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Register a new userLarge-Scale Shrinkage Estimation under Markovian Dependence
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Added by
Bowen Gang
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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We consider the problem of simultaneous estimation of a sequence of dependent parameters that are generated from a hidden Markov model. Based on observing a noise contaminated vector of observations from such a sequence model, we consider simultaneous estimation of all the parameters irrespective of their hidden states under square error loss. We study the roles of statistical shrinkage for improved estimation of these dependent parameters. Being completely agnostic on the distributional properties of the unknown underlying Hidden Markov model, we develop a novel non-parametric shrinkage algorithm. Our proposed method elegantly combines textit{Tweedie}-based non-parametric shrinkage ideas with efficient estimation of the hidden states under Markovian dependence. Based on extensive numerical experiments, we establish superior performance our our proposed algorithm compared to non-shrinkage based state-of-the-art parametric as well as non-parametric algorithms used in hidden Markov models. We provide decision theoretic properties of our methodology and exhibit its enhanced efficacy over popular shrinkage methods built under independence. We demonstrate the application of our methodology on real-world datasets for analyzing of temporally dependent social and economic indicators such as search trends and unemployment rates as well as estimating spatially dependent Copy Number Variations.
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In this article, motivated by biosurveillance and censoring sensor networks, we investigate the problem of distributed monitoring large-scale data streams where an undesired event may occur at some unknown time and affect only a few unknown data streams. We propose to develop scalable global monitoring schemes by parallel running local detection procedures and by combining these local procedures together to make a global decision based on SUM-shrinkage techniques. Our approach is illustrated in two concrete examples: one is the nonhomogeneous case when the pre-change and post-change local distributions are given, and the other is the homogeneous case of monitoring a large number of independent $N(0,1)$ data streams where the means of some data streams might shift to unknown positive or negative values. Numerical simulation studies demonstrate the usefulness of the proposed schemes.
Simultaneous, post-hoc inference is desirable in large-scale hypotheses testing as it allows for exploration of data while deciding on criteria for proclaiming discoveries. It was recently proved that all admissible post-hoc inference methods for the number of true discoveries must be based on closed testing. In this paper we investigate tractable and efficient closed testing with local tests of different properties, such as monotonicty, symmetry and separability, meaning that the test thresholds a monotonic or symmetric function or a function of sums of test scores for the individual hypotheses. This class includes well-known global null tests by Fisher, Stouffer and Ruschendorf, as well as newly proposed ones based on harmonic means and Cauchy combinations. Under monotonicity, we propose a new linear time statistic (coma) that quantifies the cost of multiplicity adjustments. If the tests are also symmetric and separable, we develop several fast (mostly linear-time) algorithms for post-hoc inference, making closed testing tractable. Paired with recent advances in global null tests based on generalized means, our work immediately instantiates a series of simultaneous inference methods that can handle many complex dependence structures and signal compositions. We provide guidance on choosing from these methods via theoretical investigation of the conservativeness and sensitivity for different local tests, as well as simulations that find analogous behavior for local tests and full closed testing. One result of independent interest is the following: if $P_1,dots,P_d$ are $p$-values from a multivariate Gaussian with arbitrary covariance, then their arithmetic average P satisfies $Pr(P leq t) leq t$ for $t leq frac{1}{2d}$.
In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods.
We consider sparse Bayesian estimation in the classical multivariate linear regression model with $p$ regressors and $q$ response variables. In univariate Bayesian linear regression with a single response $y$, shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a $p times q$ coefficients matrix $mathbf{B}$. We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when $p>n$ and even when $p$ grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive R package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.
The application of the lasso is espoused in high-dimensional settings where only a small number of the regression coefficients are believed to be nonzero. Moreover, statistical properties of high-dimensional lasso estimators are often proved under the assumption that the correlation between the predictors is bounded. In this vein, coordinatewise methods, the most common means of computing the lasso solution, work well in the presence of low to moderate multicollinearity. The computational speed of coordinatewise algorithms degrades however as sparsity decreases and multicollinearity increases. Motivated by these limitations, we propose the novel Deterministic Bayesian Lasso algorithm for computing the lasso solution. This algorithm is developed by considering a limiting version of the Bayesian lasso. The performance of the Deterministic Bayesian Lasso improves as sparsity decreases and multicollinearity increases, and can offer substantial increases in computational speed. A rigorous theoretical analysis demonstrates that (1) the Deterministic Bayesian Lasso algorithm converges to the lasso solution, and (2) it leads to a representation of the lasso estimator which shows how it achieves both $ell_1$ and $ell_2$ types of shrinkage simultaneously. Connections to other algorithms are also provided. The benefits of the Deterministic Bayesian Lasso algorithm are then illustrated on simulated and real data.
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