Invariant density estimation

This paper is concerned with the existence of optimal estimators for the unknown density function, based on a finite number of independent observations. If the statistical problem given is invariant with respect to a certain transformation group, then the invariant density estimators form an essentially complete class of decision functions. If, in particular, the sample space is an Euclidean space and if the densities in question are with respect to Lebesgue measure, an optimal density estimator exists, which is symmetric in the observations and invariant under isometric transformations. If a sufficient sub--algebra exists, under additional conditions, only -measurable density estimators are to be considered.     

On the Least Median Square Problem

We consider the exact and approximate computational complexity of the multivariate least median-of-squares (LMS) linear regression estimator. The LMS estimator is among the most widely used robust linear statistical estimators. Given a set of n points in and a parameter k, the problem is equivalent to computing the narrowest slab bounded by two parallel hyperplanes that contains k of the points. We present algorithms for the exact and approximate versions of the multivariate LMS problem. We also provide nearly matching lower bounds for these problems. These lower bounds hold under the assumptions that k is Ω(n) and that deciding whether n given points in are affinely non-degenerate requires Ω(n^d) time.     

Minimax filtering of homogeneous random fields with white noise

We consider mean-square optimal linear estimation of the transform $$A\xi = \sum\limits_{k,j = 0}^\infty {a(k,j)} \xi ( - k, - j)$$ of the homogeneous random field ξ(k, j) with density f(λ, Μ) from the observations ξ(k, j) + η(k, j) with k ≤ 0, j ≤ 0, where η(k, j) is a random field with orthogonal values uncorrelated with ξ(k, j) (white noise). We determine the worst spectral densitiesf ₀(λ, Μ) ∈ and the minimax (robust) spectral characteristics of the optimal estimator of the transform Aξ for various density classes .     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

Uniform convexity and the distribution of the norm for a Gaussian measure

Summary We show that if a Banach space E has a norm · such that the modulus of uniform convexity is bounded below by a power function, then for each Gaussian measure on E the distribition of the norm for has a bounded density with respect to Lebesgue measure. This result is optimum in the following sense:If (a _n) is an arbitrary sequence with a _n0, there exists a uniformly convex norm N(·) on the standard Hilbert space, equivalent to the usual norm such that the modulus of convexity of this norm satisfies , and a Gaussian measure on E such that the distribution of the norm for does not have a bounded density with respect to Lebesgue measure.     

Spectral Computations on Lamplighter Groups and Diestel-Leader Graphs

The Diestel-Leader graph DL(q, r) is the horocyclic product of the homogeneous trees with respective degrees q + 1 and r + 1. When q = r, it is the Cayley graph of the lamplighter group (wreath product) _q with respect to a natural generating set. For the Simple random walk (SRW) operator on the latter group, Grigorchuk and Zuk, and Dicks and Schick have determined the spectrum and the (on-diagonal) spectral measure (Plancherel measure). Here, we show that thanks to the geometric realization, these results can be obtained for all DL-graphs by directly computing an ²-complete orthonormal system of finitely supported eigenfunctions of the SRW. This allows computation of all matrix elements of the spectral resolution, including the Plancherel measure. As one application, we determine the sharp asymptotic behavior of the N-step return probabilities of SRW. The spectral computations involve a natural approximating sequence of finite subgraphs, and we study the question whether the cumulative spectral distributions of the latter converge weakly to the Plancherel measure. To this end, we provide a general result regarding Følner approximations; in the specific case of DL(q, r), the answer is positive only when r = q.     

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on -compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an almost periodic Fourier transform we recover the discrete part of the spectral measure and construct a natural, consistent estimator for the latter from samples of the process.     

Parametric density estimators and characterization of families of distributions with sufficient statistics for the location parameter

We investigate estimators of the density functionP(-). It is shown that the optimal equivariant estimator has the form (1). Characterization of families of distributions with sufficient statistics for the location parameter is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 11–16, 1978.     

Central Limit Theorems for the Number of Maxima and an Estimator of the Second Spectral Moment of a Stationary Gaussian Process, with Application to Hydroscience

Let X = (X_t, t 0) be a mean zero stationary Gaussian process with variance one, assumed to satisfy some conditions on its covariance function r. Central limit theorems and asymptotic variance formulas are provided for estimators of the square root of the second spectral moment of the process and for the number of maxima in an interval, with some applications in hydroscience. A consistent estimator of the asymptotic variance is proposed for the number of maxima.     

Distribution density of the norm of a stable vector

Let B be a Banach space,X be a stable B -valued random vector with exponentd(0,2), and P(·) be the distribution density of the norm of X. In this paper we study the question of the boundedness of P. In particular, we construct examples of a space B with a symmetric stable vector X with exponentd(1,2) with unbounded P and prove that if X is a nondegenerate strictly stable vector with exponentd(0,1), then P is bounded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 105–114, 1987.The author is grateful to Yu. A. Davydov, V. I. Paulauskas, V. Yu. Bentkus, and D. Pap for stimulating discussions of the subject of this paper. When the paper was finished the author learned that similar results are found in [9].     

