On A Surplus Process Under A Periodic Environment

Abstract

The problem of modeling claims occurring in periodic random environments is discussed in this paper. In the classical approach of risk theory, the occurrence of claims is modeled by counting processes that do not account for claims following a periodic pattern. The author discusses how the use of the classical approach to model a periodic portfolio might lead to the miscalculation of important risk indices, namely the associated ruin probability.

He presents a periodic model, in terms of nonhomogeneous Poisson processes, that has potential practical applications. The discussion is based on some properties of the modeled periodic intensities. Existing simulation techniques are adapted to this periodic model, which provides a practical way to evaluate ruin probabilities.     

On The Expected Discounted Penalty function for Lévy Risk Processes

Abstract

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes.

In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.     

Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n / m _n →γ∈(0,∞), where n denotes sample size and m _n the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.     

Uses of qualitative/quantitative terms in social and educational research

There are an increasingly greater number of authors who support the use of qualitative methodology in their research (Glaser & Strauss, 1967; Goetz & Lecompte, 1980; Bogdan & Bliken, 1982; Firestone & Herriot, 1983; Howe, 1985; Filstead, 1986, etc.). However, there are problems offering a clear conceptualization of the above mentioned notion owing to the fact that often qualitative (Q) and quantitative (C) terms are used without specifying their concrete meaning, or are even referred to polysemically. Furthermore, authors even differ in the identification of the different uses of the terms in question. With the aim of solving these problems, we thought it suitable to carry out an exploratory content analysis of the characteristics given by authors to the terms C and Q. The results illustrate that Reichardt and Cook's proposal (1986), although representing a significant advance in the field, does not go far enough, given that it does not gather all the possible uses that authors attribute to the terms C and Q. This may, in our opinion, be considered as proof of the need to search for new criteria and/or reformulate the already existing ones.     

Extension of the Wald statistic to models with dependent observations

A generalization of the Wald statistic for testing composite hypotheses is suggested for dependent data from exponential models which include Lévy processes and diffusion fields. The generalized statistic is proved to be asymptotically chi-squared distributed under regular composite hypotheses. It is simpler and more easily available than the generalized likelihood ratio statistic. Simulations in an example where the latter statistic is available show that the generalized Wald test achieves higher average power than the generalized likelihood ratio test. Received: February 29, 2000