CONTENTS1. Intuitive background. Statement of the problem...................................................................... 52. General structure of global stochastic approximation processes............................................... 73. The fundamental theorem on convergence in distribution............................................................ 104. Absolute continuity of the limit distribution 4.1. Introductory remarks................................................................................................................
0. Introduction and summary. The analysis of data from the gravitational-wave detectors that are currently under construction in several countries will be a challenging problem. The reason is that gravitational-vawe signals are expected to be extremely weak and often very rare. Therefore it will be of great importance to implement optimal statistical methods to extract all possible information about the signals from the noisy data sets. Careful statistical analysis based on correct application of...
Though widely accepted, in nonparametric models admitting asymmetric distributions the sample median, if n=2k, may be a poor estimator of the population median. Shortcomings of estimators which are not equivariant are presented.
Sometimes, e.g. in the context of estimating VaR (Value at Risk), underestimating a quantile is less desirable than overestimating it, which suggests measuring the error of estimation by an asymmetric loss function. As a loss function when estimating a parameter θ by an estimator T we take the well known Linex function exp{α(T-θ)} - α(T-θ) - 1. To estimate the quantile of order q ∈ (0,1) of a normal distribution N(μ,σ), we construct an optimal estimator in the class of all estimators of the form...
If a symmetric distribution is ε-contaminated and the contaminants have finite first moments, the median may cease to be the most robust estimator of location.
Weak laws of large numbers (WLLN), strong laws of large numbers (SLLN), and central limit theorems (CLT) in statistical models differ from those in probability theory in that they should hold uniformly in the family of distributions specified by the model. If a limit law states that for every ε > 0 there exists N such that for all n > N the inequalities |ξₙ| < ε are satisfied and N = N(ε) is explicitly given then we call the law effective. It is trivial to obtain an effective statistical...
If θ ∈ Θ is an unknown real parameter of a given distribution, we are interested in constructing an exactly median-unbiased estimator θ̂ of θ, i.e. an estimator θ̂ such that a median Med(θ̂ ) of the estimator equals θ, uniformly over θ ∈ Θ. We shall consider the problem in the case of a fixed sample size n (nonasymptotic approach).
"A high quantile is a quantile of order q with q close to one." A precise constructive definition of high quantiles is given and optimal estimates are presented.
It turns out that for standard kernel estimators no inequality like that of Dvoretzky-Kiefer-Wolfowitz can be constructed, and as a result it is impossible to answer the question of how many observations are needed to guarantee a prescribed level of accuracy of the estimator. A remedy is to adapt the bandwidth to the sample at hand.
In the paper, a numerical analysis of robustness of Student f-test, Wilcoxon-Mann-Whitney test and a sign test to some dependencies of observations is presented. A Monte Carlo approach has been applied.
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A population of N elements contains an unknown number M of marked units. Problems of estimating the fraction θ = M/N are discussed. The well known standard solution isˆθ = K/n which is the uniformly minimum variance unbiased estimator, maximum likelihood estimator, estimator obtained by the method of moments, and in consequence it shares all advantages of such estimators. In the paper some versions of the estimator are considered which are more adequate in real situations. If we know in advance...
Mierząc pewną wielkość μ (długość, ciężar, temperaturę...) otrzymujemywynik X, zwykle różniący się od μ o pewną wielkość losową (błąd losowy) ε. Rozkład F prawdopodobieństwa błędu losowego ε czasami jest znany, a czasami wiemy o nim tylko to, że jest jakimś rozkładem z ustalonej rodziny rozkładów F (np. rozkładem normalnym o średniej zero i nieznanym odchyleniu standardowym σ, albo jakimś rozkładem o cią-głej dystrybuancie). Jeżeli rozkład F ma duży rozrzut, dokładność pomiaru może być niezadowalająca....
The most powerful robust test and the robust most powerful test in a simple Gaussian model are constructed and discussed.
Przedziały ufności zostały wymyślone przez Jerzego Spławę–Neymana w 1934 [15]. Praktyczne zastosowanie teorii Neymana do przedziałowej estymacji prawdopodobieństwa sukcesu w schemacie Bernoulliego (parametru rozkładu dwumianowego) stwarzało jednak pewne trudności zarówno jeśli chodzi o ich konstrukcję (rozkład dyskretny!), jak i o ich numeryczne obliczanie. Jako panaceum wymyślono asymptotyczne przedziały ufności oparte na przybliżaniu rozkładu dwumianowego rozkładem normalnym: konstrukcja i rachunki...
An upper bound for the Kolmogorov distance between the posterior distributions in terms of that between the prior distributions is given. For some likelihood functions the inequality is sharp. Applications to assessing Bayes robustness are presented.
A homogeneous Poisson process (N(t),t ≥ 0) with the intensity function m(t)=θ is observed on the interval [0,T]. The problem consists in estimating θ with balancing the LINEX loss due to an error of estimation and the cost of sampling which depends linearly on T. The optimal T is given when the prior distribution of θ is not uniquely specified.
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