Geometric stable laws: Estimation and applications

Geometric stable laws constitute a class of limiting distributions of appropriately normalized random sums of i.i.d. random variables. We consider the problem of estimation of the parameters of univariate and multivariate geometric stable laws. Our estimation technique is based on the method of moments and yields consistent and asymptotically normal estimators. We apply our estimators to a currency exchange data and show that the geometric stable dominates Paretian stable and normal models.     

高频金融数据的标度分析

高频金融数据通常具有尖峰厚尾的非正态特征，常用稳定分布来拟合。本文采用稳定分布的性质对高频上证指数收益率的分布作了标度分析，求得特征指数为1．48，说明我们在分析证券指数分布时可以用稳定分布，而非正态分布。     

基于贝叶斯分析的中国商业银行内部欺诈分析

内部欺诈事件类型是中国商业银行最严重的操作风险类型。但由于操作风险本质特征和中国商业银行内部欺诈损失数据收集年度较短,数据匮乏,小样本数据容易导致参数结果不稳定。为了在小样本数据下进行更准确的度量,本文采用贝叶斯马尔科夫蒙特卡洛模拟方法,在损失分布法框架下,假设损失频率服从泊松-伽马分布,而损失强度服从广义帕累托-混合伽马分布,分析后验分布的形式,获得中国商业银行不同业务线的内部欺诈损失频率和损失强度的后验分布估计,并进行蒙特卡罗模拟获得不同业务线内部欺诈的风险联合分布。结果表明,拟合结果很好,与传统极值分析法相比,基于利用贝叶斯的分析获得的后验分布可以作为未来的先验分布,有利于在较小样本下获得较真实的参数估计,本方法有助于银行降低监管资本要求。     

Reliability modelling incorporating load share and frailty

The stochastic behaviour of lifetimes of a two component system is often primarily influenced by the system structure and by the covariates shared by the components. Any meaningful attempt to model the lifetimes must take into consideration the factors affecting their stochastic behaviour. In particular, for a load share system, we describe a reliability model incorporating both the load share dependence and the effect of observed and unobserved covariates. The model includes a bivariate Weibull to characterize load share, a positive stable distribution to describe frailty, and also incorporates effects of observed covariates. We investigate various interesting reliability properties of this model using cross ratio functions and conditional survivor functions. We implement maximum likelihood estimation of the model parameters and discuss model adequacy and selection. We illustrate our approach using a simulation study. For a real data situation, we demonstrate the superiority of the proposed model that incorporates both load share and frailty effects over competing models that incorporate just one of these effects. An attractive and computationally simple cross‐validation technique is introduced to reconfirm the claim. We conclude with a summary and discussion.     

Wrapped weighted exponential distributions

We have developed a new class of circular distributions named wrapped weighted exponential distributions. The estimation of unknown parameters along with some characteristics of these distributions is also investigated. Some theorems that relate the distribution to some other circular distributions are established and we clarify their modeling potential using a classical data set on movements of sea stars.     

Statistical modelling of asymmetric risk in asset returns

The purpose of this article is to provide a straightforward model for asset returns which captures the fundamental asymmetry in upward versus downward returns. We model this feature by using scale gamma distributions for the conditional distributions of positive and negative returns. By allowing the parameters for positive returns to differ from parameters for negative returns we can test the hypothesis of symmetry. Some applications of this process to expected utility and semi-variance calculations are considered. Finally we estimate the model using daily UK FT100 index and Futures data.     

Asymptotic Distribution of a Kind of Dirichlet Distribution

The Dirichlet distribution that we are concerned with in this paper is very special, in which all parameters are different from each other. We prove that the asymptotic distribution of this kind of Dirichlet distributions is a normal distribution by using the central limit theorem and Slutsky theorem.     

Robust dynamic pairs trading with cointegration

This paper investigates the robust optimal pairs trading using the concept of equivalent probability measures and a penalty function associated with the confidence in parameter estimates when the parameters in the drift term of the continuous-time cointegration model are estimated with errors. A closed-form solution is derived for the robust pairs trading rule. We compare the robust pairs trading rule against its non-robust counterpart using simulations and real data. The robust strategy is empirically more stable and less volatile.     

改进的Atlas模型下市场的稳定分布

Atlas模型在随机投资组合理论中有广泛的应用,但它的假定具有一定的局限性．我们对该模型进行了改进,证明了满足改进的Atlas模型的市场是渐进稳定的;在改进的Atlas模型下我们得到了市场稳定分布的确定性等价近似以及不同投资组合的渐近增长率与渐近超额增长率,这些结果与Atlas模型的类似结果相比有很大的优点;同时我们使用中国股票市场的交易数据对资本的稳定分布以及某些投资组合的长期平均增长率进行了实证研究,对比市场平均资本分布以及Atlas模型的相应结果,我们改进的Atlas模型在实证上比Atlas模型具有更好的适用性．     

误差项为指数白噪声的平稳AR(1)过程的贝叶斯分析

本文研究了一类误差项为指数白噪声的平稳自回归模型的参数估计问题.利用贝叶斯方法,获得了参数的后验分布及在平方损失下的贝叶斯估计结果,推广了Turkman的结果.     

