This paper is concerned with methods of sample size determination. The approach is to cover a small number of simple problems, such as estimating the mean of a normal distribution or the slope in a regression equation, and to present some key techniques. The methods covered are in two groups: frequentist and Bayesian. Frequentist methods specify a null and alternative hypothesis for the parameter of interest and then find the sample size by controlling both size and power. These methods often need to use prior information but cannot allow for the uncertainty that is associated with it. By contrast, the Bayesian approach offers a wide variety of techniques, all of which offer the ability to deal with uncertainty associated with prior information.

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar meanvariance frontier, and which may be implemented using quadratic programming.

That the returns on financial assets and insurance claims are not well described by the multivariate normal distribution is generally acknowledged in the literature. This paper presents a review of the use of the skew-normal distribution and its extensions in finance and actuarial science, highlighting known results as well as potential directions for future research. When skewness and kurtosis are present in asset returns, the skew-normal and skew-Student distributions are natural candidates in both theoretical and empirical work. Their parameterization is parsimonious and they are mathematically tractable. In finance, the distributions are interpretable in terms of the efficient markets hypothesis. Furthermore, they lead to theoretical results that are useful for portfolio selection and asset pricing. In actuarial science, the presence of skewness and kurtosis in insurance claims data is the main motivation for using the skew-normal distribution and its extensions. The skew-normal has been used in studies on risk measurement and capital allocation, which are two important research fields in actuarial science. Empirical studies consider the skew-normal distribution because of its flexibility, interpretability, and tractability. This paper comprises four main sections: an overview of skew-normal distributions; a review of skewness in finance, including asset pricing, portfolio selection, time series modeling, and a review of its applications in insurance, in which the use of alternative distribution functions is widespread. The final section summarizes some of the challenges associated with the use of skew-elliptical distributions and points out some directions for future research. C. Adcock et al.those of Arditti and Levy (1975) and Kraus and Litzenberger (1976), marks the point at which the modern theory of finance started revealing evidence of skewness analogous to that provided by Markowitz, Sharpe, Lintner, Tobin, and others.In the context of actuarial science, it is important to recognize that insurance risks have skewed distributions (see, for example, Lane 2000), which is why in many cases, the classical normal distribution is an inadequate model for insurance risks or losses. Some insurance risks also exhibit heavy tails, especially those exposed to catastrophes (see Embrechts, McNeil, and Straumann 2002). The skew-normal distribution and its extensions thus might be promising since they preserve the advantages of the normal distribution with the additional benefit of flexibility with regard to skewness and kurtosis. The skew-normal distribution and its extensions have been used recently in studies on risk management, capital allocation, and goodness-of-fit (Vernic 2006;Bolance et al. 2008;Eling 2012).This review paper is motivated by the belief that asymmetry in asset returns, and skewness in particular, is an important field of research. Such research has two clear themes. The first is the never-ending and entirely natural wish to develop better univariate models for asset returns. Such model developme...

In this paper we develop rules for determining sample sizes based on the idea of a fixed precision d at a given level of probability in the posterior distribution of the parameter of interest, θ say, which holds on the average over all possible samples x. We restrict ourselves to the important case where R(x) is a symmetric region about the posterior mean E(θ x) and require that the expectation taken over all samples, x, of the quantity Pr[θ ∈ R(x)] equals 1–α This corresponds to the sampling theory formulation and leads to results which are directly comparable with classical univariate results. The paper develops rules for the determination of sample size for Normal distributions.

No abstract