Kolmogorov Distance Research Articles

An Edgeworth expansion for functionals of Gaussian fields and its applications

This paper is concerned with the rate of convergence in the normal approximation of the sequence {Fn}, where each Fn is a functional of an infinite-dimensional Gaussian field. We develop new and powerful techniques for computing the exact rate of convergence in distribution with respect to the Kolmogorov distance. As a tool for our works, the Edgeworth expansion of general orders, with an explicitly expressed remainder, will be obtained, and this remainder term will be controlled to find upper and lower bounds of the Kolmogorov distance in the case of an arbitrary sequence {Fn}. As applications, we provide the optimal fourth moment theorem of the sequence {Fn} in the case when {Fn} is a sequence of random variables living in a fixed Wiener chaos or a finite sum of Wiener chaoses. In the former case, our results show that the conditions given in this paper seem more natural and minimal than ones appeared in the previous works.

Berry–Esseen bounds for typical weighted sums

Under correlation-type conditions, we derive upper bounds of order $\frac{1}{\sqrt{n}}$ for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.

Sharp Bounds for Exponential Approximations of NWUE Distributions

Let F be an NWUE distribution with mean 1 and G be the stationary renewal distribution of F. We would expect G to converge in distribution to the unit exponential distribution as its mean goes to 1. In this paper, we derive sharp bounds for the Kolmogorov distance between G and the unit exponential distribution, as well as between G and an exponential distribution with the same mean as G. We apply the bounds to geometric convolutions and to first passage times.

On normal approximations to symmetric hypergeometric laws

The Kolmogorov distances between a symmetric hypergeometric law with standard deviation σ \sigma and its usual normal approximations are computed and shown to be less than 1 / ( 8 π σ ) 1/(\sqrt {8\pi }\,\sigma ) , with the order 1 / σ 1/\sigma and the constant 1 / 8 π 1/\sqrt {8\pi } being optimal. The results of Hipp and Mattner (2007) for symmetric binomial laws are obtained as special cases. Connections to Berry-Esseen type results in more general situations concerning sums of simple random samples or Bernoulli convolutions are explained. Auxiliary results of independent interest include rather sharp normal distribution function inequalities, a simple identifiability result for hypergeometric laws, and some remarks related to Lévy’s concentration-variance inequality.

New Berry–Esseen bounds for functionals of binomial point processes

We obtain explicit Berry–Esseen bounds in the Kolmogorov distance for the normal approximation of nonlinear functionals of vectors of independent random variables. Our results are based on the use of Stein’s method and of random difference operators, and generalise the bounds obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry–Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes.

Bounds to the normal for proximity region graphs

In a proximity region graph G in Rd, two distinct points x,y of a point process μ are connected when the ‘forbidden region’ S(x,y) these points determine has empty intersection with μ. The Gabriel graph, where S(x,y) is the open disk with diameter the line segment connecting x and y, is one canonical example. When μ is a Poisson or binomial process, under broad conditions on the regions S(x,y), bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of G, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.

Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach

In this paper, a Bayesian nonparametric approach to the two-sample problem is proposed. Given two samples \(\text{X} = {X_1}, \ldots ,{X_{m1}}\;\mathop {\text~}\limits^{i.i.d.} F\) and \(Y = {Y_1}, \ldots ,{Y_{{m_2}}}\mathop {\text~}\limits^{i.i.d.} G\), with F and G being unknown continuous cumulative distribution functions, we wish to test the null hypothesis H 0: F = G. The method is based on computing the Kolmogorov distance between two posterior Dirichlet processes and comparing the results with a reference distance. The parameters of the Dirichlet processes are selected so that any discrepancy between the posterior distance and the reference distance is related to the difference between the two samples. Relevant theoretical properties of the procedure are also developed. Through simulated examples, the approach is compared to the frequentist Kolmogorov–Smirnov test and a Bayesian nonparametric test in which it demonstrates excellent performance.

Central Limit Theorem and Diophantine Approximations

Let $$F_n$$ denote the distribution function of the normalized sum $$Z_n = (X_1 + \cdots + X_n)/(\sigma \sqrt{n})$$ of i.i.d. random variables with finite fourth absolute moment. In this paper, polynomial rates of convergence of $$F_n$$ to the normal law with respect to the Kolmogorov distance, as well as polynomial approximations of $$F_n$$ by the Edgeworth corrections (modulo logarithmically growing factors in n), are given in terms of the characteristic function of $$X_1$$ . Particular cases of the problem are discussed in connection with Diophantine approximations.

