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Register a new userFast nonparametric near-maximum likelihood estimation of a mixing density
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Added by
Ryan Martin
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Mixture models are regularly used in density estimation applications, but the problem of estimating the mixing distribution remains a challenge. Nonparametric maximum likelihood produce estimates of the mixing distribution that are discrete, and these may be hard to interpret when the true mixing distribution is believed to have a smooth density. In this paper, we investigate an algorithm that produces a sequence of smooth estimates that has been conjectured to converge to the nonparametric maximum likelihood estimator. Here we give a rigorous proof of this conjecture, and propose a new data-driven stopping rule that produces smooth near-maximum likelihood estimates of the mixing density, and simulations demonstrate the quality empirical performance of this estimator.
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The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym derivative for a mixed fractional Brownian motion. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic normality of the maximum likelihood estimators have been established for all $Hin(0,1)$, $H eq 1/2$.
We derive Laplace-approximated maximum likelihood estimators (GLAMLEs) of parameters in our Graph Generalized Linear Latent Variable Models. Then, we study the statistical properties of GLAMLEs when the number of nodes $n_V$ and the observed times of a graph denoted by $K$ diverge to infinity. Finally, we display the estimation results in a Monte Carlo simulation considering different numbers of latent variables. Besides, we make a comparison between Laplace and variational approximations for inference of our model.
Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker & Mizera shows that modern convex optimization techniques -- in particular, interior point methods -- are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of our approach in experiments on synthetic data sets as well as a large genetic association data set. In large data sets (n = 1,000,000 observations, m = 1,000 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia, and in an R package available on CRAN (see https://CRAN.R-project.org/package=mixsqp).
In this note we provide explicit expressions and expansions for a special function which appears in nonparametric estimation of log-densities. This function returns the integral of a log-linear function on a simplex of arbitrary dimension. In particular it is used in the R-package LogCondDEAD by Cule et al. (2007).
The likelihood-informed subspace (LIS) method offers a viable route to reducing the dimensionality of high-dimensional probability distributions arisen in Bayesian inference. LIS identifies an intrinsic low-dimensional linear subspace where the target distribution differs the most from some tractable reference distribution. Such a subspace can be identified using the leading eigenvectors of a Gram matrix of the gradient of the log-likelihood function. Then, the original high-dimensional target distribution is approximated through various forms of ridge approximations of the likelihood function, in which the approximated likelihood only has support on the intrinsic low-dimensional subspace. This approximation enables the design of inference algorithms that can scale sub-linearly with the apparent dimensionality of the problem. Intuitively, the accuracy of the approximation, and hence the performance of the inference algorithms, are influenced by three factors -- the dimension truncation error in identifying the subspace, Monte Carlo error in estimating the Gram matrices, and Monte Carlo error in constructing ridge approximations. This work establishes a unified framework to analysis each of these three factors and their interplay. Under mild technical assumptions, we establish error bounds for a range of existing dimension reduction techniques based on the principle of LIS. Our error bounds also provide useful insights into the accuracy comparison of these methods. In addition, we analyze the integration of LIS with sampling methods such as Markov Chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC). We also demonstrate our analyses on a linear inverse problem with Gaussian prior, which shows that all the estimates can be dimension-independent if the prior covariance is a trace-class operator.
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