White noise driven SPDEs with reflection: Existence,uniqueness and large deviation principles

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.     

Lower deviation probabilities for supercritical Galton-Watson processes

There is a well-known sequence of constants cn describing the growth of supercritical Galton-Watson processes Zn. By lower deviation probabilities we refer to P(Zn=kn) with kn=o(cn) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Zn+1/Zn. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramér method for proving large deviations of sums of independent variables to our needs.     

Long strange segments,ruin probabilities and the effect of memory on moving average processes

We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.     

On the covariance of the asymptotic empirical copula process

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.     

A general correlation inequality and the Almost Sure Local Limit Theorem for random sequences in the domain of attraction of a stable law

In the present paper we obtain a new correlation inequality and use it for the purpose of extending the theory of the Almost Sure Local Limit Theorem to the case of lattice random sequences in the domain of attraction of a stable law. In particular, we prove ASLLT in the case of the normal domain of attraction of α$\alpha$-stable law, α∈(1,2)$\alpha \in (1,2)$.     

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . On the basis of data consisting of n observations on all p+q characteristics and an additional N−n observations on the last q characteristics, where all observations are mutually independent, denote by the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in . For the problem of shrinking to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.     

On Stein’s lemma, dependent covariates and functional monotonicity in multi-dimensional modeling

Tracking the correct directions of monotonicity in multi-dimensional modeling plays an important role in interpreting functional associations. In the presence of multiple predictors, we provide empirical evidence that the observed monotone directions via parametric, nonparametric or semiparametric fit of commonly used multi-dimensional models may entirely violate the actual directions of monotonicity. This breakdown is caused primarily by the dependence structure of covariates, with negligible influence from the bias of function estimation. To examine the linkage between the dependent covariates and monotone directions, we first generalize Stein’s Lemma for random variables which are mutually independent Gaussian to two important cases: dependent Gaussian, and independent non-Gaussian. We show that in both two cases, there is an explicit one-to-one correspondence between the monotone directions of a multi-dimensional function and the signs of a deterministic surrogate vector. Moreover, we demonstrate that the second case can be extended to accommodate a class of dependent covariates. This generalization further enables us to develop a de-correlation transform for arbitrarily dependent covariates. The transformed covariates preserve modeling interpretability with little loss in modeling efficiency. The simplicity and effectiveness of the proposed method are illustrated via simulation studies and real data application.     

Modifying estimators of ordered positive parameters under the Stein loss

This paper treats the problem of estimating positive parameters restricted to a polyhedral convex cone which includes typical order restrictions, such as simple order, tree order and umbrella order restrictions. In this paper, two methods are used to show the improvement of order-preserving estimators over crude non-order-preserving estimators without any assumption on underlying distributions. One is to use Fenchel’s duality theorem, and then the superiority of the isotonic regression estimator is established under the general restriction to polyhedral convex cones. The use of the Abel identity is the other method, and we can derive a class of improved estimators which includes order-statistics-based estimators in the typical order restrictions. When the underlying distributions are scale families, the unbiased estimators and their order-restricted estimators are shown to be minimax. The minimaxity of the restrictedly generalized Bayes estimator against the prior over the restricted space is also demonstrated in the two dimensional case. Finally, some examples and multivariate extensions are given.     

Asymptotic theory for maximum deviations of sample covariance matrix estimates

We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.     

Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP₂ dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.     

An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility

This paper proposes a general approach to obtain asymptotic lower bounds for the estimation of random functionals. The main result is an abstract convolution theorem in a non parametric setting, based on an associated LAMN property. This result is then applied to the estimation of the integrated volatility, or related quantities, of a diffusion process, when the diffusion coefficient depends on an independent Brownian motion.     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

On the ruin probability in a dependent discrete time risk model with insurance and financial risks

This paper considers the discrete-time risk model with insurance risk and financial risk in some dependence structures. Under assumptions that the insurance risks are heavy tailed (belong to the intersection of the long-tailed class and the dominatedly varying-tailed class) and the financial risks satisfy some moment conditions, the asymptotic and uniformly asymptotic relations for the finite-time and ultimate ruin probabilities are derived.     

A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions

The purpose of this paper is to investigate the effect of nonnormality upon the nonnull distributions of some MANOVA test statistics under normality. It is shown that whatever the underlying distributions, the difference of the local powers up to order N^-1 (N is the total number of observations) after either Bartlett's type adjustment or Cornish-Fisher's type adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, second ed., 1984 and third ed., 2003, Wiley, New York] under normality. The performance of higher-order results in finite samples is examined using simulation studies.     

Multivariate copulas, quasi-copulas and lattices

We investigate some properties of the partially ordered sets of multivariate copulas and quasi-copulas. Whereas the set of bivariate quasi-copulas is a complete lattice, which is order-isomorphic to the Dedekind-MacNeille completion of the set of bivariate copulas, we show that this is not the case in higher dimensions.     

Cramér large deviation expansions for martingales under Bernstein’s condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramér to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramér result and therefore are optimal. Our result implies a moderate deviation principle for martingales.     

The asymptotic behavior of linear placement statistics

Orban and Wolfe (1982) and Kim (1999) provided the limiting distribution for linear placement statistics under null hypotheses only when one of the sample sizes goes to infinity. In this paper we prove the asymptotic normality and the weak convergence of the linear placement statistics of Orban and Wolfe (1982) and Kim (1999) when the sample sizes of each group go to infinity simultaneously.