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16
A tutorial on MM algorithms
 Amer. Statist
, 2004
"... Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function ..."
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Cited by 69 (3 self)
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Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function for the loglikelihood. Iterative optimization of a surrogate function as exemplified by an EM algorithm does not necessarily require missing data. Indeed, every EM algorithm is a special case of the more general class of MM optimization algorithms, which typically exploit convexity rather than missing data in majorizing or minorizing an objective function. In our opinion, MM algorithms deserve to part of the standard toolkit of professional statisticians. The current article explains the principle behind MM algorithms, suggests some methods for constructing them, and discusses some of their attractive features. We include numerous examples throughout the article to illustrate the concepts described. In addition to surveying previous work on MM algorithms, this article introduces some new material on constrained optimization and standard error estimation. Key words and phrases: constrained optimization, EM algorithm, majorization, minorization, NewtonRaphson 1 1
Statistical inference for discretely observed Markov jump processes
 Journal of the Royal Statistical Society: Series B (Statistical Methodology
"... Likelihood inference for discretely observed Markov jump processes with finite state space is investigated. The existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated. This topic is closely related to the imbedding problem for Markov chains. It is demon ..."
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Cited by 20 (3 self)
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Likelihood inference for discretely observed Markov jump processes with finite state space is investigated. The existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated. This topic is closely related to the imbedding problem for Markov chains. It is demonstrated that the maximum likelihood estimator can be found either by the EMalgorithm or by a Markov chain Monte Carlo procedure. When the maximum likelihood estimator does not exist, an estimator can be obtained by using a penalized likelihood function or by the MCMCprocedure with a suitable prior. The theory is illustrated by a simulation study.
A survey of Monte Carlo algorithms for maximizing the likelihood of a twostage hierarchical model
, 2001
"... Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternati ..."
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Cited by 11 (4 self)
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Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo NewtonRaphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo NewtonRaphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS0072827. 1 1
Efficient estimation of transition rates between credit ratings from observations
"... at discrete time points ..."
Regressograms and meancovariance models for incomplete longitudinal data, American Statistician p. Revision Submitted
, 2011
"... Longitudinal studies are prevalent in biological and social sciences where subjects are measured repeatedly over time. Modeling the correlations and handling missing data are among the most challenging problems in analyzing such data. There are various methods for handling missing data, but databas ..."
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Cited by 1 (1 self)
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Longitudinal studies are prevalent in biological and social sciences where subjects are measured repeatedly over time. Modeling the correlations and handling missing data are among the most challenging problems in analyzing such data. There are various methods for handling missing data, but databased and graphical methods for modeling the covariance matrix of longitudinal data are relatively new. We adopt an approach based on the modified Cholesky decomposition of the covariance matrix which handles both challenges simultaneously. It amounts to formulating parametric models for the regression coefficients of the conditional mean and variance of each measurement given its predecessors. We demonstrate the roles of profile plots and regressograms in formulating joint meancovariance models for (in)complete longitudinal data. Applying these graphical tools to a casestudy of Fruit Fly Mortality data which has 22 % missing values reveals an “Sshape ” or logistic curve for the mean function and cubic polynomial models for the two factors of the modified Cholesky decomposition of the sample covariance matrix. A likelihoodbased method for estimating the parameters of the mean and covariance models using the EM algorithm is proposed. A simulation study and application to real data demonstrate that the estimation method works well for incomplete longitudinal data.
MUS81 Generates a Subset of MLH1MLH3–Independent Crossovers in Mammalian Meiosis
, 2008
"... Two eukaryotic pathways for processing doublestrand breaks (DSBs) as crossovers have been described, one dependent on the MutL homologs Mlh1 and Mlh3, and the other on the structurespecific endonuclease Mus81. Mammalian MUS81 has been implicated in maintenance of genomic stability in somatic cells ..."
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Cited by 1 (0 self)
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Two eukaryotic pathways for processing doublestrand breaks (DSBs) as crossovers have been described, one dependent on the MutL homologs Mlh1 and Mlh3, and the other on the structurespecific endonuclease Mus81. Mammalian MUS81 has been implicated in maintenance of genomic stability in somatic cells; however, little is known about its role during meiosis. Mus81deficient mice were originally reported as being viable and fertile, with normal meiotic progression; however, a more detailed examination of meiotic progression in Mus81null animals and WT controls reveals significant meiotic defects in the mutants. These include smaller testis size, a depletion of mature epididymal sperm, significantly upregulated accumulation of MLH1 on chromosomes from pachytene meiocytes in an interferenceindependent fashion, and a subset of meiotic DSBs that fail to be repaired. Interestingly, chiasmata numbers in spermatocytes from Mus81 2/2 animals are normal, suggesting additional integrated mechanisms controlling the two distinct crossover pathways. This study is the first indepth analysis of meiotic progression in Mus81nullizygous mice, and our results implicate the MUS81 pathway as a regulator of crossover
A Cautionary Note on Generalized Linear Models for Covariance of Unbalanced Longitudinal Data
"... Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positivedefiniteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positivedefiniteness cons ..."
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Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positivedefiniteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positivedefiniteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations. Keywords: modeling Cholesky decomposition, missing data, joint meancovariance 1.
Input Design for Nonlinear Stochastic Dynamic Systems A Particle Filter Approach
"... Abstract: We propose an algorithm for optimal input design in nonlinear stochastic dynamic systems. The approach relies on minimizing a function of the covariance of the parameter estimates of the system with respect to the input. The covariance matrix is approximated using a joint likelihood functi ..."
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Abstract: We propose an algorithm for optimal input design in nonlinear stochastic dynamic systems. The approach relies on minimizing a function of the covariance of the parameter estimates of the system with respect to the input. The covariance matrix is approximated using a joint likelihood function of hidden states and measurements, and a combination of state filters and smoothers. The input is parametrized using an autoregressive model. The proposed approach is illustrated through a simulation example. 1.
© 2004 Biometrika Trust Printed in Great Britain
"... In this paper we propose inference methods based on the em algorithm for estimating the parameters of a weakly parameterised competing risks model with masked causes of failure and secondstage data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause ..."
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In this paper we propose inference methods based on the em algorithm for estimating the parameters of a weakly parameterised competing risks model with masked causes of failure and secondstage data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the causespecific hazard functions and of the masking probabilities is performed via an em algorithm. Both the e and msteps can be solved in closed form under the full model and under some restricted models of interest. We illustrate the flexibility of the method by showing how grouped data and tests of common hypotheses in the literature on missing cause of death can be handled. The method is applied to a real dataset and the asymptotic and robustness properties of the estimators are investigated through simulation.