-based quantile estimation via importance resampling

Abstract

This paper presents a new method to estimate the quantiles of generic statistics by combining the concept of random weighting with importance resampling. This method converts the problem of quantile estimation to a dual problem of tail probabilities estimation. Random weighting theories are established to calculate the optimal resampling weights for estimation of tail probabilities via sequential variance minimization. Subsequently, the quantile estimation is constructed by using the obtained optimal resampling weights. Experimental results on real and simulated data sets demonstrate that the proposed random weighting method can effectively estimate the quantiles of generic statistics.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

A robust prediction error criterion for pareto modelling of upper tails

Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings. The upper tail of the distribution, where data are sparse, is typically fitted with a model, such as the Pareto model, from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. The authors develop a robust prediction error criterion for choosing k and estimating the Pareto index. A Monte Carlo study shows the good performance of the new estimator and the analysis of real data sets illustrates that a robust procedure for selection, and not just for estimation, is needed.     

Continuous threshold models with two-way interactions in survival analysis

Proportional hazards model with the biomarker–treatment interaction plays an important role in the survival analysis of the subset treatment effect. A threshold parameter for a continuous biomarker variable defines the subset of patients who can benefit or lose from a certain new treatment. In this article, we focus on a continuous threshold effect using the rectified linear unit and propose a gradient descent method to obtain the maximum likelihood estimation of the regression coefficients and the threshold parameter simultaneously. Under certain regularity conditions, we prove the consistency, asymptotic normality and provide a robust estimate of the covariance matrix when the model is misspecified. To illustrate the finite sample properties of the proposed methods, we simulate data to evaluate the empirical biases, the standard errors and the coverage probabilities for both the correctly specified models and misspecified models. The proposed continuous threshold model is applied to a prostate cancer data with serum prostatic acid phosphatase as a biomarker.     

Stochastic comparisons in frailty models

The frailty model in survival analysis accounts for unobserved heterogeneity between individuals by assuming that the hazard rate of an individual is the product of an individual specific quantity, called “frailty” and a baseline hazard rate. It is well known that the choice of the frailty distribution strongly affects the nonparametric estimate of the baseline hazard as well as that of the conditional probabilities. This paper reviews the basic concepts of a frailty model, presents various probability inequalities and other monotonicity results which may prove useful in choosing among alternative specifications. More specifically, our main result lies in seeing how well known stochastic orderings between distributions of two frailities translate into orderings between the corresponding survival functions. Some probabilistic aspects and implications of the models resulting from competing choices of the distributions of frailty or the baseline are compared.     

Gamma shared frailty model based on reversed hazard rate

Abstract

Frailty models are used in survival analysis to account for unobserved heterogeneity in individual risks to disease and death. To analyze bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), shared frailty models were suggested. Shared frailty models are frequently used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor(frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, we introduce shared gamma frailty models with reversed hazard rate. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply the proposed model to the Australian twin data set.     

Non-parametric maximum likelihood estimation of interval-censored failure time data subject to misclassification

The paper considers non-parametric maximum likelihood estimation of the failure time distribution for interval-censored data subject to misclassification. Such data can arise from two types of observation scheme; either where observations continue until the first positive test result or where tests continue regardless of the test results. In the former case, the misclassification probabilities must be known, whereas in the latter case, joint estimation of the event-time distribution and misclassification probabilities is possible. The regions for which the maximum likelihood estimate can only have support are derived. Algorithms for computing the maximum likelihood estimate are investigated and it is shown that algorithms appropriate for computing non-parametric mixing distributions perform better than an iterative convex minorant algorithm in terms of time to absolute convergence. A profile likelihood approach is proposed for joint estimation. The methods are illustrated on a data set relating to the onset of cardiac allograft vasculopathy in post-heart-transplantation patients.     

Quantile regression methods for left-truncated and right-censored data

Left-truncated and right-censored (LTRC) data are encountered frequently due to a prevalent cohort sampling in follow-up studies. Because of the skewness of the distribution of survival time, quantile regression is a useful alternative to the Cox's proportional hazards model and the accelerated failure time model for survival analysis. In this paper, we apply the quantile regression model to LTRC data and develops an unbiased estimating equation for regression coefficients. The proposed estimation methods use the inverse probabilities of truncation and censoring weighting technique. The resulting estimator is uniformly consistent and asymptotically normal. The finite-sample performance of the proposed estimation methods is also evaluated using extensive simulation studies. Finally, analysis of real data is presented to illustrate our proposed estimation methods.     

Prior distributions for stratified capture-recapture models

We consider the Arnason-Schwarz model, usually used to estimate survival and movement probabilities from capture-recapture data. A missing data structure of this model is constructed which allows a clear separation of information relative to capture and relative to movement. Extensions of the Arnason-Schwarz model are considered. For example, we consider a model that takes into account both the individual migration history and the individual reproduction history. Biological assumptions of these extensions are summarized via a directed graph. Owing to missing data, the posterior distribution of parameters is numerically intractable. To overcome those computational difficulties we advocate a Gibbs sampling algorithm that takes advantage of the missing data structure inherent in capture-recapture models. Prior information on survival, capture and movement probabilities typically consists of a prior mean and of a prior 95% credible confidence interval. Dirichlet distributions are used to incorporate some prior information on capture, survival probabilities, and movement probabilities. Finally, the influence of the prior on the Bayesian estimates of movement probabilities is examined.     

