On measuring skewness and kurtosis

The paper considers some properties of measures of asymmetry and peakedness of one dimensional distributions. It points to some misconceptions of the first and the second Pearson coefficients, the measures of asymetry and shape, that frequently occur in introductory textbooks. Also it presents different ways for obtaining the estimated values for the coefficients of skewness and kurtosis and statistical tests which include them.     

单因子利率期限结构模型的广义矩估计及对中国货币市场的实证检验

本文在多个著名的单因子利率模型的基础上，提出了一个新的一般模型，其漂移项涵盖了线性和非线性两种形式。并用广义矩方法进行参数估计，在多种指标的比较下得到了一个较好模型。此模型的漂移项为非线性形式，具有显著的均值回复效应，且利率波动对利率水平极为敏感。     

The large-sample joint distribution of key circular statistics

In this paper we derive the large-sample asymptotic joint distribution of the statistics used as fundamental measures of central location, concentration, skewness and kurtosis in the analysis of circular data. The importance of the distributional result in relation to inference for the corresponding population measures is illustrated, with various new confidence set constructions being derived and applied in the analysis of data from an animal orientation experiment.     

A Simple Derivation of Moments of the Exponentiated Weibull Distribution

The Exponentiated Weibull family is an extension of the Weibull family obtained by adding an additional shape parameter. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates which are quite common in reliability and biological studies. As with any other distribution, many of its interesting characteristics and features can be studied through moments. Presently, moments of this distribution are available only under certain restrictions. In this paper, a general derivation of moments without any restriction whatsoever is proposed. A compact expression for moments is presented.     

Tests for location-scale families based on the empirical characteristic function

A class of procedures is presented for using random samples to test the fit of location-scale families—distributions F(·;θ₁,θ₂) such that Z=(X−θ₁)/θ₂ has distribution Working with empirically standardized data, the test statistic is a measure of distance between the empirical characteristic function, and the c.f. of Z under the null hypothesis, ϕ₀(t). The closed-form test statistic is derived by integrating over the product of a weight function times Using as weight function for each location-scale family the squared modulus of ϕ₀ itself presents a unified test procedure. Included as special cases are well-known tests for normal and Cauchy laws. Small-sample powers are compared with those of Anderson-Darling tests for each of seven univariate location-scale families.     

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods.     

A new class of inverse Gaussian type distributions

The elliptical laws are a class of symmetrical probability models that include both lighter and heavier tailed distributions. These models may adapt well to the data, even when outliers exist and have other good theoretical properties and application perspectives. In this article, we present a new class of models, which is generated from symmetrical distributions in and generalize the well known inverse Gaussian distribution. Specifically, the density, distribution function, properties, transformations and moments of this new model are obtained. Also, a graphical analysis of the density is provided. Furthermore, we estimate parameters, propose asymptotic inference and discuss influence diagnostics by using likelihood methods for the new distribution. In particular, we show that the maximum likelihood estimates parameters of the new model under the t kernel are down-weighted for the outliers. Thus, smaller weights are attributed to outlying observations, which produce robust parameter estimates. Finally, an illustrative example with real data shows that the new distribution fits better to the data than some other well known probabilistic models.     

Analytic moments for GJR-GARCH (1, 1) processes

For a GJR-GARCH(1, 1) specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moments for the most commonly used GARCH models are stated as special cases. We also derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments. A simulation study using these analytic moments produces approximate predictive distributions which are free from the bias affecting simulations. An empirical study using almost 30 years of daily equity index, exchange rate and interest rate data applies Johnson SU and Edgeworth expansion distribution fitting to our closed-form formulae for higher moments of returns.     

Variance-Gamma and Normal-Inverse Gaussian models: Goodness-of-fit to Chinese high-frequency index returns

In this study Variance-Gamma (VG) and Normal-Inverse Gaussian (NIG) distributions are compared with the benchmark of generalized hyperbolic distribution in terms of their fit to the empirical distribution of high-frequency stock market index returns in China. First, we estimate the considered models in a Markov regime switching framework for the identification of different volatility regimes. Second, the goodness-of-fit results are compared at different time scales of log-returns. Third, the goodness-of-fit results are validated through bootstrapping experiments. Our results show that as the time scale of log-returns decrease NIG model outperforms the VG model consistently and the difference between the goodness-of-fit statistics increase. For high-frequency Chinese index returns, NIG model is more robust and provides a better fit to the empirical distributions of returns at different time scales.