Integral models of reductive groups and integral Mumford-Tate groups

Let $G$ be a reductive algebraic group over a $p$-adic field or number field $K$, and let $V$ be a $K$-linear faithful representation of $G$. A lattice $\Lambda$ in the vector space $V$ defines a model $\hat{G}_{\Lambda}$ of $G$ over $\mathscr{O}_K$. One may wonder to what extent $\Lambda$ is determined by the group scheme $\hat{G}_{\Lambda}$. In this paper we prove that up to a natural equivalence relation on the set of lattices there are only finitely many $\Lambda$ corresponding to one model $\hat{G}_{\Lambda}$. Furthermore, we relate this fact to moduli spaces of abelian varieties as follows: let $\mathscr{A}_{g,n}$ be the moduli space of principally polarised abelian varieties of dimension $g$ with level $n$ structure. We prove that there are at most finitely many special subvarieties of $\mathscr{A}_{g,n}$ with a given integral generic Mumford-Tate group

A Kiefer-Wolfowitz type of result in a general setting, with an application to smooth monotone estimation

We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-\tau)}$, for some $\tau\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_n$. In typical examples, the approximating process is Gaussian and $\tau=1$, in which case the convergence rate is $n^{-2/3}(\log n)^{2/3}$ is in the same spirit as the one obtained by Kiefer and Wolfowitz (1976) for the special case of estimating a decreasing density. We also obtain a similar result for the primitive of $F_n$, in which case $\tau=2$, leading to a faster rate $n^{-1}\log n$, also found by Wang and Woodfroofe (2007). As an application in our general setup, we show that a smoothed Grenander type estimator and its derivative are asymptotically equivalent to the ordinary kernel estimator and its derivative in first order

Asymptotic expansion of the minimum covariance determinant estimators

In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for multivariate distributions that have a density and provide an explicit expression. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the minimum covariance determinant (MCD) estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators.Comment: 21 page

The behavior of the NPMLE of a decreasing density near the boundaries of the support

We investigate the behavior of the nonparametric maximum likelihood estimator $\hat{f}_n$ for a decreasing density $f$ near the boundaries of the support of $f$. We establish the limiting distribution of $\hat{f}_n(n^{-\alpha})$, where we need to distinguish between different values of $0<\alpha<1$. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of $f$ at the boundaries of the support. The limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator.Comment: Published at http://dx.doi.org/10.1214/009053606000000100 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic normality of the LkL_k-error of the Grenander estimator

We investigate the limit behavior of the $L_k$-distance between a decreasing density $f$ and its nonparametric maximum likelihood estimator $\hat{f}_n$ for $k\geq1$. Due to the inconsistency of $\hat{f}_n$ at zero, the case $k=2.5$ turns out to be a kind of transition point. We extend asymptotic normality of the $L_1$-distance to the $L_k$-distance for $1\leq k<2.5$, and obtain the analogous limiting result for a modification of the $L_k$-distance for $k\geq2.5$. Since the $L_1$-distance is the area between $f$ and $\hat{f}_n$, which is also the area between the inverse $g$ of $f$ and the more tractable inverse $U_n$ of $\hat{f}_n$, the problem can be reduced immediately to deriving asymptotic normality of the $L_1$-distance between $U_n$ and $g$. Although we lose this easy correspondence for $k>1$, we show that the $L_k$-distance between $f$ and $\hat{f}_n$ is asymptotically equivalent to the $L_k$-distance between $U_n$ and $g$.Comment: Published at http://dx.doi.org/10.1214/009053605000000462 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org