Distributions slanted to the right

We solve Problem 234 in Statistica Neerlandica by introducing the concept of slantedness. Distributions with a decreasing Lebesgue density are slanted to the right. This is no longer true for distributions on a lattice with decreasing density. Both kinds of distributions have positive central odd moments.     

Maximum likelihood estimation of the double exponential jump-diffusion process

The double exponential jump-diffusion (DEJD) model, recently proposed by Kou (Manage Sci 48(8), 1086–1101, 2002) and Ramezani and Zeng (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=606361, 1998), generates a highly skewed and leptokurtic distribution and is capable of matching key features of stock and index returns. Moreover, DEJD leads to tractable pricing formulas for exotic and path dependent options (Kou and Wang Manage Sci 50(9), 1178–1192, 2004). Accordingly, the double exponential representation has gained wide acceptance. However, estimation and empirical assessment of this model has received little attention to date. The primary objective of this paper is to fill this gap. We use daily returns for the S&;P-500 and the NASDAQ indexes and individual stocks, in conjunction with maximum likelihood estimation (MLE) to fit the DEJD model. We utilize the BIC criterion to assess the performance of DEJD relative to log-normally distributed jump-diffusion (LJD) and the geometric brownian motion (GBM). We find that DEJD performs better than these alternatives for both indexes and individual stocks.     

Probabilistic Models for Bacterial Taxonomy

We give a survey of different partitioning methods that have been applied to bacterial taxonomy. We introduce a theoretical framework, which makes it possible to treat the various models in a unified way. The key concepts of our approach are prediction and storing of microbiological information in a Bayesian forecasting setting. We show that there is a close connection between classification and probabilistic identification and that, in fact, our approach ties these two concepts together in a coherent way.     

An approximation to the limit distribution of the epps-pulley test statistic for normality

Summary The approach of Epps and Pulley (1983) based on the empirical characteristic function is one of the most powerful tools for detecting any departures from normality. We obtain the first four moments of the limiting null distribution of the Epps-Pulley Statistic. Johnson- and Pearson curve fitting yields excellent approximations to simulated quantiles, and by modifying the test statistic the procedure may be carried out easily without the use of extensive tables for all sample sizes. Research done while the author was on leave at the University of Gie?en.     

Pricing of a structured product on the SX5E when the uncertainty of returns is modeled as a log-stable process

This work presents the participation factor and the valuation of a first-generation structured product with European call options on the Eurostoxx, when the uncertainty of the yields is modeled through log-stable processes. The basic statistics of the index yields are also exposed, the α-stable parameters are estimated, and the valuation of the of the structured models is compared through the log-stable and log-Gaussian models using inputs from the bond markets; concluding that investors obtain higher yields than those of the bond market through both models, and that the differences of the yields depend on the participation factor and on the value of the index at the time of liquidation.     

THE BIVARIATE EXPONENTIAL DISTRIBUTION; A REVIEW AND SOME NEW RESULTS

Various models have been proposed as bivariate forms of the exponential distribution. A brief but comprehensive review is presented which classifies, interrelates and contrasts the different models and outlines what is known about distributional properties, applicability and estimation and testing of parameters (particularly the association parameter). Some new results are presented for one particular model. Maximum likelihood, and moment–type, estimators of the association parameter are examined. Asymptotic variances are derived and attention is given to the relative efficiency of the estimators and to problems of their evaluation.     

Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.     

Estimating Discrete Joint Probability Distributions for Demographic Characteristics at the Store Level Given Store Level Marginal Distributions and a City-Wide Joint Distribution

This paper provides a solution to the problem of estimating a joint distribution using the associated marginal distributions and a related joint distribution. The particular application we have in mind is estimating joint distributions of demographic characteristics corresponding to market areas for individual retail stores. Marginal distributions are generally available at the census tract level, but joint distributions are only available for Metropolitan Statistical Areas which are generally much larger than the market for a single retail store. Joint distributions over demographics are an important input into mixed logit demand models for aggregate data. Market shares that vary systematically with demographics are essential for relieving the restrictions imposed by the Independence from Irrelevant Alternative property of the logit model.We approach this problem by formulating a parametric function that incorporates both the city-wide joint distributional information and marginal information specific to the retail stores market area. To estimate the function, we form moment conditions equating the moments of the parametric function to observed data, and we input these into a GMM objective. In one of our illustrations we use four marginal demographic distributions from each of eight stores in Dominicks Finer Foods data archive to estimate a four dimensional joint distribution for each store. Our results show that our GMM approach produces estimated joint distributions that differ substantially from the product of marginal distributions and emit marginals that closely match the observed marginal distributions. Mixed logit demand estimates are also presented which show the estimates to be sensitive to the formulation of the demographics distribution.The views expressed are not purported to reflect those of the United States Department of Justice     

Unconditional distributions obtained from conditional specification models with applications in risk theory

Bivariate distributions, specified in terms of their conditional distributions, provide a powerful tool to obtain flexible distributions. These distributions play an important role in specifying the conjugate prior in certain multi-parameter Bayesian settings. In this paper, the conditional specification technique is applied to look for more flexible distributions than the traditional ones used in the actuarial literature, as the Poisson, negative binomial and others. The new specification draws inferences about parameters of interest in problems appearing in actuarial statistics. Two unconditional (discrete) distributions obtained are studied and used in the collective risk model to compute the right-tail probability of the aggregate claim size distribution. Comparisons with the compound Poisson and compound negative binomial are made.     

Option Pricing with Heterogeneous Expectations

This paper re-derives the finite mixture option pricing model of Ritchey (1990), based on the assumption that the option investors hold heterogeneous expectations about the parameters of the lognormal process of the underlying asset price. By proving that the model admits no riskless arbitrage, this paper justifies that the entire family of finite mixture of lognormal distributions is a desirable candidate set for recovering the risk-neutral probability distributions from contemporaneous options quotes. The parametric method derived from the model is significantly simpler than the nonparametric method of Rubinstein (1994) for recovering the risk-neutral probability distributions from contemporaneous option prices.