Sufficient Conditions For Asymptotic Normality Research Articles

Quasi‐maximum likelihood estimation of conditional autoregressive Wishart models

In this article, we consider a quasi‐maximum likelihood (QML) estimation of conditional autoregressive Wishart models, which generalize the conditional autoregressive Wishart models by not restricting the conditional distribution of covariances to follow the Wishart distribution. Strong consistency is established under the existence of the expectation of the log of the determinant. Sufficient conditions for asymptotic normality of the QML estimator are derived. Monte Carlo experiments show an inefficiency caused by using non‐Wishart distributions, which are negligible for the sample size T = 500. We use the daily covariance matrix of the returns of the Nikkei 225 index and its futures for the QML estimation of the conditional autoregressive Wishart model. The results indicate its appropriateness for the QML estimation.

Asymptotic Properties of One-Step Weighted $M$-Estimators with Applications to Regression

We study the asymptotic behavior of one-step weighted $M$-estimators based on independent not necessarily identically distributed observations, which approximate consistent weighted $M$-estimators. We find sufficient conditions for asymptotic normality of these estimators. As an application, we consider some known regression models where the one-step estimation under consideration allows us to construct explicit asymptotically optimal estimators having the same accuracy as the least-squares or quasi-likelihood estimators.

A high quantile estimator based on the log-generalized Weibull tail limit

The estimation of high quantiles for very low probabilities of exceedance pn much smaller than 1/n (with n the sample size) remains a major challenge. For this purpose, the log-Generalized Weibull (log-GW) tail limit was recently proposed as regularity condition as an alternative to the Generalized Pareto (GP) tail limit, in order to avoid potentially severe bias in applications of the latter. Continuing in this direction, a new estimator for the log-GW tail index and a related quantile estimator are introduced. Both are constructed using the Hill estimator as building block. Sufficient conditions for asymptotic normality are established. These results, together with the results of simulations and an application, indicate that the new estimator fulfils the potential of the log-GW tail limit as a widely applicable model for high quantile estimation, showing a substantial reduction in bias as well as improved precision when compared to an estimator based on the GP tail limit.

Sufficient m-out-of-n (m/n) bootstrap

ABSTRACTTraditional resampling methods for estimating sampling distributions sometimes fail, and alternative approaches are then needed. For example, if the classical central limit theorem does not hold and the naïve bootstrap fails, the m/n bootstrap, based on smaller-sized resamples, may be used as an alternative. An alternative to the naïve bootstrap, the sufficient bootstrap, which uses only the distinct observations in a bootstrap sample, is another recently proposed bootstrap approach that has been suggested to reduce the computational burden associated with bootstrapping. It works as long as naïve bootstrap does. However, if the naïve bootstrap fails, so will the sufficient bootstrap. In this paper, we propose combining the sufficient bootstrap with the m/n bootstrap in order to both regain consistent estimation of sampling distributions and to reduce the computational burden of the bootstrap. We obtain necessary and sufficient conditions for asymptotic normality of the proposed method, and propose new values for the resample size m. We compare the proposed method with the naïve bootstrap, the sufficient bootstrap, and the m/n bootstrap by simulation.

Universal limit theorems in graph coloring problems with connections to extremal combinatorics

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (nonnormal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erdős and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

Asymptotic Properties of Generalized Multivariate Rank Statistics

This article discusses generalization of the well-known multivariate rank statistics under right-censored data case. Empirical process representation used to get the generalization. The marginal distribution functions are estimated by Kaplan–Meier estimators. Sufficient conditions for asymptotic normality of the generalized multivariate rank statistics under independently right censored data are specified. Several auxiliary results on sup-norm convergence of Kaplan–Meier estimators in randomly exhausting regions are given too.

Asymptotic normality of the number of values of m-dependent random variables which occur a given number of times

We consider a stationary sequence x 1 , x 2 , . . . , x t , . . . of random variables each of which takes values in the set . Let μ r = μ r ( n, N ) be the number of values in the set which occur r times in the first n elements of the sequence. In this paper, we obtain asymptotic formulas for the mean value E μ r ( n, N ) and the variance D μ r ( n, N ) and give sufficient conditions for asymptotic normality of the random variables μ r ( n, N ) as n, N → ∞ under the condition that the sequence consists of m -dependent random variables and the parameters of the distribution of the sequence vary in the so-called central domain.

Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium

We consider the asymptotic normality of a continuous procedure of stochastic approximation in the case where the regression function contains a singularly perturbed term depending on the external medium described by a uniformly ergodic Markov process. Within the framework of the scheme of diffusion approximation, we formulate sufficient conditions for asymptotic normality in terms of the existence of a Lyapunov function for the corresponding averaged equation.

A sufficient condition for asymptotic normality of the normalized quadratic form [formula omitted

Many sufficient conditions of asymptotic normality of the normalized quadratic form Ψ n ( f , g ) have been proposed since 1958. The less restrictive was given in the paper of L. Giraitis and D. Surgailis (1990). Using a linear operator approach, it is possible to produce an even less restrictive sufficient condition on the couple of functions ( f , g ) . To cite this article: V. Solev, L. Gerville-Reache, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Maximum Likelihood Estimation of Asymmetric Laplace Parameters

Maximum likelihood estimators (MLE's) are presented for the parameters of a univariate asymmetric Laplace distribution for all possible situations related to known or unknown parameters. These estimators admit explicit form in all but two cases. In these exceptions effective algorithms for computing the estimators are provided. Asymptotic distributions of the estimators are given. The asymptotic normality and consistency of the MLE's for the scale and location parameters are derived directly via representations of the relevant random variables rather than from general sufficient conditions for asymptotic normality of the MLE's.

