A product Pareto distribution

The Pareto distributions are becoming increasing prominent in several applied areas. In this note, a new Pareto distribution is introduced. It takes the form of the product of two Pareto probability density functions. Various structural properties of this distribution are derived, including its cumulative distribution function, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, method of moments estimates, maximum likelihood estimates and the Fisher information matrix. The calculations involve the use of several special functions.     

On the two-sided power distribution

This paper concerns a two-parameter two-sided power distribution introduced by van Dorp and Kotz on the interval [0,1]. We introduce a reformulated two-sided power distribution (with the same number of parameters) and provide evidence to prove that it is more flexible than the one suggested by van Dorp and Kotz. We derive various properties of the new distribution as well as provide several hitherto unknown properties of the distribution due to van Dorp and Kotz. We also discuss estimation by the method of moments and the method of maximum likelihood.Received September 2003     

Sociological Models Based on Fréchet Random Variables

Products of random variables are of both practical and theoretical significance to social scientists. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product XY is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.     

Models for stock returns

Historically, the normal variance model has been used to describe stock return distributions. This model is based on taking the conditional stock return distribution to be normal with its variance itself being a random variable. The form of the actual stock return distribution will depend on the distribution for the variance. In practice, the distributions chosen for the variance appear to be very limited. In this note, we derive a comprehensive collection of formulas for the actual stock return distribution, covering some sixteen flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. We feel that this work could serve as a useful reference and lead to improved modelling with respect to stock market returns.     

A note on “Modelling exchange rate returns: which flexible distribution to use?”

Corlu and Corlu [Quant. Finance, 2014, doi: 10.1080/14697688.2014.942231] provided a novel modelling of exchange rate data for nine currencies using five flexible distributions. They stated that the generalized lambda, skew t and normal inverse Gaussian distributions ‘do a good job’. Here, we reanalyse the data and show that a distribution simpler than all of these fits at least as well as these distributions. We also find that the normal inverse Gaussian distribution provides good fits for only one of the data-sets.     

Compound Mixed Poisson Distributions II

Market cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process: a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied.     

Compound mixed Poisson distributions I

The concept of compound mixed Poisson distributions in actuarial science is used to represent such variables as the total amount of claims or losses payable by an insurer. In this paper, comprehensive collections of approximate forms are derived for the compound mixed Poisson distribution. The calculations involve use of several special functions and their properties. We believe that the results will serve as an important reference in actuarial science.     

A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model $$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$ for ?1 < β ^p ≤ 0 ≤ α ^p  < 1 and α ^p ? β ^p  > 0 when X _t has the two-parameter beta distribution B₂(p, q) with parameters q > 1 and ${p \in \mathcal P(u,v)}$ , where $$\mathcal P(u,v) = \left\{u/v : u < v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p \in \mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model.     

Asymptotic properties of M-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in \(\mathbb{R }^q\) with shift parameter \(\alpha \) in \(\mathbb{R }^q\) and other parameters \(\beta \) in \(\mathbb{R }^p\) . Residuals are assumed to be from an unknown distribution function (d.f.). Let \(\widehat{\phi }\) be a smooth \(M\) -estimator of \(\phi = {{\beta }\atopwithdelims (){\alpha }}\) and \(T(\phi )\) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of \(T(\widehat{\phi })\) and an estimator of \(T(\phi )\) with bias \(\sim n^{-2}\) requiring \(\sim n\) calculations. (In contrast, the jackknife and bootstrap estimators require \(\sim n^2\) calculations.) For a linear regression with random covariates of low skewness, if \(T(\phi ) = \nu \beta \) , then \(T(\widehat{\phi })\) has bias \(\sim n^{-2}\) (not \(n^{-1}\) ) and skewness \(\sim n^{-3}\) (not \(n^{-2}\) ), and the usual approximate one-sided confidence interval (CI) for \(T(\phi )\) has error \(\sim n^{-1}\) (not \(n^{-1/2}\) ). These results extend to random covariates.