Normal approximation for stabilizing functionals

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

Closeness of Lindley distribution to Weibull and gamma distributions

In this paper we consider the problem of the model selection/discrimination among three different positively skewed lifetime distributions. Lindley, Weibull, and gamma distributions have been used to effectively analyze positively skewed lifetime data. This paper assesses how much closer the Lindley distribution gets to Weibull and gamma distributions. We consider three techniques that involve the likelihood ratio test, asymptotic likelihood ratio test, and minimum Kolmogorov distance as optimality criteria to diagnose the appropriate fitting model among the three distributions for a given data set. Monte Carlo simulation study is performed for computing the probability of correct selection based on the considered optimality criteria among these families of distributions for various choices of sample sizes and shape parameters. It is observed that overall, the Lindley distribution is closer to Weibull distribution in the sense of likelihood ratio and Kolmogorov criteria. A real data set is presented and analyzed for illustrative purposes.

Approximate Central Limit Theorems

We refine the classical Lindeberg-Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parametrized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg's condition. This allows us to continue to provide information in non-standard settings in which the classical central limit theorem fails to hold. Stein's method plays a key role in the development of this theory.

Boundary density and Voronoi set estimation for irregular sets

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension s > d − 1 s>d-1 in R d \mathbb {R}^{d} . These quantities turn out to be crucial in some problems of set estimation, as we show here for the Voronoi approximation of the set with a random input constituted by n n iid points in some larger bounded domain. We prove that some classes of such sets have positive inner and outer boundary density, and therefore satisfy Berry-Esseen bounds in n − s / 2 d n^{-s/2d} for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counterexample. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.

Central Limit Theorem for the Volume of Random Polytopes with Vertices on the Boundary

Given a convex body K with smooth boundary \(\partial K\), select a fixed number n of uniformly distributed random points from \(\partial K\). The convex hull \(K_n\) of these points is a random polytope having all its vertices on the boundary of K. The closeness of the volume of \(K_n\) to a Gaussian random variable is investigated in terms of the Kolmogorov distance by combining a version of Stein’s method with geometric estimates for the surface body of K.

Optimal Berry–Esseen bound for an estimator of parameter in the Ornstein–Uhlenbeck process

This paper is concerned with the study of the rate of central limit theorem for the maximum likelihood estimator θˆT of the unknown parameter θ>0, based on the observation X={Xt,0≤t≤T}, occurring in the drift coefficient of an Ornstein–Uhlenbeck process dXt=−θXtdt+dWt,X0=0 for 0≤t≤T, where {Wt,t≥0} is a standard Brownian motion. The tool we use is an Edgeworth expansion with an explicitly expressed remainder. We prove that upper and lower bounds, obtained by controlling the remainder term, give an optimal rate 1T in Kolmogorov distance for normal approximation of θˆT.

Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where $n$ is the sample size. However, the corresponding bounds on the Kolmogorov distance were only of the order $O(1/n^{1/4})$. In this paper, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform counterparts, which work better in tail zones of the distribution of the MLE. These results are based in part on previously obtained general optimal-order bounds on the rate of convergence to normality in the multivariate delta method. The crucial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of independent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to $M$-estimators.

Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of [Formula: see text], where R is the offered load. Futhermore, these error bounds are universal, valid in any load condition from lightly loaded to heavily loaded.

Proximity of probability distributions in terms of Fourier–Stieltjes transforms

A survey is given of some results on smoothing inequalities for various probability metrics (in particular, for the Kolmogorov distance), and some analogues of these results in the class of functions of bounded variation are presented. Bibliography: 61 titles.

The Forecast Ability of Option-implied Densities from Emerging Markets Currencies

This paper empirically evaluates Risk-Neutral Densities (RND) and Real-World Densities (RWD) as predictors of emerging markets currencies. The dataset consists of volatility surfaces from 11 emerging market currencies, with approximately six years of daily data, using options with one-month expiration. Therefore, there is a strong overlapping in data, which is tackled with specific econometric techniques. Results of the out-of-sample assessment show that both RND and RWD underweight the tails of the actual distribution. This is probably due to the lack of options with extreme strikes. Although the RWDs perform better than RND in terms of Kolmogorov distance, they still have problems in fitting the tails of actual data. Thus, the risk-aversion adjustment may improve the forecast ability, but it does not solve the tails misfitting.

Berry–Esseen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

Abstract Let B be a bifractional Brownian motion with parameters H ∈ (0,1) and K ∈ (0,1]. For any n ≥ 1, set Z n = ∑ i = 0 n - 1 [ n 2 H K ( B ( i + 1 ) / n - B i / n ) 2 - 𝔼 ( ( B i + 1 - B i ) 2 ) ] ${Z_n = \sum _{i=0 }^{n-1} [n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\mathbb {E} ((B_{i+1}-B_{i})^2 ) ]}$ . We use Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati [11] to derive, in the case when 0 < H K ≤ 3/4, Berry–Esseen-type bounds for the Kolmogorov distance between the law of the correct renormalization Vn of Zn and the standard normal law. Finally, we study almost sure central limit theorems for the sequence Vn .

Stein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models

This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore, these error bounds are \emph{universal}, valid in any load condition from lightly loaded to heavily loaded.