Edgeworth Type Research Articles

Distribution of the product of a Wishart matrix and a normal vector

We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.

On the validity of the formal Edgeworth expansion for posterior densities

We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth type.

Second Order Approximations for Slightly Trimmed Means

We investigate the second order asymptotic behavior of distributions of statistics $T_n=\frac 1n \sum_{i=k_{n}+1}^{n-m_{n}}{X_{i:n}}$, where $k_n$, $m_n$ are sequences of integers, $0\le k_n < n-m_n \le n$, and $r_n:=\min(k_n, m_n) \to \infty$ as ${n \to \infty}$, and the ${X_{i:n}}$'s denote the order statistics corresponding to a sample $X_1,\dots,X_n$ of $n$ independent identically distributed random variables. In particular, we focus on the case of slightly trimmed means with vanishing trimming percentages; i.e., we assume that $\max(k_n,m_n)/n\to 0$ as ${n \to \infty}$, and heavy tailed distribution $F$; i.e., the common distribution of the observations $F$ is assumed to have an infinite variance. We derive optimal bounds of Berry--Esseen type of the order $O(r_n^{-1/2})$ for the normal approximation to $T_n$ and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed means and their Studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csörgö, Haeusler, and Mason [Ann. Probab., 16 (1988), pp. 672--699] and on second order approximations for (Studentized) trimmed means with fixed trimming percentages by Gribkova and Helmers [Math. Methods Statist., 15 (2006), pp. 61--87; Math. Methods Statist., 16 (2007), pp. 142--176].

Edgeworth type expansion of ruin probability under Lévy risk processes in the small loading asymptotics

This paper presents an asymptotic expansion of the ultimate ruin probability under Lévy insurance risks as the loading factor tends to zero. The expansion formula is obtained via the Edgeworth type expansion for compound geometric distributions. We give higher-order expansion of the ruin probability, any order of which is available in explicit form, and discuss a certain type of validity of the expansion. We shall also give applications to evaluation of the VaR-type risk measure due to ruin, and the scale function of spectrally negative Lévy processes.

Asymptotic expansions in the CLT in free probability

We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution.

Approximations to distribution of median in stratified samples

We consider an Edgeworth type approximation to the distribution function of sample median in the case of stratified samples drawn without replacement. We give explicit expression of this approximation, and also its empirical version based on bootstrap. We compare their accuracy with that of the normal approximation by numerical examples.&#x0D;

Edgeworth Type Expansion of Ruin Probability Under Levy Risk Processes in the Small Loading Asymptotics

This paper presents an asymptotic expansion of the ultimate ruin probability under Levy insurance risks as the loading factor tends to zero. The expansion formula is obtained via the Edgeworth type expansion for the compound geometric sum. We give higher-order expansion of the ruin probability, any order of which is available in explicit form, and discuss a certain type of validity of the expansion. We shall also give applications to evaluation of the VaR-type risk measure due to ruin, and the scale function of spectrally negative Levy processes.

Asymptotic Expansions of MM-type and QML Estimators for the MA(1) with Mean Models

The second order asymptotic expansions, of the Edgeworth type, of an MM-type and the QML estimators for the MA(1) with mean model are given. We also derive the second order expansions of the sample autocorrelation and the autocorrelation based on the QML estimator of the MA parameter. By employing Nagar type expansions, we compare the estimators in terms of bias and mean squared error. The results presented here are important for deciding on the estimation method we choose, as well as for bias reduction and increasing the efficiency of the estimators. The theoretical results are generally confirmed by a Monte-Carlo exercise.

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.

Higher Order Asymptotic Cumulants of Studentized Estimators in Covariance Structures

In covariance structure analysis, the Studentized pivotal statistic of a parameter estimator is often used since the statistic is asymptotically normally distributed with mean zero and unit variance. For more accurate asymptotic distribution, the first and third asymptotic cumulants can be used to have the single-term Edgeworth, Cornish-Fisher, and Hall type asymptotic expansions. In this paper, the higher order asymptotic variance and the fourth asymptotic cumulant of the statistic are obtained under nonnormality when the partial derivatives of a parameter estimator with respect to sample variances and covariances up to the third order and the moments of the associated observed variables up to the eighth order are available. The result can be used to have the two-term Edgeworth expansion. Simulations are performed to see the accuracy of the asymptotic results in finite samples.

