Inference For Mixtures Research Articles

Laplacian‐P‐splines for Bayesian inference in the mixture cure model

The mixture cure model for analyzing survival data is characterized by the assumption that the population under study is divided into a group of subjects who will experience the event of interest over some finite time horizon and another group of cured subjects who will never experience the event irrespective of the duration of follow‐up. When using the Bayesian paradigm for inference in survival models with a cure fraction, it is common practice to rely on Markov chain Monte Carlo (MCMC) methods to sample from posterior distributions. Although computationally feasible, the iterative nature of MCMC often implies long sampling times to explore the target space with chains that may suffer from slow convergence and poor mixing. Furthermore, extra efforts have to be invested in diagnostic checks to monitor the reliability of the generated posterior samples. A sampling‐free strategy for fast and flexible Bayesian inference in the mixture cure model is suggested in this article by combining Laplace approximations and penalized B‐splines. A logistic regression model is assumed for the cure proportion and a Cox proportional hazards model with a P‐spline approximated baseline hazard is used to specify the conditional survival function of susceptible subjects. Laplace approximations to the posterior conditional latent vector are based on analytical formulas for the gradient and Hessian of the log‐likelihood, resulting in a substantial speed‐up in approximating posterior distributions. The spline specification yields smooth estimates of survival curves and functions of latent variables together with their associated credible interval are estimated in seconds. A fully stochastic algorithm based on a Metropolis‐Langevin‐within‐Gibbs sampler is also suggested as an alternative to the proposed Laplacian‐P‐splines mixture cure (LPSMC) methodology. The statistical performance and computational efficiency of LPSMC is assessed in a simulation study. Results show that LPSMC is an appealing alternative to MCMC for approximate Bayesian inference in standard mixture cure models. Finally, the novel LPSMC approach is illustrated on three applications involving real survival data.

Estimation and inference for mixture of partially linear additive models

In this paper, a semiparametric mixture of regression models is proposed, where the regression functions are partially linear additive while the mixing proportions and variances are unknown constant. The asymptotic normality of the SBK estimators and the model selection consistency of the proposed BIC based on B-spline estimators are established. Simulations and applications are presented to illustrate the performance of the proposed.

Empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout

In this paper, empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout is investigated. We estimate the non-parametric function based on the estimating equations and the local linear profile-kernel method. An empirical log-likelihood ratio statistic for parametric components is proposed to construct confidence regions and is shown to be an asymptotically chi-squared distribution. The non-parametric version of Wilk's theorem is also derived. A simulation study is undertaken to illustrate the finite sample performance of the proposed method.

Empirical likelihood inference in mixtures of semiparametric varying coefficient EV models for longitudinal data with nonignorable dropout

In this paper, empirical likelihood inference in mixtures of semiparametric varying coefficient errors-in-variables (EV) models for longitudinal data with nonignorable dropout is investigated. The empirical log-likelihood ratio statistic for the fixed-effects parameters and the mean parameters of random effects are proposed. The proposed statistic at the true parameters is proven to be asymptotically χq+r2, where q and r are the dimensions of the fixed and random effects respectively, and the corresponding confidence regions for the parameters of interest are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters, and prove that it is asymptotically normal under some suitable conditions. Simulation studies are undertaken to assess the finite sample performance of the proposed method.

Selection sampling from large data sets for targeted inference in mixture modeling

One of the challenges in using Markov chain Monte Carlo for model analysis in studies with very large datasets is the need to scan through the whole data at each iteration of the sampler, which can be computationally prohibitive. Several approaches have been developed to address this, typically drawing computationally manageable subsamples of the data. Here we consider the specific case where most of the data from a mixture model provides little or no information about the parameters of interest, and we aim to select subsamples such that the information extracted is most relevant. The motivating application arises in flow cytometry, where several measurements from a vast number of cells are available. Interest lies in identifying specific rare cell subtypes and characterizing them according to their corresponding markers. We present a Markov chain Monte Carlo approach where an initial subsample of the full dataset is used to guide selection sampling of a further set of observations targeted at a scientifically interesting, low probability region. We define a Sequential Monte Carlo strategy in which the targeted subsample is augmented sequentially as estimates improve, and introduce a stopping rule for determining the size of the targeted subsample. An example from flow cytometry illustrates the ability of the approach to increase the resolution of inferences for rare cell subtypes.

Inference for Mixtures

In this article, we consider the problem of estimating certain “parameters” in a mixture of probability measures. We show that a single sample is typically suitable for estimating the component measures, but not suitable for estimating the mixing measures, especially when consistency is required. To have consistent estimators of the mixing measure, several samples with increasing size are needed in general.

Likelihood Inference for Mixtures: Geometrical and Other Constructions of Monotone Step-Length Algorithms

Likelihood Inference for Mixtures: Geometrical and Other Constructions of Monotone Step-Length Algorithms

Bayesian inference in mixtures of two exponentials

This paper provides the Bayes estimators of scale parameters in two exponential distributions mixed in proportion p, which is assumed to be known. For deriving the Bayes estimators the prior distributions are chosen such that they are centered at the known prior values of the scale parameters. The Bayes estimators thus obtained are found to be equivalent to Thompson's shrinkage estimators. The Bayes estimator of reliability function and another estimator of reliability using the Bayes estimators of scale parameters are given. The efficiency of the proposed estimator of reliability with respect to its maximum likelihood estimator (mle) is studied by Monte Carlo simulation. Lastly, the Bayes confidence interval of reliability is obtained and its validity is examined by a Monte Carlo study of 500 samples.