Search Results (17)

Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4. Full article

Edgeworth–Cornish–Fisher expansions are hugely important, as they give the distribution, density and quantiles of any standard estimate. Here we show that the sample mean of a univariate or multivariate stationary process is a standard estimate, so that all the known results for standard estimates can be applied. We also show how to allow for missing data and weighted means. Full article

Expectiles were introduced to statistics around 40 years ago, but have recently gained renewed interest due to their relevance in financial risk management. In particular, the 2007–2009 global financial crisis highlighted the need for more robust risk evaluation tools, leading to the adoption of inference methods for expectiles. While first-order asymptotic inference results for expectiles are well established, higher-order asymptotic results remain underdeveloped. This study aims to fill that gap by deriving higher-order asymptotic results for expectiles, ultimately improving the accuracy of confidence intervals. The paper outlines the derivation of the Edgeworth expansion for both the standardized and studentized versions of the kernel-based estimator of the expectile, using large deviation results on U-statistics. The expansion is then inverted to construct more precise confidence intervals for the expectile. These theoretical results were applied to moderate sample sizes ranging from 20 to 200. To demonstrate the advantages of this methodology, an example from risk management is presented. The enhanced confidence intervals consistently outperformed those based on the first-order normal approximation. The methodology introduced in this paper can also be extended to other contexts. Full article

In context-aware decision analysis, mean can be an important measure, even when the distribution is skewed. Previous comparative studies showed that it is a real challenge to construct a confidence interval that performs well for highly skewed data. In this study, we propose new confidence intervals for the population mean based on Edgeworth expansion that include both skewness and kurtosis corrections. We compared existing and newly proposed confidence intervals for a range of samples from symmetric and skewed distributions of varying levels of kurtosis. Using Monte Carlo simulations, we evaluated the performance of these intervals based on the coverage probability, mean length, and standard deviation of the length. The proposed bootstrap Edgeworth-based confidence interval outperformed other confidence intervals in terms of coverage probability for both symmetric and skewed distributions and can be recommended for general use in practice. Full article

Tool holders are one of the most important structures in transferring machine tools and energy for manufacturing in CNC lathe. Power servo tool holders influence kinematic accuracy and machining accuracy and so are vital to the transposition system. Reliability evaluation is also critical to guaranteeing and maintaining the accuracy of the transposition system. The first four statistical moments are derived to depict the transmission error and system characteristics. Considering the Edgeworth expansion with higher terms, reliability and reliability-based sensitivity evaluations using moments are proposed to assess system accuracy. Compared with different methods, the proposed method can represent higher statistical characteristics, helping to avoid underestimations of system reliability. Also, results calculated with the proposed method for the transposition system are in agreement with the results from the Monte Carlo simulation with 10⁷ samples. The relative error of failure probability is 4.32%. Considering the plus–minus sign and values of results, reliability-based sensitivity represents the effects of the parameters’ dispersions on system reliability. The reliability-based sensitivity indices can be utilized to optimize the system structure and to improve system accuracy, which can increase the system reliability from 98.34% to 99.99% in the transposition system of the power servo tool holder. Full article

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in $n^{-1/2}$ about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate $\widehat{w}$ of an unknown vector w in $R^p$, as a standard estimate, if $E\widehat{w}\to w$ as $n\to \infty$, and for $r\ge 1$ the rth-order cumulants of $\widehat{w}$ have magnitude $n^{1-r}$ and can be expanded in $n^{-1}.$ Here we present a significant extension. We give the expansion of the distribution of any smooth function of $\widehat{w}$, say $t\left(\widehat{w}\right)$ in $R^q,$ giving its distribution to $n^{-5/2}$. We do this by showing that $t\left(\widehat{w}\right)$, is a standard estimate of $t\left(w\right)$. This provides far more accurate approximations for the distribution of $t\left(\widehat{w}\right)$ than its asymptotic normality. Full article

The dynamic and static characteristics of hydrostatic thrust bearings are significantly affected by the bearing surface topography. Previous studies on hydrostatic thrust bearings have focused on Gaussian distribution models of bearing surface topography. However, based on actual measurements, the non-Gaussianity of the distribution characteristics of bearing surface topography is clear. To accurately characterize the non-Gaussian distribution of bearing surface topography, the traditional probability density function of Gaussian distribution was modified by introducing Edgeworth expansion. The non-Gaussian surface was then reflected by two parameters: kurtosis and skewness. This had an effect on the static characteristics of hydrostatic thrust bearings with both circumferential and radial surface topographies. The comparison between the Gaussian distribution results and those of the non-Gaussian model showed that errors between the two models could reach more than 10%. Therefore, it is important to take into account the non-Gaussianity of bearing surface when discussing static characteristics of hydrostatic thrust bearings considering the surface topography. Full article