Expansions for log densities of asymptotically normal estimates

Edgeworth–type expansions are given for the log density and also for the derivatives of the density of an asymptotically normal random variable having the standard expansions for its cumulants. Expansions for the log density are much simpler than for the density. In fact the rth term of the expansion for the log density is a polynomial of degree only r + 2, while that for the density is a polynomial of degree 3r.

Edgeworth expansion of the largest eigenvalue distribution function of Gaussian unitary ensemble revisited

We derive expansions of the resolvent Rn(x,y;t)=(Qn(x;t)Pn(y;t)−Qn(y;t)Pn(x;t))∕(x−y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian unitary ensemble (GUEn) and the finite n-expansion of Qn(x;t) and Pn(x;t). Using these large n-expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. These large n-expansions are essential ingredients in the derivation of our results for Gaussian orthogonal ensemble (GOEn) (Choup, L. N., arXiv:0801.2620v1) where we give explicit n−1∕3 and n−2∕3 correction terms to the limiting GOE Tracy–Widom distribution function.

Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.

Calculation of a formal moment generating function by using a differential operator

A differential form in a formal moment generating function is given by the decomposition of powers in terms of the Hermite polynomials. This paper shows that this differential form for calculating the expectation of normal and χ 2 distributions has the benefit of avoiding divergence for Edgeworth type approximations from the viewpoint of a formal power series ring. A symbolic computational algorithm is also discussed, within the distribution theory of statistics.

Uniform Markov renewal theory and ruin probabilities in Markov random walks

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time \tau=\tau(b)=inf{n:S_n>b}. When a drift \mu of the random walk S_n is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities P_{\pi}{\tau<m} and P_{\pi}{\tau<m,S_m<c}, where m\leq\infty, c\leq b and P_{\pi} denotes the probability under the initial distribution \pi. When \mu\neq0, Brownian approximations for the first passage probabilities with correction terms are derived.

On Euler products and multi-variate Gaussians

In this Note, we extend a recent result of A. Selberg concerning the asymptotic value distribution of Euler products to a multi-dimensional setting. Under certain conditions, an asymptotic development of Edgeworth type is found. To cite this article: D.A. Hejhal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Edgeworth type expansions for Euler schemes for stochastic differential equations.

Edgeworth type expansions for Euler schemes for stochastic differential equations.

Study on the short-term characteristics of the wind field in Keelung, Taiwan

Short-term characteristics of the wind field and temperature measured in Keelung, Taiwan were studied. The fluctuating components of measured wind speeds and the temperature were fitted with statistical models found in the literature. In general, the distributions of the fluctuating components are skewed, having more pronounced peaks as compared with the symmetric Gaussian model. The Edgeworth's type A Gram–Charlier series expansion was found to be more adequate in describing the statistical properties of the winds. Spectra of the turbulent wind fluctuations are found to follow the − 5 3 -power law at the high-frequency tail. The spectra of the temperature, on the other hand, are usually flat in this region, resembling that of the white noise.

THE ACCURACY OF CAUCHY APPROXIMATION FOR THE WINDINGS OF PLANAR BROWNIAN MOTION

Our method of proof relies upon the skew–product representation of planar Brownian motion, and changes of probabilities between the laws of Bessel processes, as indicated in [8], [9], [10]; in particular, this paper is an improvement of [10], which discusses only the case k = 2. The spirit of our expansion is close to the Edgeworth type expansions in the central limit theorems, that is, to the asymptotic expansions of the distribution

Distribution and density approximation of the co variance matrix in the growth curve model

Approximations of the distribution for the maximum likelihood estimator of the dispersion matrix in the Growth Curve model due to Potthoff and Roy 0964) are considered. Three different approaches are presented. Two are based on an Edgeworth type expansion where as approximating distribution two different Wishart distributions are used. In a third approach we utilize the structure of the estimator and combine an Edgeworth type expansion with an approach put forward by Fujikoshi (1985).