We study mixed models with a single grouping factor where inference about unknown parameters requires optimizing a marginal likelihood defined by an intractable integral. Low-dimensional numerical integration techniques are regularly used to approximate these integrals with inferences about parameters based on the resulting approximate marginal likelihood. For a generic class of mixed models that satisfy explicit regularity conditions we derive the stochastic relative error rate incurred for both the likelihood and maximum likelihood estimator when adaptive numerical integration is used to approximate the marginal likelihood. We then specialize the analysis to well-specified generalized linear mixed models having exponential family response and multivariate Gaussian random effects, verifying that the regularity conditions hold and hence that the convergence rates apply. We also prove that for models with likelihoods satisfying very weak concentration conditions that the maximum likelihood estimators from non-adaptive numerical integration approximations of the marginal likelihood are not consistent, further motivating adaptive numerical integration as the preferred tool for inference in mixed models. Code to reproduce the simulations in this paper is provided at https://github.com/awstringer1/aq-theory-paper-code.
Several identities, involving the Lebesgue measure of data-based simplices or parallelotopes, have been obtained forfunctionals involving the sample covariance matrixSor its population counterpart Sigma. This is the case in particularfor Wilks' generalized variance det(S), which allowed one to obtain an explicit expression for E[det(S)]wheneverobservations are randomly sampled from a distribution with finite second-order moments. To date, however, allsuch results are limited to scalar functionals. In this paper, we obtain geometric identities for the adjugate adj(S) of S and for other functionals involving adj(S)and the sample mean vector (X) over bar. This allows us in particular to define uniformly minimum risk unbiased (UMRU) estimators of the corresponding population quantities. Just asthe results from (Ann. Statist. 36(2008) 2261-2283) find applications when conditional independence is of interest,our results are relevant in an elliptical framework (resp., in a general framework) where conditional independenceis replaced with partial uncorrelatedness (resp., with an original concept of partial median-uncorrelatedness).
Shuffled regression and unlinked regression represent intriguing challenges that have garnered considerable attentionin many fields, including ecological regression, multi-target tracking problems, image denoising, and others.However, a notable gap exists in the existing literature, particularly in vanishing noise, i.e., how the rate ofestimation of the underlying signal scales with the error variance. This paper aims to bridge this gap by delvinginto the monotone function estimation problem under vanishing noise variance, i.e., we allow the error varianceto go to 0 as the number of observations increases. Our investigation reveals that, asymptotically, the shuffledregression problem is comparatively simpler than the unlinked regression; if the error variance is smaller than athreshold, then the minimax risk of the shuffled regression is smaller than that of the unlinked regression. On theother hand, the minimax estimation error is of the same order in the two problems if the noise level is larger thanthat threshold. Our analysis is quite general in that we do not assume smoothness assumptions on the link functionm0(which only needs to be left-continuous, but may even have jump discontinuities). Because these problems arerelated to deconvolution, we also provide bounds for deconvolution in a similar context. Through this exploration,we contribute to understanding the intricate relationships between these statistical problems and shed light on theirbehaviors under vanishing noise.
We investigate a multivariate Hilbertian additive model in which the response variable is Hilbert-space-valuedand predictors are multi-dimensional Euclidean. We allow for the scenario where both variables are unobservablebut they are estimable. This scenario includes the case of principal or singular component scores, the case ofdensity-valued responses and the case of semiparametric regression. For such cases, we provide estimation errorsfor the variables, which are of importance in their own right. Additionally, we derive the full non-asymptotic andasymptotic properties of our regression estimator under such estimation errors. This allows us to handle variousnovel regression problems. We demonstrate the strong performance of our regression estimator via simulationstudies and a real data application
We investigate the asymptotic behavior of the spiked eigenvalues and the largest non-spiked eigenvalues of thesample covariance matrixTXX & lowast;T & lowast;, where the entries ofXfollow the elliptical distributions. We discuss thismodel under a general framework that allows for the existence of both divergent spiked eigenvalues and boundedones, while the total number of spikes can grow to infinity. We show that some well-known results which havebeen proved forXwith independent components still hold under elliptical distributions. Specifically, the divergentspiked sample eigenvalues converge in distribution to Gaussian limits after proper centralization and scaling.The asymptotic means depend not only on the population spikes but also on the non-spikes, while the asymptoticvariances depend solely on the radius ofX. For the bounded sample spiked eigenvalues, they converge in probabilityto certain typical locations of the limiting spectral distribution. We also provide a central limit theorem for a class ofbounded spiked eigenvalues for potential statistical use. Additionally, for the largest non-spiked sample eigenvalues,the limiting Tracy-Widom law is obtained. We note that the results above are derived under the assumption that theradius of the entries inXhas finite fourth moments.