Improving on the mle of a bounded location parameter for spherical distributions

For the problem of estimating under squared error loss the location parameter of a p-variate spherically symmetric distribution where the location parameter lies in a ball of radius m, a general sufficient condition for an estimator to dominate the maximum likelihood estimator is obtained. Dominance results are then made explicit for the case of a multivariate student distribution with d degrees of freedom and, in particular, we show that the Bayes estimator with respect to a uniform prior on the boundary of the parameter space dominates the maximum likelihood estimator whenever and d?p. The sufficient condition matches the one obtained by Marchand and Perron (Ann. Statist. 29 (2001) 1078) in the normal case with identity covariance matrix. Furthermore, we derive an explicit class of estimators which, for , dominate the maximum likelihood estimator simultaneously for the normal distribution with identity covariance matrix and for all multivariate student distributions with d degrees of freedom, d?p. Finally, we obtain estimators which dominate the maximum likelihood estimator simultaneously for all distributions in the subclass of scale mixtures of normals for which the scaling random variable is bounded below by some positive constant with probability one.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

Multiple Stable Stochastic Integrals: Series Representation and Absolute Continuity of Their Law

We study multiple stable stochastic integrals. We begin with a definition of these integrals using a generalization of the LePage representation as Samorodnitsky et al. in Ref. 15 but with a simplified probabilistic background and for general stable measure when 0<<1 and for symmetric stable measure when 1<2. This representation allows to study the law of multiple stable integrals. We prove they are absolutely continuous using a stratification method on the Skorokhod space on which we have previously taken back the problem.     

Large Deviations Behavior for the Quadratic Error of Density Estimate

For a projection estimator f_n of an unknown density f we investigate the behavior of large deviations probability P{T_n > r_n} when r_n , where T_n is appropriately centered and normed quadratic error f_n-f)².     

Change-point Estimation of a Mean Shift in Moving-average Processes Under Dependence Assumptions

In this paper we discuss the least-square estimator of the unknown change point in a mean shift for moving-average processes of ALNQD sequence. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained. The results are also true for ρ-mixing, φ-mixing, α-mixing sequences under suitable conditions. These results extend those of Bai, who studied the mean shift point of a linear process of i.i.d, variables, and the condition ∑j=0^∞j｜aj｜ 〈 ∞ in Bai is weakened to ∑j=0^∞｜aj｜〈∞.     

Estimates for then-widths of compact sets of differentiate functions in spaces with weight functions

We obtain two-sided estimates of the widths defined by A. N. Kolmogorov for the unit spheres of the anisotropic Sobolev-Slobodetskii spaces in L_q() for an arbitrary measure, guaranteeing the compactness of the corresponding embedding. As shown by examples, these estimates turn out to be exact as far as the order is concerned for measures with a strong singularity and, in addition, they allow us to justify the formula of the classical spectral asymptotics under very weak (close to necessary) restrictions of the measure .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 117–132, 1976.     

A new approach to the single point catalytic super-Brownian motion

Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst _c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure ^c ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX _t is determined from ^c by an explicit representation formula. Heuristically, a mass ^c (ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure ^c . In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure ^c .     

A Robust Heuristic Estimator for the Period of a Poisson Intensity Function

While there are a number of methods for estimating the intensity of a cyclic point process, these assume prior estimation of the period itself. The standard method for the latter is the periodogram, or spectral analysis, approach. This is a parametric method which is sensitive to the form, in particular the number of peaks per cycle, of the intensity. We construct a family of nonparametric estimators for the period of a cyclic Poisson process, with the object of robustness against the form of the intensity. These are tested, along with the standard periodogram estimate and an earlier nonparametric estimator, on simulated data from a range of intensity functions. While the nonparametric estimators presently lack the well-developed asymptotic and statistical properties of the periodogram, the best of them is almost as accurate as the periodogram for the unimodal intensity cycles on which the latter is based. Whereas the periodogram cannot handle multimodal cycles at all, the better nonparametric estimators are reasonably accurate, and sometimes err by estimating multiples of the period rather than divisors, errors that are arguably less damaging. We conclude with some remarks concerning the derivation of asymptotic properties for our nonparametric estimator.     

The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum

We realize the lamplighter group /2 as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [–1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [–1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e^–1/1–x of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially.     

Biased estimation and hypothesis testing in linear regression

We consider a test of the simple hypothesis =₀ based on some biased estimator. Under a certain condition the corresponding test statistic coincides with the usualF-statistic based on the least squares estimator. Surprisingly, this condition is met by several well-known biased estimators.