Uniform Hölder exponent of a stationary increments Gaussian process: Estimation starting from average values

Let {X(t)}_t∈R be a stationary increments Gaussian process satisfying some assumptions. By using the notion of generalized quadratic variation we build a strongly consistent and asymptotically normal estimator of the uniform Hölder exponent of X, over a compact interval. Our estimator is obtained starting from average values of the process over a regular grid.     

Validity of Edgeworth expansions of minimum contrast estimators for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. To estimate θ we propose a minimum contrast estimation method which includes the maximum likelihood method and the quasi-maximum likelihood method as special cases. Let θ̂_τ be the minimum contrast estimator of θ. Then we derive the Edgewroth expansion of the distribution of θ̂_τ up to third order, and prove its valldity. By this Edgeworth expansion we can see that this minimum contrast estimator is always second-order asymptotically efficient in the class of second-order asymptotically median unbiased estimators. Also the third-order asymptotic comparisons among minimum contrast estimators will be discussed.     

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Goodness-of-fit Tests for Continuous-time Stationary Processes

The paper considers the following problem of hypotheses testing: based on a finite realization {X(t)}, 0 ≤ t ≤ T of a zero mean real-valued mean square continuous stationary Gaussian process X(t), t ? R, construct goodness-of-fit tests for testing a hypothesis H₀ that the hypothetical spectral density of the process X(t) has the specified form. We show that in the case where the hypothetical spectral density of X(t) does not depend on unknown parameters (the hypothesis H₀ is simple), then the suggested test statistic has a chi-square distribution. In the case where the hypothesis H₀ is composite, that is, the hypothetical spectral density of X(t) depends on an unknown p–dimensional vector parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic, which is similar to that of obtained by Chernov and Lehman in the case of independent observations. The testing procedure works both for short- and long-memory models.     

Sur la convergence uniforme presque complète dans l'estimation de la densité spectrale d'un processus à temps continu après échantillonnage du temps

Let $\text{X={X(t)}}{}_{\mathrm{<mtext>t\in </mtext><mtext>R</mtext>}}$ be a continuous-time strictly stationary and strongly mixing process. In this paper, we prove in the setting of spectral density estimation, at first, under some hard conditions on the spectral density φ_X (because of aliasing phenomenon), the uniformly complete convergence of the spectral density estimate from periodic sampling. Afterwards, to overcome aliasing, we consider the sampled process $\text{{X(t}{}_{\mathrm{n}}\text{)}}{}_{\mathrm{<mtext>n\in </mtext><mtext>Z</mtext>}}$, where {t_n} is a stationary point process independent from X. The uniform complete convergence of the spectral estimate based on the discrete time observations {X(t_k),t_k} is also obtained. The convergence rates are also established. To cite this article: M. Rachdi, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Shrinkage estimation in the frequency domain of multivariate time series

In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L₂ risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of “Kolmogorov” asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as “effective sample size” into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L₂ risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.     

Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

Let {X _n } be an integer-valued Markov chain with finite state space. Let $S_{n}=\sum_{k=0}^{n}X_{k}$ and let L _n (x) be the number of times S _k hits x∈? up to step n. Define the normalized local time process l _n (t,x) by The subject of this paper is to prove a functional weak invariance principle for the normalized sequence l _n (t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.     

On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations _X0,_XΔ,…,_XnΔ$X_0,X_{\Delta },\dots ,X_{n\Delta }$, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞$n\Delta \to \infty$ and Δ→0$\Delta \to 0$) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.     

A generalization of Hiraguchi's: Inequality for posets

For a poset X, Dim(X) is the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains. Hiraguchi proved that if | X | ? 4, then Dim(X) ? [| X |/2]. For each k ? 2, we define Dim_k(X) as the smallest positive integer t for which X is isomorphic to a subposet of the cartesian product of t chains, each of length k. We then prove that if | X | ? 5, Dim₃(X) ? {| X |/2} and if | X | ? 6, then Dim₄(X) ? [| X |/2].     

