An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

Jiří Anděl

Displaying similar documents to “An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process”

On the strong Brillinger-mixing property of $α$ -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

Similarity:

First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C (x, y)$ defining an $α$ -determinantal point process (DPP). Assuming absolute integrability of the function $C_{0} (x) = C (o, x)$ , we show that a stationary $α$ -DPP with kernel function $C_{0} (x)$ is “strongly” Brillinger-mixing, implying, among others, that its tail- $σ$ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

Estimating the conditional expectations for continuous time stationary processes

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

Similarity:

One of the basic estimation problems for continuous time stationary processes $X_{t}$ , is that of estimating $E {X_{t + β} | X_{s} : s \in [0, t]}$ based on the observation of the single block ${X_{s} : s \in [0, t]}$ when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.

On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$ -spaces

Jakub Olejnik (2010)

Banach Center Publications

Similarity:

In this paper we consider processes Xₜ with values in $L^{p}$ , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$ . The state space of $(Z, S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z, S)$ is the product of the uniform probability measure and a Gaussian distribution.

Universal rates for estimating the residual waiting time in an intermittent way

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

Similarity:

A simple renewal process is a stochastic process ${X_{n}}$ taking values in ${0, 1}$ where the lengths of the runs of $1$ ’s between successive zeros are independent and identically distributed. After observing $X_{0}, X_{1}, ... X_{n}$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional...

An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

Charles-Elie Rabier (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval $[0, T]$ representing a chromosome. The originality is in the fact that some genotypes are missing. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on $[0, T]$ and under local alternatives with a QTL at $t^{☆}$ on $[0, T]$ . We show that the LRT process is asymptotically...

Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

Branching Processes in Random Environment (BPREs) $(Z_{n} : n \geq 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $ℙ (1 \leq Z_{n} \leq k | Z_{0} = i)$ , $k, i \in ℕ$ as $n \to \infty$ . More precisely, we characterize...

Estimation of the density of a determinantal process

Yannick Baraud (2013)

Confluentes Mathematici

Similarity:

We consider the problem of estimating the density $Π$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish non-asymptotic risk bounds with respect to the Hellinger loss and deduce, when $n$ goes to infinity, uniform rates of convergence over classes of densities $Π$ of interest.

Bernstein inequality for the parameter of the pth order autoregressive process AR(p)

Samir Benaissa (2006)

Applicationes Mathematicae

Similarity:

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X ̃_{t} = θ ̃ X ̃_{t - 1} + ε ̃_{t}$ . In this paper we study the convergence in distribution of the linear operator $I (θ ̃_{T}, θ ̃) = (θ ̃_{T} - θ ̃) θ ̃^{T - 2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator. ...

Estimator selection in the gaussian setting

Yannick Baraud, Christophe Giraud, Sylvie Huet (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $σ^{2}$ . Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $𝔽$ of estimators of $f$ based on $Y$ and, with the same data $Y$ , aim at selecting an estimator among $𝔽$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown....

Density estimation via best $L^{2}$ -approximation on classes of step functions

Dietmar Ferger, John Venz (2017)

Kybernetika

Similarity:

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^{2}$ -approximation of a probability density function $f$ . If $f$ itself is a step-function the number of jumps may be unknown.

Minimax nonparametric prediction

Maciej Wilczyński (2001)

Applicationes Mathematicae

Similarity:

Let U₀ be a random vector taking its values in a measurable space and having an unknown distribution P and let U₁,...,Uₙ and $V ₁, . . ., V_{m}$ be independent, simple random samples from P of size n and m, respectively. Further, let $z ₁, . . ., z_{k}$ be real-valued functions defined on the same space. Assuming that only the first sample is observed, we find a minimax predictor d⁰(n,U₁,...,Uₙ) of the vector $Y^{m} = \sum_{j = 1}^{m} {(z ₁ (V_{j}), . . ., z_{k} (V_{j}))}^{T}$ with respect to a quadratic errors loss function.