Scale estimators for symmetric stable distributions

This paper reports on a simulation comparison of scale estimators for symmetric stable distributions in terms of their ability to identify the population with the greater scale. The modified geometric mean is found to be superior to the sample standard deviation and the Fama-Roll estimator for the larger values of the characteristic exponent, while the Fama-Roll estimator is judged superior for the smaller values. Further, this study sheds some light on the question of the appropriate sample size for discriminating risk measurement in investment analysis when the samples are taken from symmetric stable distributions.     

A note on the Cauchy-type mixture distributions

An interesting class of continuous distributions, called Cauchy-type mixture, with potential applications in modelling erratic phenomena is introduced by Soltani and Tafakori [A class of continuous kernels and Cauchy type heavy tail distributions. Statist Probab Lett. 2013;83:1018–1027]. In this work, we provide more insights into the Cauchy-type mixture distributions, involving certain characterizations, connections with the generalized Linnik distributions and the class of discrete distributions induced by stable laws. We also prove that the Laplace transform of Cauchy-type mixture distributions when normalized by constant terms become as a density functions in terms of distributional conjugate property.     

Multi-modal tempered stable distributions and prosses with applications to finance

Abstract

The assumption of underlying return distribution plays an important role in asset pricing models. While the return distribution used in the traditional theories of asset pricing is the unimodal distribution, numerous studies which have investigated the empirical behavior of asset returns in financial markets use multi-modal distribution. We introduce a new parsimonious multi-modal distribution, referred to as the multi-modal tempered stable (MMTS) distribution. In this article we also generate the exponential Lévy market models and derive the value-at-risk (VaR) induced from them. To demonstrate the advantages, we will present the results of the parameter estimation and the VaRs for financial data.     

Triple sampling procedure for estimating the scale parameters of pareto distributions

This paper sheds light on the large sample performance of the three stage sam- pling procedure, as it pertains to estimating the scale parameter(s) of the Pareto distribution(s). This group sampling procedure merges the efficiency of the purely sequential procedure of Anscombe (1953) and Chow and Robbins (1965) with substan-tial savings in the number of sampling operations, as noted by Hall (1981). Both its simplicity and its economical features provide visible advantages over the one-by-one sampling as an alternative. In this paper we develop some asymptotic properties for the final stage sample size of the triple stage sampling originated by Hall (1981). These results are used to study both the point and the interval estimation problems for the scale parameters of the Pareto distributions. Since our results are asymptotic in nature, a simulation study is given to discuss the moderate sample size peformance of the proposed procedures.     

Minimum variance unbiased estimation in natural exponential families generated by stable distributions

The class of nature exponential families generated by stable distributions has been introduced in different contexts by several authors. Tweedie (1984) and Jorgensen (1987) studied this class in the context of generalized liner models and exponential dispersion models. Bar-Lev and Enis (1986) introduced this class in the context of the property of reproducibility in natural exponential families and Hougaard (1986) found the distributions in this class to be natural candidates for applications as survival distributions in life tables for heterogeneous populations. In this note, we consider such a class in the context of minimum variance unbiased estimation. For each family in this class, we obtain an explicit expression for the uniformly minimum variance unbiased estimator for the r-th cumlant, the density function, and the reliability function.     

Asymptotic expansions for the distributions of statistics based on a correlation matrix

An asymptotic expansion is given for the distribution of the α-th largest latent root of a correlation matrix, when the observations are from a multivariate normal distribution. An asymptotic expansion for the distribution of a test statistic based on a correlation matrix, which is useful in dimensionality reduction in principal component analysis, is also given. These expansions hold when the corresponding latent root of the population correlation matrix is simple. The approach here is based on a perturbation method.     

Further asymptotic expansions of the distribution of the sphericity test

In this paper new asymptotic expansions of the distributions of the sphericity test criterion are obtained in the null and the non-null case when the alternatives are close to the hypothesis. These expansions are obtained for the first time in terms of beta distributions. These appear to be better than the ones available in the literature.     

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.     

Umvu estimation for certain family of exponential distributions

In this paper we study certain properties of estimable and UMUV-estimable functions in a subfamily of the one-parameter exponential family of distributions for which there exists a sufficient and complete statistic following a Gamma distribution. These results are applied to the problem of estimation in the transformed chi-square family.     

A Moment Preserving Finitization Across the Power Series Family of Probability Distributions

Finitization transforms a discrete distribution into a distribution with smaller support of specified size. In special cases finitization preserves moments (moments of the order n finitization coincide with those of the parent distribution). We create a moment preserving finitization method for power series distributions by introducing an alternative representation and showing how to finitize members of this new class in a manner that preserves moments of the parent distribution. We provide results on convolutions and a reproductive property for power series distributions that have been finitized in this manner, and show how these finitized distributions accelerate variate generation in simulation.     