This article completes our studies on the formal construction of asymptotic approximations for statistics based on a random number of observations. Second order Chebyshev–Edgeworth expansions of asymptotically normally or chi-squared distributed statistics from samples with negative binomial or Pareto-like distributed random sample sizes are obtained. The results can have applications for a wide spectrum of asymptotically normally or chi-square distributed statistics. Random, non-random, and mixed scaling factors for each of the studied statistics produce three different limit distributions. In addition to the expected normal or chi-squared distributions, Student’s t-, Laplace, Fisher, gamma, and weighted sums of generalized gamma distributions also occur. Full article

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of $1/\sqrt{n}$ and $1/n$ in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations. Full article

We develop a technique for obtaining the fourth moment bound on the normal approximation of F, where F is an ${\mathbb{R}}^d$-valued random vector whose components are functionals of Gaussian fields. This study transcends the case of vectors of multiple stochastic integrals, which has been the subject of research so far. We perform this task by investigating the relationship between the expectations of two operators $\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}^{*}$. Here, the operator $\mathsf{\Gamma }$ was introduced in Noreddine and Nourdin (2011) [On the Gaussian approximation of vector-valued multiple integrals. J. Multi. Anal.], and ${\mathsf{\Gamma }}^{*}$ is a muilti-dimensional version of the operator used in Kim and Park (2018) [An Edgeworth expansion for functionals of Gaussian fields and its applications, stoch. proc. their Appl.]. In the specific case where F is a random variable belonging to the vector-valued multiple integrals, the conditions in the general case of F for the fourth moment bound are naturally satisfied and our method yields a better estimate than that obtained by the previous methods. In the case of $d=1$, the method developed here shows that, even in the case of general functionals of Gaussian fields, the fourth moment theorem holds without conditions for the multi-dimensional case. Full article

This paper analytically derives a stability test for the probability distribution of a random variable that follows the Edgeworth–Sargan density, also called Gram–Charlier. The distribution of the test is a weighted sum of Chi-squared densities of increasing degrees of freedom, starting with the standard equivalent Chi-squared under the same conditions. The weights turn out to be linear combinations of the parameters of the distribution and the moments of a Gaussian density, and can be computed exactly. This is a convenient result, since then the probability intervals can be easily calculated from existing Chi-squared distribution tables. The test is applied to assess the weekly solar irradiance data stability for a twelve-year period. It shows that the density is acceptably stable overall, except for some eventual and localised dates. It is also shown that the usual probability intervals implemented in stability testing are larger than those of the equivalent Chi-squared distribution under comparable conditions. This implies that the common upper tail interval values for rejecting the null stability hypothesis are larger. Full article

This paper is concerned with the rate of convergence of the distribution of the sequence $\{F_n/G_n\}$, where $F_n$ and $G_n$ are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of $F_n/G_n$. As a tool for our work, an Edgeworth expansion for the distribution of $F_n/G_n$, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs. Full article

The probability density function (pdf) valid for the Gaussian case is often applied for describing the convolutional noise pdf in the blind adaptive deconvolution problem, although it is known that it can be applied only at the latter stages of the deconvolution process, where the convolutional noise pdf tends to be approximately Gaussian. Recently, the deconvolutional noise pdf was approximated with the Edgeworth Expansion and with the Maximum Entropy density function for the 16 Quadrature Amplitude Modulation (QAM) input but no equalization performance improvement was seen for the hard channel case with the equalization algorithm based on the Maximum Entropy density function approach for the convolutional noise pdf compared with the original Maximum Entropy algorithm, while for the Edgeworth Expansion approximation technique, additional predefined parameters were needed in the algorithm. In this paper, the Generalized Gaussian density (GGD) function and the Edgeworth Expansion are applied for approximating the convolutional noise pdf for the 16 QAM input case, with no need for additional predefined parameters in the obtained equalization method. Simulation results indicate that improved equalization performance is obtained from the convergence time point of view of approximately 15,000 symbols for the hard channel case with our new proposed equalization method based on the new model for the convolutional noise pdf compared to the original Maximum Entropy algorithm. By convergence time, we mean the number of symbols required to reach a residual inter-symbol-interference (ISI) for which reliable decisions can be made on the equalized output sequence. Full article

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized $T_0^2$ statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples. Full article

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples. Full article