Asymptotically optimal selection of a piecewise polynomial estimator of a regression function

Let (X, Y) be a pair of random variables such that X = (X₁,…, X_d) ranges over a nondegenerate compact d-dimensional interval C and Y is real-valued. Let the conditional distribution of Y given X have mean θ(X) and satisfy an appropriate moment condition. It is assumed that the distribution of X is absolutely continuous and its density is bounded away from zero and infinity on C. Without loss of generality let C be the unit cube. Consider an estimator of θ having the form of a piecewise polynomial of degree k_n based on m_n^d cubes of length 1/m_n, where the m_n^d(_d^k_n+d) coefficients are chosen by the method of least squares based on a random sample of size n from the distribution of (X, Y). Let (k_n, m_n) be chosen by the FPE procedure. It is shown that the indicated estimator has an asymptotically minimal squared error of prediction if θ is not of the form of piecewise polynomial.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Maximum likelihood estimation of a survival function under the koziol-green proportional hazards model

Let X and Y be two independent competing lifetimes wih survival functions S_F(t) and S_G(t). In this paper we, study a proportional-hazards competing risks model characterized by Armitage (1959): S_G(t) = S_G(t)]^β for certain positive constant β. The maximum likelihood estimator for S_F is proposed, examined and compared with the product limit estimator of Kaplan and Meier (1958). The MLE is shown to be asymptotically more efficient that the PLE under the specified model.     

A definition of estimator efficiency ink-parameter case

Summary The paper introduces a new definition of efficiency in the multiparameter case (θ₁,...,θ_k) when the variance-covariance matrix of the vector estimator (t ₁, ...t _k) exists. The definition is also applicable to the asymptotically unbiased estimators. The basic idea is that, as we want in general to estimate some function g(θ₁,...θ_k) of the parameters, efficiency of the vector estimator shall be defined as the smallest efficiency of the estimatorg(t ₁, ...t _k),g being regular. It is shown that this definition is asymptotically equivalent to the one obtained by any linear combination of the estimators, as it happens, naturally, for quantile estimation in the location-dispersion case. This efficiency is larger than Cramér efficiency which is, thus, not attained, apart from a very exceptional case. Finally, a lower bound for the asymptotic variance is obtained.     

Generalized profile LSE in varying-coefficient partially linear models with measurement errors

This paper is concerned with the estimating problem of a semiparametric varying-coefficient partially linear errors-in-variables model Yi=Xτiβ+Zτiα(Ui)+εi , Wi=Xi+ξi,i=1, ··· , n. Due to measurement errors, the usual profile least square estimator of the parametric component, local polynomial estimator of the nonparametric component and profile least squares based estimator of the error variance are biased and inconsistent. By taking the measurement errors into account we propose a generalized profile least squares estimator for the parametric component and show it is consistent and asymptotically normal. Correspondingly, the consistent estimation of the nonparametric component and error variance are proposed as well. These results may be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of these proposed estimations.     

Kernel estimation of density level sets

Let f be a multivariate density and f_n be a kernel estimate of f drawn from the n-sample X₁,…,X_n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f?t} and its plug-in estimator {f_n?t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.     

Estimation for the change point of volatility in a stochastic differential equation

We consider a multidimensional Itô process Y=(Y_t)_t∈[0,T] with some unknown drift coefficient process b_t and volatility coefficient σ(X_t,θ) with covariate process X=(X_t)_t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t^∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.     

Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Let X ₁,..., X _n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H ₀: X ₁,..., X _n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H ₀ versus the “at most one change (AMOC) in the mean” alternative hypothesis H _A: there is an integer k*, 1 ≤ k* < n, such that EX ₁ = · · · = EX_k* ≠ EX_k*+1 = ··· = EX _n. A natural way of testing for H ₀ versus H _A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ_k = |S _k/k - (S _n - S _k)/(n - k)|, where S _k:= X ₁ + ··· + X _k. Asymptotic distributions of these test statistics under H ₀ as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z _n(t)|, where {Z _n(t), 0 ≤ t ≤ 1}_n≥1 are the tied-down partial sums processes such that

$${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$

if 0 ≤ t < 1, and Z _n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H ₀ versus H _A via sup-functionals of appropriately weighted |Z _n(t)|. Simply considering max_1?k<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z _n(t)|/(t(1 - t)) under H ₀ and assuming also that E|X|^r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z _n(t)| has not been considered before.

     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

On set systems with a threshold property

For n,k and t such that 1<t<k<n, a set F of subsets of [n] has the (k,t)-threshold property if every k-subset of [n] contains at least t sets from F and every (k-1)-subset of [n] contains less than t sets from F. The minimal number of sets in a set system with this property is denoted by m(n,k,t). In this paper we determine m(n,4,3)exactly for n sufficiently large, and we show that m(n,k,2) is asymptotically equal to the generalized Turán number T_k-1(n,k,2).