Limit laws for power sums and norms of i.i.d. samples

We study the limit behavior of power sums and norms of i.i.d. positive samples from the max domain of attraction of an extreme value distribution. To this end, we combine limit theorems for sums and for maxima and use a link between extreme value theory and the Lévy measures of certain infinitely divisible laws, which are limit distributions of power sums. In connection with the von Mises representation of the Gumbel max domain of attraction, this new approach allows us to extend the limit results for power sums found in Ben Arous et al. (Probab Theory Relat Fields 132:579–612, 2005) and Bogachev (J Theor Probab 19:849–873, 2006). Furthermore, our findings shed a new light on the results of Schlather (Ann Probab 29:862–881, 2001) and treat the Gumbel case which is missing there.     

Nonparametric estimation for i.i.d. Gaussian continuous time moving average models

Statistical Inference for Stochastic Processes - We consider a Gaussian continuous time moving average model $$X(t)=\int _0^t a(t-s)dW(s)$$ where W is a standard Brownian motion and a(.) a...     

Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments

The aim of this paper is to derive consequences of a result of G?tze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d -valued random vectors ξ _j having finite moments E IIξ _j II^γ, γ>2. Bibliography: 13 titles.     

Classification methods for random utility models with i.i.d. disturbances under the most probable alternative rule

In this paper random utility maximization based on maximization of correct classification of the choice decisions over a given data set is considered. It is shown that if the disturbance vector in the random utility model is independent and identically distributed, then preference determination based on the most probable alternative reduces to deterministic utility maximization. As a consequence of the above equivalence, the form of the error distribution (normal, Weibull, uniform etc.) plays no role in the determination of the preferred alternative. Parameter estimation under the most probable alternative rule is carried out using two methods. The first is based on the solution of an appropriately defined system of linear inequalities and the second one is based on the function optimization of a newly proposed function, whose optimum is achieved when the number of correctly classified individuals is maximized. The ability to use these algorithms in the framework of pattern recognition and machine learning is pointed out. Simulations and a real case study involving intercity travel behavior are employed to assess the proposed methods.     

Limit theorems for the minimum interpoint distance between any pair of i.i.d. random points in R d

The limit theorem for the minimum interpoint distance between any pair of i.i.d. random points in R ^d with common density fL ² was studied by a method which makes use of the convergence of point processes. Some one-dimensional examples with fL ² (including the cases Beta and Gamma distributions) were also considered.     

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

Estimates for the rate of strong Gaussian approximation for sums of i.i.d. multidimensional random vectors

The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d-valued random variables ξ_j that have finite moments of the form EH (‖ξ_j‖), where H (x) is a monotone function growing not slower than x² and not faster than e^cx. We obtain some generalization and improvements of results of U. Einmahl (1989). Bibliography: 28 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 141–158.     

A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

We consider a random walk on $\mathbb{Z }^d,\ d\ge 2$ , in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from $x\in \mathbb{Z }^d$ to nearest neighbor $x+e$ is the same as to nearest neighbor $x-e$ . Assuming that the environment is genuinely $d$ -dimensional and balanced we show a quenched invariance principle: for $P$ almost every environment, the diffusively rescaled random walk converges to a Brownian motion with deterministic non-degenerate diffusion matrix. Within the i.i.d. setting, our result extend both Lawler’s uniformly elliptic result (Comm Math Phys, 87(1), pp 81–87, 1982/1983) and Guo and Zeitouni’s elliptic result (to appear in PTRF, 2010) to the general (non elliptic) case. Our proof is based on analytic methods and percolation arguments.     

i.i.d情况下非参数回归模型的误差密度估计的收敛性质

本文研究了i．i．d情况下非参数回归的误差密度估计的一致收敛和均方收敛，给出了一定条件下误差密度的估计量f^n(x)的一致收敛速度和均方收敛速度。     

A two-sided estimate in the hsu-robbins-erdös law of large numbers for i.i.d. random variables sequence

The Paley-type inequalities for the complete convergence and lower and upper bounds in the Baum-Katz law of large numbers for i.i.d. random variables sequence are obtained. The results obtained generalize the result of Pruss [8].     

