arXiv subjects

Gourab Mukherjee

Publications and source records attributed to Gourab Mukherjee.

21 records · Page 2Linked to original sources

arXiv2016

Empirical Bayes Estimates for a 2-Way Cross-Classified Additive Model

We develop an empirical Bayes procedure for estimating the cell means in an unbalanced, two-way additive model with fixed effects. We employ a hierarchical model, which reflects exchangeability of the effects within treatment and within block but not necessarily between them, as suggested before by Lindley and Smith (1972). The hyperparameters of this hierarchical model, instead of considered fixed, are to be substituted with data-dependent values in such a way that the point risk of the empirical Bayes estimator is small. Our method chooses the hyperparameters by minimizing an unbiased risk estimate and is shown to be asymptotically optimal for the estimation problem defined above. The usual empirical Best Linear Unbiased Predictor (BLUP) is shown to be substantially different from the proposed method in the unbalanced case and therefore performs sub-optimally. Our estimator is implemented through a computationally tractable algorithm that is scalable to work under large designs. The case of missing cell observations is treated as well. We demonstrate the advantages of our method over the BLUP estimator through simulations and in a real data example, where we estimate average nitrate levels in water sources based on their locations and the time of the day.

stat.ME↗

arXiv2015

Exact minimax estimation of the predictive density in sparse Gaussian models

We consider estimating the predictive density under Kullback-Leibler loss in an $\ell_0$ sparse Gaussian sequence model. Explicit expressions of the first order minimax risk along with its exact constant, asymptotically least favorable priors and optimal predictive density estimates are derived. Compared to the sparse recovery results involving point estimation of the normal mean, new decision theoretic phenomena are seen. Suboptimal performance of the class of plug-in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk. We find that minimax optimal strategies lie outside the Gaussian family but can be constructed with threshold predictive density estimates. Novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates.

math.ST↗

arXiv2012

On the within-family Kullback-Leibler risk in Gaussian Predictive models

We consider estimating the predictive density under Kullback-Leibler loss in a high-dimensional Gaussian model. Decision theoretic properties of the within-family prediction error -- the minimal risk among estimates in the class $\mathcal{G}$ of all Gaussian densities are discussed. We show that in sparse models, the class $\mathcal{G}$ is minimax sub-optimal. We produce asymptotically sharp upper and lower bounds on the within-family prediction errors for various subfamilies of $\mathcal{G}$. Under mild regularity conditions, in the sub-family where the covariance structure is represented by a single data dependent parameter $\Shat=\dhat \cdot I$, the Kullback-Leiber risk has a tractable decomposition which can be subsequently minimized to yield optimally flattened predictive density estimates. The optimal predictive risk can be explicitly expressed in terms of the corresponding mean square error of the location estimate, and so, the role of shrinkage in the predictive regime can be determined based on point estimation theory results. Our results demonstrate that some of the decision theoretic parallels between predictive density estimation and point estimation regimes can be explained by second moment based concentration properties of the quadratic loss.

math.ST↗