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

Václav Kůs; Domingo Morales; Jitka Hrabáková; Iva Frýdlová

Displaying similar documents to “Existence, Consistency and computer simulation for selected variants of minimum distance estimators”

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...

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....

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...

$M$ -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...

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...

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...

Spatially adaptive density estimation by localised Haar projections

Florian Gach, Richard Nickl, Vladimir Spokoiny (2013)

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

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Given a random sample from some unknown density $f_{0} : ℝ \to [0, \infty)$ we devise Haar wavelet estimators for $f_{0}$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny ( (1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_{0}$ , simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in...

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

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This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_{i}$ and outputs $y_{i}$ , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_{z}$ on the base of given data $z : = ((x ₁, y ₁), . . ., (x_{m}, y_{m}))$ that approximates well the regression function...

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

Gusztáv Morvai, Benjamin Weiss (2020)

Kybernetika

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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...

Orthogonal series estimation of band-limited regression functions

Waldemar Popiński (2014)

Applicationes Mathematicae

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The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_{k}$ , k = 0,±1,..., for the observation model $y_{j} = f (u_{j}) + η_{j}$ , j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_{j}$ are independent random variables uniformly distributed in the observation interval [-T,T], $η_{j}$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions...

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

Lothar Heinrich (2016)

Applications of Mathematics

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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...

The Bayes choice of an experiment in estimating a success probability

Alicja Jokiel-Rokita, Ryszard Magiera (2002)

Applicationes Mathematicae

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A Bayesian method of estimation of a success probability p is considered in the case when two experiments are available: individual Bernoulli (p) trials-the p-experiment-or products of r individual Bernoulli (p) trials-the $p^{r}$ -experiment. This problem has its roots in reliability, where one can test either single components or a system of r identical components. One of the problems considered is to find the degree r̃ of the $p^{r ̃}$ -experiment and the size m̃ of the p-experiment such that the...