Estimation for seasonal fractional ARIMA with stable innovations via the empirical characteristic function method

Seasonal fractional ARIMA (ARFISMA) model with infinite variance innovations is used in the analysis of seasonal long-memory time series with large fluctuations (heavy-tailed distributions). Two methods, which are the empirical characteristic function (ECF) procedure developed by Knight and Yu [The empirical characteristic function in time series estimation. Econometric Theory. 2002;18:691–721] and the Two-Step method (TSM) are proposed to estimate the parameters of stable ARFISMA model. The ECF method estimates simultaneously all the parameters, while the TSM considers in the first step the Markov Chains Monte Carlo–Whittle approach introduced by Ndongo et al. [Estimation of long-memory parameters for seasonal fractional ARIMA with stable innovations. Stat Methodol. 2010;7:141–151], combined with the maximum likelihood estimation method developed by Alvarez and Olivares [Méthodes d'estimation pour des lois stables avec des applications en finance. Journal de la Société Française de Statistique. 2005;1(4):23–54] in the second step. Monte Carlo simulations are also used to evaluate the finite sample performance of these estimation techniques.     

Estimation of the parameters of symmetric stable ARMA and ARMA–GARCH models

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.     

Edgeworth expansions for the conditional distributions in logistic regression models

Edgeworth expansions are derived for conditional distributions of sufficient statistics as well as conditional maximum likelihood estimators of log odds ratios in logistic regression models assuming that the risk factors are not almost equally distanced. Expansions are given in several special cases. Similar results are obtained for models with polytomous outcomes.     

Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures

An asymptotic expansion of the null distribution of the chi-square statistic based on the asymptotically distribution-free theory for general covariance structures is derived under non-normality. The added higher-order term in the approximate density is given by a weighted sum of those of the chi-square distributed variables with different degrees of freedom. A formula for the corresponding Bartlett correction is also shown without using the above asymptotic expansion. Under a fixed alternative hypothesis, the Edgeworth expansion of the distribution of the standardized chi-square statistic is given up to order O(1/n). From the intermediate results of the asymptotic expansions for the chi-square statistics, asymptotic expansions of the joint distributions of the parameter estimators both under the null and fixed alternative hypotheses are derived up to order O(1/n).     

Some orthogonal expansions for the logistic distribution

We expand a continuous random variable as a sum of a sequence of un-correlated random variables. These variables are principal components of a Bernoulli process, as well as principal dimensions in continuous metric scaling on a particular distance function. We obtain expansions for the uniform, exponential and logistic distributions. A goodness-of-fit application is given.     

Asymptotic normality for chi-bar-square distributions

Chi-bar-square distributions, which are mixtures of chi-square distributions, mixed over their degrees of freedom, often occur when testing hypotheses that involve inequality constraints. Here, necessary and sufficient conditions on the mixing or weighting distribution are found to ensure asymptotic normality of the corresponding chi-bar-square distribution. Essentially, asymptotic normality occurs for the chi-bar-square distribution if either the ratio of the mean to the variance of the mixing distribution goes to infinity, or the weighting distribution itself is asymptotically normal. Other than a combination of these two phenomena, this is also the only way for asymptotic normality to hold. Several examples of pertinent chi-bar-square distributions are shown to be asymptotically normal by the results in this paper.     

Regression with an infinite number of observations applied to estimating the parameters of the stable distribution using the empirical characteristic function

ABSTRACT

The parameters of stable law parameters can be estimated using a regression based approach involving the empirical characteristic function. One approach is to use a fixed number of points for all parameters of the distribution to estimate the characteristic function. In this work the results are derived where all points in an interval is used to estimate the empirical characteristic function, thus least squares estimators of a linear function of the parameters, using an infinite number of observations. It was found that the procedure performs very good in small samples.     

Moment inequalities for truncated multivariate distributions

In this note some properties of the absolute moments of a doubly truncated arbitrary multivariate distribution are studied and several moment inequalities are derived.     

A generalization for two-sided power distributions and adjusted method of moments

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.     

On the Harris extended family of distributions

A new method for generating new classes of distributions based on the probability-generating function is presented in Aly and Benkherouf [A new family of distributions based on probability generating functions. Sankhya B. 2011;73:70–80]. In particular, they focused their interest to the so-called Harris extended family of distributions. In this paper, we provide several general results regarding the Harris extended models such as the general behaviour of the failure rate function. We also derive a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution. Additionally, some stochastic order relations are established and limiting distributions of sample extremes are also considered for this model. These general results are illustrated in several special Harris extended models. Finally, we discuss estimation of the model parameters by the method of maximum likelihood and provide an application to real data for illustrative purposes.