基于回归的时变偏度和时变峰度识别检验

资产收益率时变高阶矩建模的首要前提是资产收益率的偏度和峰度具有时变性,即资产收益率存在类似于异方差性的异偏度和异峰度特征。目前文献中的时变偏度和时变峰度识别检验存在适用性较差且检验功效较低等不足。本文提出基于回归的检验方法识别资产收益率偏度和峰度的时变性。该检验一方面利用概率积分变换缓解了拉格朗日乘数检验对资产收益率序列的高阶矩存在性的限制,另一方面考虑了检验统计量中参数估计的不确定性对其统计性质的影响,具有良好的渐近统计性质且适用性更广。蒙特卡洛模拟表明该检验具有良好的有限样本性质,具有合适的检验水平和较高的检验功效。最后,将基于回归的检验运用于上证综指和深圳成指收益率的时变建模研究中。     

Recurrence formula on the joint distribution of sample skewness and kurtosis under normality

Abstract

Two recurrence relations with respect to sample size are given concerning the joint distribution of skewness and kurtosis of random observations from a normal population: one between the probability density functions and the other between the product moments. As a consequence, the latter yields a recurrence formula for the moments of sample kurtosis. The exact moments of Jarque-Bera statistic is also given.     

Practical implications of higher moments in risk management

In this paper the out-of-sample prediction of Value-at-Risk by means of models accounting for higher moments is studied. We consider models differing in terms of skewness and kurtosis and, in particular, the GARCHDSK model, which allows for constant and dynamic skewness and kurtosis. The issue of VaR prediction performance is approached first from a purely statistical viewpoint, studying the properties concerning correct coverage rates and independence of VaR violations. Then, financial implications of different VaR models, in terms of market risk capital requirements, as defined by the Basel Accord, are considered. Our results, based on the analysis of eight international stock indexes, highlight the presence of conditional skewness and kurtosis, in some case time-varying, and point out that asymmetry plays a significant role in risk management.     

Moment-Implied Densities: Properties and Applications

Suppose one uses a parametric density function based on the first four (conditional) moments to model risk. There are quite a few densities to choose from and depending on which is selected, one implicitly assumes very different tail behavior and very different feasible skewness/kurtosis combinations. Surprisingly, there is no systematic analysis of the tradeoff one faces. It is the purpose of the article to address this. We focus on the tail behavior and the range of skewness and kurtosis as these are key for common applications such as risk management.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.     

Some developments on the log-Dagum distribution

Skewed and fat-tailed distributions frequently occur in many applications. Models proposed to deal with skewness and kurtosis may be difficult to treat because the density function cannot usually be written in a closed form and the moments might not exist. The log-Dagum distribution is a flexible and simple model obtained by a logarithmic transformation of the Dagum random variable. In this paper, some characteristics of the model are illustrated and the estimation of the parameters is considered. An application is given with the purpose of modeling kurtosis and skewness that mark the financial return distribution.      

上证50ETF隐含高阶矩风险对股票收益的预测研究

高阶矩是刻画资产收益涨跌非对称和“尖峰厚尾”现象中不可忽略的系统性风险。本文基于我国上证50ETF期权数据采用无模型方法估计隐含波动率、隐含偏度和隐含峰度,通过自回归滑动平均模型提取期权隐含高阶矩新息(Innovations),将它们作为高阶矩风险的度量,探讨其对股票收益的预测作用。研究表明:①在控制换手率和股息率等变量后,隐含波动率对于上证50指数和市场未来4周的超额收益有显著负向的预测作用;②隐含偏度新息越低,上证50指数和市场的超额收益越高,这种预测能力在未来1周和未来4周均显著,但随着时间的推移,隐含偏度新息的预测能力逐渐下降;③隐含偏度风险对于我国股市横截面收益也有显著的解释能力,投资组合在隐含偏度风险因子上的风险暴露越大即因子载荷值越大,则未来的收益会越低;④隐含峰度新息总体上与股票收益负相关。     

Measures of multivariate skewness and kurtosis for tests of nonnormality

Measures of multivariate skewness and kurtosis are proposed that are based on the skewness and kurtosis of individual components of standardized sample vectors. Asymptotic properties and small sample critical values of tests for nonnormality based on these measures are provided. It is demonstrated that the tests have favorable power properties. Extensions to time series data are pointed out.     

On some characteristics of bivariate chi-square distribution

Product moments of bivariate chi-square distribution have been derived in closed forms. Finite expressions have been derived for product moments of integer orders. Marginal and conditional distributions, conditional moments, coefficient of skewness and kurtosis of conditional distribution have also been discussed. Shannon entropy of the distribution is also derived. We also discuss the Bayesian estimation of a parameter of the distribution. Results match with the independent case when the variables are uncorrelated.     

Kurtosis of the logistic-exponential survival distribution

ABSTRACT

In this article, the kurtosis of the logistic-exponential distribution is analyzed. All the moments of this survival distribution are finite, but do not possess closed-form expressions. The standardized fourth central moment, known as Pearson’s coefficient of kurtosis and often used to describe the kurtosis of a distribution, can thus also not be expressed in closed form for the logistic-exponential distribution. Alternative kurtosis measures are therefore considered, specifically quantile-based measures and the L-kurtosis ratio. It is shown that these kurtosis measures of the logistic-exponential distribution are invariant to the values of the distribution’s single shape parameter and hence skewness invariant.     

