Charles R. Doss

GLOBAL RATES OF CONVERGENCE OF THE MLES OF LOG-CONCAVE AND S CONCAVE DENSITIES

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when -1 < s < ∞ where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s < -1.

Markov chain Monte Carlo estimation of quantiles

We consider quantile estimation using Markov chain Monte Carlo and establish conditions under which the sampling distribution of the Monte Carlo error is approximately Normal. Further, we investigate techniques to estimate the associated asymptotic variance, which enables construction of an asymptotically valid interval estimator. Finally, we explore the finite sample properties of these methods through examples and provide some recommendations to practitioners.

Fitting birth–death processes to panel data with applications to bacterial DNA fingerprinting

Continuous-time linear birth-death-immigration (BDI) processes are frequently used in ecology and epidemiology to model stochastic dynamics of the population of interest. In clinical settings, multiple birth-death processes can describe disease trajectories of individual patients, allowing for estimation of the effects of individual covariates on the birth and death rates of the process. Such estimation is usually accomplished by analyzing patient data collected at unevenly spaced time points, referred to as panel data in the biostatistics literature. Fitting linear BDI processes to panel data is a nontrivial optimization problem because birth and death rates can be functions of many parameters related to the covariates of interest. We propose a novel expectation-maximization (EM) algorithm for fitting linear BDI models with covariates to panel data. We derive a closed-form expression for the joint generating function of some of the BDI process statistics and use this generating function to reduce the E-step of the EM algorithm, as well as calculation of the Fisher information, to one-dimensional integration. This analytical technique yields a computationally efficient and robust optimization algorithm that we implemented in an open-source R package. We apply our method to DNA fingerprinting of Mycobacterium tuberculosis, the causative agent of tuberculosis, to study intrapatient time evolution of IS6110 copy number, a genetic marker frequently used during estimation of epidemiological clusters of Mycobacterium tuberculosis infections. Our analysis reveals previously undocumented differences in IS6110 birth-death rates among three major lineages of Mycobacterium tuberculosis, which has important implications for epidemiologists that use IS6110 for DNA fingerprinting of Mycobacterium tuberculosis.

Inference for a two-component mixture of symmetric distributions under log-concavity

In this article, we reconsider the problem of estimating the unknown symmetric density in a two-component location mixture model under the assumption that the symmetric density is log-concave. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are p n-consistent, we establish that these MLE’s converge to the truth at the rate n 2=5 in the L1 distance. To estimate the shift locations and mixing probability, we use the estimators proposed by Hunter et al. (2007). The unknown symmetric density is eciently computed using the R package logcondens.mode.

Observations on abundance of bluntnose sixgill sharks, Hexanchus griseus, in an urban waterway in Puget Sound, 2003-2005.

The bluntnose sixgill shark, Hexanchus griseus, is a widely distributed but poorly understood large, apex predator. Anecdotal reports of diver-shark encounters in the late 1990’s and early 2000’s in the Pacific Northwest stimulated interest in the normally deep-dwelling shark and its presence in the shallow waters of Puget Sound. Analysis of underwater video documenting sharks at the Seattle Aquarium’s sixgill research site in Elliott Bay and mark-resight techniques were used to answer research questions about abundance and seasonality. Seasonal changes in relative abundance in Puget Sound from 2003–2005 are reported here. At the Seattle Aquarium study site, 45 sixgills were tagged with modified Floy visual marker tags, along with an estimated 197 observations of untagged sharks plus 31 returning tagged sharks, for a total of 273 sixgill observations recorded. A mark-resight statistical model based on analysis of underwater video estimated a range of abundance from a high of 98 sharks seen in July of 2004 to a low of 32 sharks seen in March of 2004. Both analyses found sixgills significantly more abundant in the summer months at the Seattle Aquarium’s research station.

Concave regression: value-constrained estimation and likelihood ratio-based inference

We propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function, based on the shape restriction of concavity (alternatively, convexity). Dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis, that is, studying a concave least-squares estimator satisfying a further equality constraint. We study this null hypothesis least-squares estimator (NLSE) here, and use it to study our likelihood ratio statistic. The NLSE is the solution to a convex program, and we find a set of inequality and equality constraints that characterize the solution. We also study a corresponding limiting version of the convex program based on observing a Brownian motion with drift. The solution to the limit problem is a stochastic process. We study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem. This allows us to show the limit stochastic process yields the limit distribution of the (finite sample) NLSE. We conjecture that the likelihood ratio statistic is asymptotically pivotal, meaning that it has a limit distribution with no nuisance parameters to be estimated, which makes it a very effective tool for this difficult inference problem. We provide a partial proof of this conjecture, and we also provide simulation evidence strongly supporting this conjecture.