We study nonparametric maximum likelihood estimation of a log-concave density function f 0 which is known to satisfy further constraints, where either (a) the mode m of f 0 is known, or (b) f 0 is known to be symmetric about a fixed point m. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE's), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE's pointwise limit distribution at m (either the known mode or the known center of symmetry) and at a point x 0 = m. Software to compute the constrained estimators is available in the R package logcondens.mode.The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of f 0 . These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters.

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.

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.

In this article, we revisit the problem of estimating the unknown zerosymmetric distribution in a two-component location mixture model, considered in previous works, now under the assumption that the zero-symmetric distribution has a log-concave density. When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric logconcave Maximum Likelihood estimator (MLE) of both the mixed density and that of the unknown zero-symmetric component are consistent in the Hellinger distance. In case the estimators for the shift locations and mixing probability are ? n-consistent, we establish that these MLE's converge to the truth at the rate n´2 {5 in the L 1 distance. To estimate the shift locations and mixing probability, we use the estimators proposed by Hunter et al. (2007). The unknown zero-symmetric density is efficiently computed using the R package logcondens.mode.

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