Limit laws for the cumulative number of ties for the maximum in a random sequence

Let {_Xn,n?1}$\{X_n,n?1\}$ be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation _Xn$X_n$ is a tie for the maximum if _Xn=max{_X1,…,_Xn-1}$X_n=\max \{X_1,\dots ,X_{n-1}\}$. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n$n$ observations.     

On the maximum of periodic integer-valued sequences with exponential type tails via max-semistable laws

In this work we study the limiting distribution of the maximum term of periodic integer-valued sequences with marginal distribution belonging to a particular class where the tail decays exponentially. This class does not belong to the domain of attraction of any max-stable distribution. Nevertheless, we prove that the limiting distribution is max-semistable when we consider the maximum of the first k_n observations, for a suitable sequence {_kn}$\{k_n\}$ increasing to infinity. We obtain an expression for calculating the extremal index of sequences satisfying certain local conditions similar to conditions D^(m)(_un)$D^{(m)}(u_n)$, m∈N$m\in \mathbb{N}$, defined by Chernick et al. (1991). We apply the results to a class of max-autoregressive sequences and a class of moving average models. The results generalize the ones obtained for the stationary case.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

On the extremal behavior of a nonstationary normal random field

Let X={_Xn}n?1$\mathbf{X}=\{X_{\mathbf{n}}{\}}_{\mathbf{n}?\mathbf{1}}$ be a nonstationary random field satisfying a long range weak dependence for each coordinate at a time and a local dependence condition that avoids clustering of exceedances of high values. For these random fields, the probability of no exceedances of high values can be approximated by exp(−τ)$\exp (-\tau )$, where τ$\tau$ is the limiting mean number of exceedances. We present a class of nonstationary normal random fields for which this result can be applied.     

Laplace record data

Basic properties of upper record values _XT(1),_XT(2),…,_XT(n)$X_{T(1)},X_{T(2)},\dots ,X_{T(n)}$ from a symmetric two-parameter Laplace distribution are established. In particular, unimodality of the density function and the exact expression of the mode are derived. Moreover, we obtain approximations of the first and second moment and the variance of _XT(k)$X_{T(k)}$ which provide close approximations even for moderate k. Additionally, limit laws and simulation of Laplace records are considered. Finally, we discuss maximum likelihood estimation in a location-scale family of Laplace distributions. We obtain nice representations of the estimators provided that the location parameter is unknown and present interesting properties of the established estimators. Some illustrative examples complete the presentation.     

Regression models for functional data by reproducing kernel Hilbert spaces methods

Non-parametric regression models are developed when the predictor is a function-valued random variable X={_Xt}t∈T$X=\{X_t{\}}_{t\in T}$. Based on a representation of the regression function f(X)$f(X)$ in a reproducing kernel Hilbert space such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variable are performed on stock-exchange and medical data. The results of different regression models are compared.     

Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n   consistent plug-in estimators of conditional expectations E(h(_Xn+1)|_X1,…,_Xn)$E(h(X_{n+1})|X_1,\dots ,X_n)$. More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F. For fixed h it suffices that h   is locally of bounded variation and locally Lipschitz in _L2(F)$L_2(F)$, and that the convolution of h and F   is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P(_Xn+1?·|_X1,…,_Xn)$P(X_{n+1}?·|X_1,\dots ,X_n)$ holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F-Donsker, assuming some smoothness of F. The proofs use empirical process arguments.     

Likelihood ratio ordering of order statistics,mixtures and systems

Let X=(_X1,_X2,…,_Xn)$\mathit{X}=(X_1,X_2,\dots ,X_n)$ be an exchangeable random vector, and denote _X1:i=min{_X1,_X2,…,_Xi}$X_{1:i}=\min \{X_1,X_2,\dots ,X_i\}$ and _Xi:i=max{_X1,_X2,…,_Xi}$X_{i:i}=\max \{X_1,X_2,\dots ,X_i\}$, 1?i?n$1?i?n$. These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes _Xi$X_i$. In this paper we obtain conditions under which _X1:i$X_{1:i}$ (or _Xi:i$X_{i:i}$) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

Combinatorially composing Chebyshev polynomials

We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m,n?0$m,n?0$, _Tm(_Tn(x))=_Tmn(x)$T_m(T_n(x))=T_{\mathit{mn}}(x)$ and _Um−1(_Tn(x))_Un−1(x)=_Umn−1(x).$U_{m-1}(T_n(x))U_{n-1}(x)=U_{\mathit{mn}-1}(x).$     

Adaptive estimation of the conditional cumulative distribution function from current status data

Consider a positive random variable of interest Y depending on a covariate X, and a random observation time T independent of Y given X. Assume that the only knowledge available about Y is its current status at time T  : δ=_I{Y≤T}$\delta ={\mathbb{I}}_{\{Y\le T\}}$ with I$\mathbb{I}$ the indicator function. This paper presents a procedure to estimate the conditional cumulative distribution function F of Y given X   from an independent identically distributed sample of (X,T,δ)$(X,T,\delta )$.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Nonparametric estimation of conditional expectation

Denote the integer lattice points in the N  -dimensional Euclidean space by Z^N${\mathbb{Z}}^N$ and assume that (_Xi,_Yi)$(X_{\mathbf{i}},Y_{\mathbf{i}})$, i∈Z^N$\mathbf{i}\in {\mathbb{Z}}^N$ is a mixing random field. Estimators of the conditional expectation r(x)=E[_Yi|_Xi=x]$r(\mathbf{x})=\mathrm{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=\mathbf{x}]$ by nearest neighbor methods are established and investigated. The main analytical result of this study is that, under general mixing assumptions, the estimators considered are asymptotically normal. Many difficulties arise since points in higher dimensional space N?2$N?2$ cannot be linearly ordered. Our result applies to many situations where parametric methods cannot be adopted with confidence.