Asymptotic normality of convergent estimates of conditional quantiles

We state sufficient conditions for asymptotic normality of convergent estimates of conditional quantiles, irrespective of data dependence and consider the particular case of α-mixing stationary processes under optimal condition of convergence. We apply this result to confidence intervals building for time series predictors based on nonparametric estimates of the conditional median.

A Normal Law for Matchings

For a simple graph G, let be the size of a matching drawn uniformly at random from the set of all matchings of G. Motivated by work of C. Godsil [11], we give, for a sequence and , several necessary and sufficient conditions for asymptotic normality of the distribution of , for instance (where E and is the size of a largest matching in ). In particular this gives asymptotic normality for any sequence of regular graphs (of positive degree) or graphs with perfect matchings. When tends to a finite limit, a sufficient (and probably necessary) condition is given for to be asymptotically Poisson. The material presented here suggests numerous related questions, some of which are discussed in the last section of the paper.

Propriétés asymptotiques d'estimateurs convergents des quantiles conditionnels

We state sufficient conditions for asymptotic normality of convergent estimates of the conditional quantiles, irrespective of data dependence, and give an application to α-mixing stationary processes, under optimal conditions. As an application, we use asymptotic normality to construct confidence bands for predictors based on nonparametric estimates of the conditional median.

On asymptotic normality of the hill estimator

For iid observations from a common distribution Fwith regularly varying tail , a popular estimator of α is the Hill estimator. Regular variation of the distribution tail is equivalent to weak consistency of the Hill estimator in a manner made precise in Mason (1982) but necessary and sufficient conditions for asymptotic normality of this estimator are still somewhat shrouded in confusion. This is in part due to the different possibilities for a centering in the asymptotic normality statement. We clarify the roles played by smoothness conditions such as Von Mises conditionsfor the asymptotic normality and give a minimal condition under which a non constant centering can be used

Uniform CLT, WLLN, LIL and bootstrapping in a data analytic approach to trimmed L-statistics

Suppose the score function J and its approximation J n satisfy very mild conditions, and let d K = J d F −1 define K. Then a necessary and sufficient condition for asymptotic normality of the normalized trimmed L-statistic T n , which places no other restriction on the numbers k n and k′ n of trimmed observations, is given. In the untrimmed case the condition reduces to K having finite variance or K being in D(Normal), depending on whether one uses √ n norming or general norming. Any distribution in the domain of attraction of any stable is brought into the domain of attraction of the normal by trimming slowly growing numbers k n and k′ n . Consistent variance estimation is established in the same generality. Random trimming is also allowed. Uniformity in F and k n and k′ n is developed for K beneath an envelope K 0, both for normality and consistency of the variance estimate. Bootstrapped versions of these theorems are developed, including a CLT for data analysts with conditions formulated solely in terms of the data. Also, the WLLN holds uniformly in K and ( k n , k′ n ) if and only if the envelope K 0 satisfies the known necessary and sufficient condition for X ̄ n under a fixed qf K 0. A strong law of large numbers for the variance estimator is required, as it is intimately tied to bootstrapping. A short extra step would give the bounded LIL (that is, a.s. consistency of the estimator at the best possible rate). The strong law for variances and this LIL could be treated in the same generality as above, and uniform versions developed. However, all this is only in the technical report version of this paper. The power of the related family of inequalities for qfs on which these proofs rest is fully displayed herein, though the inequalities themselves are in a separate paper.

REML estimation: asymptotic behavior and related topics

The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require normality or boundedness of the rank p of design matrix of fixed effects. Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an application, we show for all unconfounded balanced mixed models of the analysis of variance the REML (ANOVA) estimates are consistent; and are also A.N. provided the models are nondegenerate; the MLE are consistent (A.N.) if and only if certain constraints on p are satisfied.

On sufficient conditions for asymptotic normality of decomposable statistics in a non-homogeneous allocation scheme

On sufficient conditions for asymptotic normality of decomposable statistics in a non-homogeneous allocation scheme

On the asymptotic normality of self-normalized sums

AbstractLet X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

Error inference for nonparametric regression

This study examines means for inferring the distribution of the er- ror in nonparametric regression. The central objective is to develop confidence intervals for nonparametric regression. Our computational study would seem to affirm that our methods are potentially useful in cases of small sample size or heterogeneously distributed error. Theoretical developments offer sufficient conditions for asymptotic normality. exists but, aside from some smoothness properties, has unknown structure. Currently, the primary emphasis of the nonparametric regression literature is on providing estimators mn(x) of re(x), the subscript indicating that mn(z) is constructed from the first n terms of the data sequence. The thrust of the present study is in a different direction: a nonparametric regression construct mn (x) having been specified, our wish is to infer the distribution of mn (x) -m(x). A central motivation for inference of the error law was to construct confidence intervals for m(x). Another potential use is for hypothesis testing. Bhattacharya (1976) has given a definitive method for testing whether the regression function is constant. Our methodology, while not as focussed, could be adapted to test whether re(x) is positive, for instance. Another potential use is for estimating

Necessary and sufficient conditions for asymptotic normality of trimmed L-statistics

Necessary and sufficient conditions are obtained for the asymptotic normality of L-statistics when the weights are formed by means of a general class of smooth score functions J. This asymptotic normality problem is solved for trimming fixed fractions, vanishing fractions and fixed numbers of observations. In the process of obtaining the solution, all possible limit laws of such statistics are described; these include a generalization of the class of infinitely divisible laws that may be of independent interest. A further set of sufficient conditions is derived for the asymptotic normality of L-statistics that rival, in terms of mildness of regularity, those conditions required by all previous theorems on the subject. Asymptotic normality for a generalized trimmed mean form of student's t-statistic is also included. It is asymptotically normal whenever the observations are in the domain of attraction of any stable law.