A lifetime model with increasing failure rate

This paper deals with a new two-parameter lifetime distribution with increasing failure rate. This distribution is constructed as a distribution of a random sum of independent exponential random variables when the sample size has a zero truncated binomial distribution. Various statistical properties of the distribution are derived. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. Simulation studies show the performance of the estimators. Also, estimation of the parameters is considered in the presence of censoring. A real data set is analyzed for illustrative purposes and it is noted that the distribution is a good competitor to the gamma, Weibull, exponentiated exponential, weighted exponential and Poisson-exponential distributions for this data set.     

Skew normal measurement error models

In this paper we define a class of skew normal measurement error models, extending usual symmetric normal models in order to avoid data transformation. The likelihood function of the observed data is obtained, which can be maximized by using existing statistical software. Inference on the parameters of interest can be approached by using the observed information matrix, which can also be computed by using existing statistical software, such as the Ox program. Bayesian inference is also discussed for the family of asymmetric models in terms of invariance with respect to the symmetric normal distribution showing that early results obtained for the normal distribution also holds for the asymmetric family. Results of a simulation study and an analysis of a real data set analysis are provided.     

On the central limit theorem for modulus trimmed sums

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter process where one of the parameters is time and the second one is the fraction of truncation.     

Pareto严格稳定分布在保险理赔中的应用

本文研究了Pareto严格稳定分布在保险中的应用.利用极大似然估计的方法得到了Pareto严格稳定分布,正态分布和Pareto分布的参数估计.根据信息准则,表明Pareto严格稳定分布能够较好地拟合保险数据.     

正态分布和瑞利分布混合情形下的参数估计及分类问题

学者往往用单一的分布模拟和拟合杂波,如正态分布、瑞利分布和威布尔分布等。然而在实际中,雷达杂波由多种类型的杂波组成,单一分布通常不能精确刻画雷达杂波规律,因此,应用混合分布模型对雷达杂波数据建模更准确。本文考虑用正态分布和瑞利分布的混合分布拟合杂波,并应用矩估计方法和基于EM算法的极大似然估计方法估计模型参数,最后,应用最大后验概率分类准则验证2种估计方法的分类准确率。通过数据模拟,得出极大似然估计的效果和分类准确率都要优于矩估计的估计效果和分类准确率。     

Method of Moments Using Monte Carlo Simulation

Abstract

We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the log-likelihood. We present a simple implementation using the Newton-Raphson algorithm with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to least squares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow “moments” that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by specifying moments of a distribution using prior information. We present two examples—specification of a multivariate prior distribution in a constrained-parameter family and estimation of parameters in an image model. The former example, used for an application in pharmacokinetics, motivated this work. This work is similar to Ruppert's method in stochastic approximation, combines Monte Carlo simulation and the Newton-Raphson algorithm as in Penttinen, uses computational ideas and importance sampling identities of Gelfand and Carlin, Geyer, and Geyer and Thompson developed for Monte Carlo maximum likelihood, and has some similarities to the maximum likelihood methods of Wei and Tanner.     

The exponentiated generalized inverse Gaussian distribution

The modeling and analysis of lifetime data is an important aspect of statistical work in a wide variety of scientific and technological fields. Good (1953) introduced a probability distribution which is commonly used in the analysis of lifetime data. For the first time, based on this distribution, we propose the so-called exponentiated generalized inverse Gaussian distribution, which extends the exponentiated standard gamma distribution (Nadarajah and Kotz, 2006). Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters. The usefulness of the new model is illustrated by means of a real data set.     

Bivariate Birnbaum–Saunders distribution and associated inference

Univariate Birnbaum–Saunders distribution has been used quite effectively to model positively skewed data, especially lifetime data and crack growth data. In this paper, we introduce bivariate Birnbaum–Saunders distribution which is an absolutely continuous distribution whose marginals are univariate Birnbaum–Saunders distributions. Different properties of this bivariate Birnbaum–Saunders distribution are then discussed. This new family has five unknown parameters and it is shown that the maximum likelihood estimators can be obtained by solving two non-linear equations. We also propose simple modified moment estimators for the unknown parameters which are explicit and can therefore be used effectively as an initial guess for the computation of the maximum likelihood estimators. We then present the asymptotic distributions of the maximum likelihood estimators and use them to construct confidence intervals for the parameters. We also discuss likelihood ratio tests for some hypotheses of interest. Monte Carlo simulations are then carried out to examine the performance of the proposed estimators. Finally, a numerical data analysis is performed in order to illustrate all the methods of inference discussed here.     

Pair-copula constructions of multiple dependence

Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.     

Gamma shared frailty model based on reversed hazard rate for bivariate survival data

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between the individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. In this paper, we suggest the shared gamma frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using MCMC technique to estimate the parameters involved in the model and compare the frailty model with the baseline model. We apply the proposed models to Australian twin data set and suggest a better model.