A characterization of limiting distributions of estimators in an autoregressive process

Summary Let the random variablesX ₁,X ₂, ...,X _n be generated by the first-order autoregressive modelX _i =θX _i−1 +e _i wheree _i ,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ², and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators of θ as well as a local asymptotic minimax bound of the risks of estimators.     

Stability of testing hypotheses

The stability of testing hypotheses is discussed. Differing from the usual tests measured by Neyman-Pearson lemma, the regret and correction of the tests are considered. After the decision is made based on the observationsX ₁,X ₂, ⋅⋅⋅,X _n, one more piece of datumX _n+1 is picked and the test is done again in the same way but based onX ₁,X ₂, ⋅⋅⋅,X _n,X _n+l There are three situations: (i) The previous decision is right but the new decision is wrong; (ii) the previous decision is wrong but the new decision is right; (iii) both of them are right or both of them are wrong. Of course, it is desired that the probability of the occurrence of (i) is as small as possible and the probability of the occurrence of (ii) is as large as possible. Since the sample size is sometimes not chosen very precisely after the type I error and the type II error are determined in practice, it seems more urgent to consider the above problem. Some optimal plans are also given. Project supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation.     

Strong convergence rates for the estimation of a covariance operator for associated samples

Let X _n , n ≥ 1, be a strictly stationary associated sequence of random variables, with common continuous distribution function F. Using histogram type estimators we consider the estimation of the two-dimensional distribution function of (X ₁,X _k+1) as well as the estimation of the covariance function of the limit empirical process induced by the sequence X _n , n ≥ 1. Assuming a convenient decrease rate of the covariances Cov(X ₁,X _n+1), n ≥ 1, we derive uniform strong convergence rates for these estimators. The condition on the covariance structure of the variables is satisfied either if Cov(X ₁,X _n+1) decreases polynomially or if it decreases geometrically, but as we could expect, under the latter condition we are able to establish faster convergence rates. For the two-dimensional distribution function the rate of convergence derived under a geometrical decrease of the covariances is close to the optimal rate for independent samples.      

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums , whereX _n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX _n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.     

Estimation in a model with infinite-dimensional nuisance parameter

Let X₁ be a random variable with density function f(t), Ψ(t) be an increasing absolutely continuous function, Φ(t) be the inverse function to Ψ(t), and X₂ be the random variable X₂ = Φ(X₁). We consider the maximum likelihood estimator for the density ψ of the function Ψ in the case when we observe two independent samples from the distributions of X₁ and X₂. Under appropriate conditions on the involved distributions, we prove the consistency of the maximum likelihood estimator. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 160–165. An erratum to this article is available at .     

Stable Laws and Products of Positive Random Matrices

Let S be the multiplicative semigroup of q×q matrices with positive entries such that every row and every column contains a strictly positive element. Denote by (X _n ) _n≥1 a sequence of independent identically distributed random variables in S and by X ⁽ⁿ⁾=X _n ⋅⋅⋅ X ₁,  n≥1, the associated left random walk on S. We assume that (X _n ) _n≥1 satisfies the contraction property

Estimation of the variance for strongly mixing sequences

Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u,     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

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     

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter ‘residual’ probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.      

Tail asymptotics for the sum of two heavy-tailed dependent risks

Let X ₁ , X ₂ denote positive heavy-tailed random variables with continuous marginal distribution functions F ₁ and F ₂, respectively. The asymptotic behavior of the tail of X ₁ +X ₂ is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F ₁ , F ₂ and the underlying dependence structure of X ₁ and X ₂, we survey explicit asymptotic results available in the literature and add several new cases.Supported by the Austrian Science Fund Project P-18392.     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

Small deviation probabilities for sums of independent positive random variables

In this note, we give estimates of small deviation probabilities of the sum ∑_j≥1 λ_j X_j, where {λ_j} are nonnegative numbers and {X_j} are i.i.d. positive random variables that satisfy mild assumptions at zero and infinity. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 151–167.     

Elementary Symmetric Polynomials in Random Variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ _k of arbitrary degree k in random variables X _i (i=1,2,…,m) defined on special subsets of commutative rings ℛ _m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ _k (X ₁,…,X _m ) tend to a limit when m→∞ if X ₁,…,X _m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ _m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.      

Polynomial Regression with Censored Data based on Preliminary Nonparametric Estimation

Consider the polynomial regression model , where σ²(X)=Var(Y|X) is unknown, and ε is independent of X and has zero mean. Suppose that Y is subject to random right censoring. A new estimation procedure for the parameters β₀,...,β _p is proposed, which extends the classical least squares procedure to censored data. The proposed method is inspired by the method of Buckley and James (1979, Biometrika, 66, 429–436), but is, unlike the latter method, a noniterative procedure due to nonparametric preliminary estimation of the conditional regression function. The asymptotic normality of the estimators is established. Simulations are carried out for both methods and they show that the proposed estimators have usually smaller variance and smaller mean squared error than the Buckley–James estimators. The two estimation procedures are also applied to a medical and an astronomical data set.     

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y ₁, ..., Y _n are modeled in dependence of 1-periodic, second order stationary random functions X ₁, ...,X _n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L ²-risk. This means that we provide adaptive estimators of both β and its derivatives.     

Efficient algorithms for heavy-tail analysis under interval uncertainty

Most applications of statistics to science and engineering are based on the assumption that the corresponding random variables are normally distributed, i.e., distributed according to Gaussian law in which the probability density function ρ(x) exponentially decreases with x: ρ(x)∼exp (−k⋅x ²). Normal distributions indeed frequently occur in practice. However, there are also many practical situations, including situations from mathematical finance, in which we encounter heavy-tailed distributions, i.e., distributions in which ρ(x) decreases as ρ(x)∼x ^−α . To properly take this uncertainty into account when making decisions, it is necessary to estimate the parameters of such distributions based on the sample data x ₁,…,x _n —and thus, to predict the size and the probabilities of large deviations. The most well-known statistical estimates for such distributions are the Hill estimator H for α and the Weismann estimator W for the corresponding quantiles.     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field X _i , i ∈ Z ^d from the density is contaminated by measurement errors. In particular, the observations Y _i , i ∈ Z ^d are such that Y _i = X _i + ε _i , where the errors ε _i are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Almost sure central limit theorems for random functions

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.