Prior distributions for stratified capture-recapture models

We consider the Arnason-Schwarz model, usually used to estimate survival and movement probabilities from capture-recapture data. A missing data structure of this model is constructed which allows a clear separation of information relative to capture and relative to movement. Extensions of the Arnason-Schwarz model are considered. For example, we consider a model that takes into account both the individual migration history and the individual reproduction history. Biological assumptions of these extensions are summarized via a directed graph. Owing to missing data, the posterior distribution of parameters is numerically intractable. To overcome those computational difficulties we advocate a Gibbs sampling algorithm that takes advantage of the missing data structure inherent in capture-recapture models. Prior information on survival, capture and movement probabilities typically consists of a prior mean and of a prior 95% credible confidence interval. Dirichlet distributions are used to incorporate some prior information on capture, survival probabilities, and movement probabilities. Finally, the influence of the prior on the Bayesian estimates of movement probabilities is examined.     

Maximum likelihood estimation of bivariate survival probabilities and its existence and uniqueness

Nonparametric maximum likelihood estimation of bivariate survival probabilities is developed for interval censored survival data. We restrict our attention to the situation where response times within pairs are not distinguishable, and the univariate survival distribution is the same for any individual within any pair. Campbell's (1981) model is modified to incorporate this restriction. Existence and uniqueness of maximum likelihood estimators are discussed. This methodology is illustrated with a bivariate life table analysis of an angioplasty study where each patient undergoes two procedures.     

Flexible semi-parametric regression of state occupational probabilities in a multistate model with right-censored data

Inference for the state occupation probabilities, given a set of baseline covariates, is an important problem in survival analysis and time to event multistate data. We introduce an inverse censoring probability re-weighted semi-parametric single index model based approach to estimate conditional state occupation probabilities of a given individual in a multistate model under right-censoring. Besides obtaining a temporal regression function, we also test the potential time varying effect of a baseline covariate on future state occupation. We show that the proposed technique has desirable finite sample performances and its performance is competitive when compared with three other existing approaches. We illustrate the proposed methodology using two different data sets. First, we re-examine a well-known data set dealing with leukemia patients undergoing bone marrow transplant with various state transitions. Our second illustration is based on data from a study involving functional status of a set of spinal cord injured patients undergoing a rehabilitation program.     

On confidence intervals for semiparametric expectile regression

In regression scenarios there is a growing demand for information on the conditional distribution of the response beyond the mean. In this scenario quantile regression is an established method of tail analysis. It is well understood in terms of asymptotic properties and estimation quality. Another way to look at the tail of a distribution is via expectiles. They provide a valuable alternative since they come with a combination of preferable attributes. The easy weighted least squares estimation of expectiles and the quadratic penalties often used in flexible regression models are natural partners. Also, in a similar way as quantiles can be seen as a generalisation of median regression, expectiles offer a generalisation of mean regression. In addition to regression estimates, confidence intervals are essential for interpretational purposes and to assess the variability of the estimate, but there is a lack of knowledge regarding the asymptotic properties of a semiparametric expectile regression estimate. Therefore confidence intervals for expectiles based on an asymptotic normal distribution are introduced. Their properties are investigated by a simulation study and compared to a boostrap-based gold standard method. Finally the introduced confidence intervals help to evaluate a geoadditive expectile regression model on childhood malnutrition data from India.     

Quantile estimation in the proportional hazards model of random censorship

The asymptotic theory is given for quantile estimation in the proportional hazards model of random censorship. In this model, the tail of the censoring distribution function is some power of the tail of the survival distribution function. The quantile estimator is based on the maximum likelihood estimator for the survival time distribution, due to Abdushukurov, Cheng and Lin.     

Small sample comparison of six bivariate survival curve estimators 1

We study the performance of six proposed bivariate survival curve estimators on simulated right censored data. The performance of the estimators is compared for data generated by three bivariate models with exponential marginal distributions. The estimators are compared in their ability to estimate correlations and survival functions probabilities. Simulated data results are presented so that the proposed estimators in this relatively new area of analysis can be explicitly compared to the known distribution of the data and the parameters of the underlying model. The results show clear differences in the performance of the estimators.     

Analysis of Bivariate Survival Data using Shared Inverse Gaussian Frailty Model

In this article, we consider shared frailty model with inverse Gaussian distribution as frailty distribution and log-logistic distribution (LLD) as baseline distribution for bivariate survival times. We fit this model to three real-life bivariate survival data sets. The problem of analyzing and estimating parameters of shared inverse Gaussian frailty is the interest of this article and then compare the results with shared gamma frailty model under the same baseline for considered three data sets. Data are analyzed using Bayesian approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same group. The variance component estimation provides the estimated dispersion of the random effects. We carried out a test for frailty (or heterogeneity) using Bayes factor. Model comparison is made using information criteria and Bayes factor. We observed that the shared inverse Gaussian frailty model with LLD as baseline is the better fit for all three bivariate data sets.     

The use of auxiliary variables in capture-recapture modelling: An overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

The use of auxiliary variables in capture-recapture modelling: an overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

Statistical learning theory for fitting multimodal distribution to rainfall data: an application

The promising methodology of the “Statistical Learning Theory” for the estimation of multimodal distribution is thoroughly studied. The “tail” is estimated through Hill's, UH and moment methods. The threshold value is determined by nonparametric bootstrap and the minimum mean square error criterion. Further, the “body” is estimated by the nonparametric structural risk minimization method of the empirical distribution function under the regression set-up. As an illustration, rainfall data for the meteorological subdivision of Orissa, India during the period 1871–2006 are used. It is shown that Hill's method has performed the best for tail density. Finally, the combined estimated “body” and “tail” of the multimodal distribution is shown to capture the multimodality present in the data.