Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models

S. Valère Bitseki Penda; Hacène Djellout

Displaying similar documents to “Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models”

Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

João Tiago Mexia, João Lita da Silva (2007)

Discussiones Mathematicae Probability and Statistics

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Let $Y_{i} = x_{i}^{T} β + e_{i}$ , 1 ≤ i ≤ n, n ≥ 1 be a linear regression model and suppose that the random errors e₁, e₂, ... are independent and α-stable. In this paper, we obtain sufficient conditions for the strong consistency of the least squares estimator β̃ of β under additional assumptions on the non-random sequence x₁, x₂,... of real vectors.

$M$ -estimators of structural parameters in pseudolinear models

Friedrich Liese, Igor Vajda (1999)

Applications of Mathematics

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Real valued $M$ -estimators ${\hat{θ}}_{n} : = min \sum_{1}^{n} ρ (Y_{i} - τ (θ))$ in a statistical model with observations $Y_{i} \sim F_{θ_{0}}$ are replaced by $ℝ^{p}$ -valued $M$ -estimators ${\hat{β}}_{n} : = min \sum_{1}^{n} ρ (Y_{i} - τ (u (z_{i}^{T} β)))$ in a new model with observations $Y_{i} \sim F_{u (z_{i}^{t} β_{0})}$ , where $z_{i} \in ℝ^{p}$ are regressors, $β_{0} \in ℝ^{p}$ is a structural parameter and $u : ℝ \to ℝ$ a structural function of the new model. Sufficient conditions for the consistency of ${\hat{β}}_{n}$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” ${\hat{θ}}_{n}$ . The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical...

Existence, Consistency and computer simulation for selected variants of minimum distance estimators

Václav Kůs, Domingo Morales, Jitka Hrabáková, Iva Frýdlová (2018)

Kybernetika

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The paper deals with sufficient conditions for the existence of general approximate minimum distance estimator (AMDE) of a probability density function $f_{0}$ on the real line. It shows that the AMDE always exists when the bounded $φ$ -divergence, Kolmogorov, Lévy, Cramér, or discrepancy distance is used. Consequently, $n^{- 1 / 2}$ consistency rate in any bounded $φ$ -divergence is established for Kolmogorov, Lévy, and discrepancy estimators under the condition that the degree of variations of the corresponding...

Rank theory approach to ridge, LASSO, preliminary test and Stein-type estimators: Comparative study

A. K. Md. Ehsanes Saleh, Radim Navrátil (2018)

Kybernetika

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In the development of efficient predictive models, the key is to identify suitable predictors for a given linear model. For the first time, this paper provides a comparative study of ridge regression, LASSO, preliminary test and Stein-type estimators based on the theory of rank statistics. Under the orthonormal design matrix of a given linear model, we find that the rank based ridge estimator outperforms the usual rank estimator, restricted R-estimator, rank-based LASSO, preliminary...

Instrumental weighted variables under heteroscedasticity. Part I – Consistency

Jan Ámos Víšek (2017)

Kybernetika

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The proof of consistency instrumental weighted variables, the robust version of the classical instrumental variables is given. It is proved that all solutions of the corresponding normal equations are contained, with high probability, in a ball, the radius of which can be selected - asymptotically - arbitrarily small. Then also $\sqrt{n}$ -consistency is proved. An extended numerical study (the Part II of the paper) offers a picture of behavior of the estimator for finite samples under various...

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

Diane M. Donovan, Mike Grannell, Emine Ş. Yazıcı (2020)

Commentationes Mathematicae Universitatis Carolinae

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We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2 t$ mutually orthogonal Latin squares of order $n^{t}$ .

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

Jiří Anděl (1998)

Applications of Mathematics

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Let $𝕖_{t} = {(e_{t 1}, \dots, e_{t p})}^{'}$ be a $p$ -dimensional nonnegative strict white noise with finite second moments. Let $h_{i j} (x)$ be nondecreasing functions from $[0, \infty)$ onto $[0, \infty)$ such that $h_{i j} (x) \leq x$ for $i, j = 1, \dots, p$ . Let $𝕌 = (u_{i j})$ be a $p \times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process $𝕏_{t} = {(X_{t 1}, \dots, X_{t p})}^{'}$ by $X_{t j} = u_{j 1} h_{1 j} (X_{t - 1, 1}) + \dots + u_{j p} h_{p j} (X_{t - 1, p}) + e_{t j}$ for $j = 1, \dots, p$ . A method for estimating $𝕌$ from a realization $𝕏_{1}, \dots, 𝕏_{n}$ is proposed. It is proved that the estimators are strongly consistent.

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

Samir Benaissa (2006)

Applicationes Mathematicae

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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

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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....

Orthogonal series regression estimation under long-range dependent errors

Waldemar Popiński (2001)

Applicationes Mathematicae

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This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_{i} = f (X_{i}) + η_{i}$ , i=1,...,n, where $X_{i} \in A \subset ℝ^{d}$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and $η_{i}$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{- 1} \sum_{i = 1}^{n} (f (X_{i}) - f ̂_{N} (X_{i})) ²$ for the estimator $f ̂_{N} (x) = \sum_{k = 1}^{N} c ̂_{k} e_{k} (x)$ , constructed using an orthonormal system...

Compact hypothesis and extremal set estimators

João Tiago Mexia, Pedro Corte Real (2003)

Discussiones Mathematicae Probability and Statistics

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In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when...