On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models

While much used in practice, latent variable models raise challenging estimation problems due to the intractability of their likelihood. Monte Carlo maximum likelihood (MCML), as proposed by Geyer & Thompson (1992 ), is a simulation-based approach to maximum likelihood approximation applicable to general latent variable models. MCML can be described as an importance sampling method in which the likelihood ratio is approximated by Monte Carlo averages of importance ratios simulated from the complete data model corresponding to an arbitrary value of the unknown parameter. This paper studies the asymptotic (in the number of observations) performance of the MCML method in the case of latent variable models with independent observations. This is in contrast with previous works on the same topic which only considered conditional convergence to the maximum likelihood estimator, for a fixed set of observations. A first important result is that when is fixed, the MCML method can only be consistent if the number of simulations grows exponentially fast with the number of observations. If on the other hand, is obtained from a consistent sequence of estimates of the unknown parameter, then the requirements on the number of simulations are shown to be much weaker.     

Statistical and Combinatorial Aspects of Comparative Genomics*

Abstract. This document presents a survey of the statistical and combinatorial aspects of four areas of comparative genomics: gene order based measures of evolutionary distances between species, construction of phylogenetic trees, detection of horizontal transfer of genes, and detection of ancient whole genome duplications.     

The analysis of incomplete data using stochastic covariates

In the development of many diseases there are often associated random variables which continuously reflect the progress of a subject towards the final expression of the disease (failure). At any given time these processes, which we call stochastic covariates, may provide information about the current hazard and the remaining time to failure. Likewise, in situations when the specific times of key prior events are not known, such as the time of onset of an occult tumour or the time of infection with HIV-1, it may be possible to identify a stochastic covariate which reveals, indirectly, when the event of interest occurred. The analysis of carcinogenicity trials which involve occult tumours is usually based on the time of death or sacrifice and an indicator of tumour presence for each animal in the experiment. However, the size of an occult tumour observed at the endpoint represents data concerning tumour development which may convey additional information concerning both the tumour incidence rate and the rate of death to which tumour-bearing animals are subject. We develop a stochastic model for tumour growth and suggest different ways in which the effect of this growth on the hazard of failure might be modelled. Using a combined model for tumour growth and additive competing risks of death, we show that if this tumour size information is used, assumptions concerning tumour lethality, the context of observation or multiple sacrifice times are no longer necessary in order to estimate the tumour incidence rate. Parametric estimation based on the method of maximum likelihood is outlined and is applied to simulated data from the combined model. The results of this limited study confirm that use of the stochastic covariate tumour size results in more precise estimation of the incidence rate for occult tumours.     

Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models

ABSTRACT.  This paper develops a new contrast process for parametric inference of general hidden Markov models, when the hidden chain has a non-compact state space. This contrast is based on the conditional likelihood approach, often used for ARCH-type models. We prove the strong consistency of the conditional likelihood estimators under appropriate conditions. The method is applied to the Kalman filter (for which this contrast and the exact likelihood lead to asymptotically equivalent estimators) and to the discretely observed stochastic volatility models.     

Bootstrap Estimation of Benchmark Doses and Confidence Limits with Clustered Quantal Data

The benchmark dose (BMD) is an exposure level that would induce a small risk increase (BMR level) above the background. The BMD approach to deriving a reference dose for risk assessment of noncancer effects is advantageous in that the estimate of BMD is not restricted to experimental doses and utilizes most available dose-response information. To quantify statistical uncertainty of a BMD estimate, we often calculate and report its lower confidence limit (i.e., BMDL), and may even consider it as a more conservative alternative to BMD itself. Computation of BMDL may involve normal confidence limits to BMD in conjunction with the delta method. Therefore, factors, such as small sample size and nonlinearity in model parameters, can affect the performance of the delta method BMDL, and alternative methods are useful. In this article, we propose a bootstrap method to estimate BMDL utilizing a scheme that consists of a resampling of residuals after model fitting and a one-step formula for parameter estimation. We illustrate the method with clustered binary data from developmental toxicity experiments. Our analysis shows that with moderately elevated dose-response data, the distribution of BMD estimator tends to be left-skewed and bootstrap BMDL s are smaller than the delta method BMDL s on average, hence quantifying risk more conservatively. Statistically, the bootstrap BMDL quantifies the uncertainty of the true BMD more honestly than the delta method BMDL as its coverage probability is closer to the nominal level than that of delta method BMDL. We find that BMD and BMDL estimates are generally insensitive to model choices provided that the models fit the data comparably well near the region of BMD. Our analysis also suggests that, in the presence of a significant and moderately strong dose-response relationship, the developmental toxicity experiments under the standard protocol support dose-response assessment at 5% BMR for BMD and 95% confidence level for BMDL.     

A Numerical Solution to the Nonhomogeneous Two-Stage MVK Model of Cancer

In this article, we describe a straightforward method for solving the probability of at least one malignant cell by time t, and the associated hazard function, in the general (i.e., nonhomogeneous) two-stage Moolgavkar-Venzon-Knudson (MVK) model of cancer. The method consists of solving four coupled ordinary differential equations derived from the Kolmogorov backward equations for this process. The relationship of this method to previously proposed solutions is discussed.     

Binary Dynamic Logit for Correlated Ordinal: estimation,application and simulation

We evaluate the estimation performance of the Binary Dynamic Logit model for correlated ordinal variables (BDLCO model), and compare it to GEE and Ordinal Logistic Regression performance in terms of bias and Mean Absolute Percentage Error (MAPE) via Monte Carlo simulation. Our results indicate that when the proportional-odds assumption does not hold, the proposed BDLCO method is superior to existing models in estimating correlated ordinal data. Moreover, this method is flexible in terms of modeling dependence and allows unequal slopes for each category, and can be used to estimate an apple bloom data set where the proportional-odds assumption is violated. We also provide a function in R to implement BDLCO.     

A multivariate zero-inflated binomial model for the analysis of correlated proportional data

In this paper, a new multivariate zero-inflated binomial (MZIB) distribution is proposed to analyse the correlated proportional data with excessive zeros. The distributional properties of purposed model are studied. The Fisher scoring algorithm and EM algorithm are given for the computation of estimates of parameters in the proposed MZIB model with/without covariates. The score tests and the likelihood ratio tests are derived for assessing both the zero-inflation and the equality of multiple binomial probabilities in correlated proportional data. A limited simulation study is performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates and to compare the nominal levels and powers of both score tests and likelihood ratio tests. The whitefly data is used to illustrate the proposed methodologies.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

Some characterizations of discrete distributions based on weak records

Let X ₁, X ₂,... be iid random variables (rv's) with the support on nonnegative integers and let (W _n , n≥0) denote the corresponding sequence of weak record values. We obtain new characterization of geometric and some other discrete distributions based on different forms of partial independence of rv's W _n and W _n+r —W _n for some fixed n≥0 and r≥1. We also prove that rv's W ₀ and W _n+1 —W _n have identical distribution if and only if (iff) the underlying distribution is geometric.