A New Measure of Kurtosis Adjusted for Skewness

Studies of kurtosis often concentrate on only symmetric distributions. This paper identifies a process through which the standardized measure of kurtosis based on the fourth moment about the mean can be written in terms of two parts: (i) an irreducible component, about L₄, which can be seen to occur naturally in the analysis of fourth moments; (ii) terms that depend only on moments of lower order, in particular including the effects of asymmetry attached to the third moment about the mean. This separation of the effect of skewness allows definition of an improved measure of kurtosis. This paper calculates and discusses examples of the new measure of kurtosis for a range of standard distributions.     

Distributional properties of simpson's index of diversity

Closed expressions for the first four moments of Simpson's index of diversity are derived using techniaues suggested by Haldane (1937). As the samole size increases the behavior of the skewness and kurtosis is studied for several Dopulations with varying degrees of diversity, If the populationproportions decrease accordinq to a geometric progression, graphs of β₁and β₂ indicate that convergence to normality in general is more rapid for populations which are less diverse.     

Distribution properties and estimation of the ratio of independent Weibull random variables

The important problem of the ratio of Weibull random variables is considered. Two motivating examples from engineering are discussed. Exact expressions are derived for the probability density function, cumulative distribution function, hazard rate function, shape characteristics, moments, factorial moments, skewness, kurtosis and percentiles of the ratio. Estimation procedures by the methods of moments and maximum likelihood are provided. The performances of the estimates from these methods are compared by simulation. Finally, an application is discussed for aspect and performance ratios of systems.     

Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis

Estimates based on L-moments are less non-robust than estimates based on ordinary moments because the former are linear combinations of order statistics for all orders, whereas the later take increasing powers of deviations from the mean as the order increases. Estimates based on L-moments can also be more efficient than maximum likelihood estimates. Similarly, L-skewness and L-kurtosis are less non-robust and more informative than the traditional measures of skewness and kurtosis. Here, we give nonparametric bias-reduced estimates of both types of skewness and kurtosis. Their asymptotic computational efficiency is infinitely better than that of corresponding bootstrapped estimates.     

Computational Examples of a New Method for Distribution Selection in the Pearson System

A considerable problem in statistics and risk management is finding distributions that capture the complex behaviour exhibited by financial data. The importance of higher order moments in decision making has been well recognized and there is increasing interest in modelling with distributions that are able to account for these effects. The Pearson system can be used to model a wide scale of distributions with various skewness and kurtosis. This paper provides computational examples of a new easily implemented method for selecting probability density functions from the Pearson family of distributions. We apply this method to daily, monthly, and annual series using a range of data from commodity markets to macroeconomic variables.     

Uniform-Geometric distribution

In this paper, a new discrete distribution called Uniform-Geometric distribution is proposed. Several distributional properties including survival function, moments, skewness, kurtosis, entropy and hazard rate function are discussed. Estimation of distribution parameter is studied by methods of moments, proportions and maximum likelihood. A simulation study is performed to compare the performance of the different estimates in terms of bias and mean square error. Two real data applications are also presented to see that new distribution is useful in modelling data.     

A lagrange expansion for the parameters of johnson's Su

A series, based on Lagrange's formula, is given for the kurtosis in terms of the skewness and the ω-parameter of S_u. Also a Maclaunn expansion for a function of ω in terms of the skewness and kurtosis is described.     

The general expressions for the moments of lawless and wang's ordinary ridge regression estimator

In this paper, we derive the exact general expressions for the moments of an ordinary ridge regression (ORR) estimator for individual regression coefficients in a different way from Firinguetti (1987). Using the derived expressions, we evaluate numerically the first four moments of the ORR estimator, and examine its bias, mean square error, skewness and kurtosis. Further, Monte Carlo experiments are carried out in order to examine the shape of the density function of the ORR estimator.     

A New Fatigue Life Model Based on the Family of Skew-Elliptical Distributions

ABSTRACT

Fatigue is structural damage produced by cyclic stress and tension. An important statistical model for fatigue life is the Birnbaum–Saunders distribution, which was developed to model ruptured lifetimes of metals that had been subjected to fatigue. This model has been previously generalized and in this article we extend it starting from a skew-elliptical distribution, the incorporation of the elliptical aspect makes the kurtosis flexible, and the skewness makes the asymmetry flexible. In this work we found the probability density, reliability, and hazard functions; as well as its moments and variation, skewness, and kurtosis coefficients. In addition, some properties of this new distribution were found.     

Exponentiated Sinh Cauchy Distribution with Applications

The exponentiated sinh Cauchy distribution is characterized by four parameters: location, scale, symmetry, and asymmetry. The symmetry parameter preserves the symmetry of the distribution by producing both bimodal and unimodal densities having coefficient of kurtosis values ranging from one to positive infinity. The asymmetry parameter changes the symmetry of the distribution by producing both positively and negatively skewed densities having coefficient of skewness values ranging from negative infinity to positive infinity. Bimodality, skewness, and kurtosis properties of this regular distribution are presented. In addition, relations to some well-known distributions are examined in terms of skewness and kurtosis by constructing aliases of the proposed distribution on the symmetry and asymmetry parameter plane. The maximum likelihood parameter estimation technique is discussed, and examples are provided and analyzed based on data from astronomy and medical sciences to illustrate the flexibility of the distribution for modeling bimodal and unimodal data.