On the Number of i.i.d. Samples Required to Observe All of the Balls in an Urn

Suppose an urn contains m distinct balls, numbered 1,...,m, and let τ denote the number of i.i.d. samples required to observe all of the balls in the urn. We generalize the partial fraction expansion type arguments used by Pólya (Z Angew Math Mech 10:96–97, 1930) for approximating \mathbbE(t)\mathbb{E}(\tau) in the case of fixed sample sizes to obtain an approximation of \mathbbE(t)\mathbb{E}(\tau) when the sample sizes are i.i.d. random variables. The approximation agrees with that of Sellke (Ann Appl Probab 5(1):294–309, 1995), who made use of Wald’s equation and a Markov chain coupling argument. We also derive a new approximation of \mathbbV(t)\mathbb{V}(\tau), provide an (improved) bound on the error in these approximations, derive a recurrence for \mathbbE(t)\mathbb{E}(\tau), give a new large deviation type result for tail probabilities, and look at some special cases.     

I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Let F(s, t) = P(X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that

Full-size image

, where {ζ_i(s, t)} is i.i.d. mean zero process and R_n(s, t) is of the order O((n⁻¹log n)^3/4) a.s. uniformly on compact sets. Weak convergence of the process {n⁻¹ Σ_{i = 1}ⁿ ζ_i(s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator is also modified so that the modified estimator is closer to the true survival function than in supnorm.     

Strong approximations for partial sums of i.i.d.B-valued r.v.'s in the domain of attraction of a Gaussian law

Summary We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985).     

Optimal designs for estimating the control values in multi-univariate regression models

This paper considers a linear regression model with a one-dimensional control variable x and an m-dimensional response vector . The components of are correlated with a known covariance matrix. Based on the assumed regression model, it is of interest to obtain a suitable estimation of the corresponding control value for a given target vector on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration on the performance of an estimator for the control value includes the difference of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of control point, say x₀, is defined to be the one which minimizes the weighted sum of squares of those standardized differences within the range of x. The objective of this study is to find a locally optimal design for estimating x₀, which minimizes the mean squared error of the estimator of x₀. It is shown that the optimality criterion is equivalent to a c-criterion under certain conditions and explicit solutions with dual response under linear and quadratic polynomial regressions are obtained.     

On the existence of universal numberings for finite families of d.c.e. sets

We investigate properties of universal numberings of finite families of d.c.e. sets. We show different cases of finite families of d.c.e. sets for which there is a universal numbering and for which there is not.     

An effective strategy for initializing the EM algorithm in finite mixture models

Finite mixture models represent one of the most popular tools for modeling heterogeneous data. The traditional approach for parameter estimation is based on maximizing the likelihood function. Direct optimization is often troublesome due to the complex likelihood structure. The expectation–maximization algorithm proves to be an effective remedy that alleviates this issue. The solution obtained by this procedure is entirely driven by the choice of starting parameter values. This highlights the importance of an effective initialization strategy. Despite efforts undertaken in this area, there is no uniform winner found and practitioners tend to ignore the issue, often finding misleading or erroneous results. In this paper, we propose a simple yet effective tool for initializing the expectation–maximization algorithm in the mixture modeling setting. The idea is based on model averaging and proves to be efficient in detecting correct solutions even in those cases when competitors perform poorly. The utility of the proposed methodology is shown through comprehensive simulation study and applied to a well-known classification dataset with good results.     

Modified Kent's statistics for testing goodness of fit for the Fisher distribution in small concentrated samples

The limiting null distribution of Kent's (1982) statistic, to test whether a sample comes from the Fisher distribution is derived when κ, the concentration parameter, goes to ∞. A modification is suggested, the limiting null distribution of which is ξ₂² when either κ or n, the sample size, goes to ∞. Tests of Fisherness based on the eigenvalues of the sample cross product matrix are also considered. Numerical examples are presented.