Gamma Mixing Distribution Research Articles

Shock models governed by an inverse gamma mixed Poisson process

AbstractWe study three classes of shock models governed by an inverse gamma mixed Poisson process (IGMP), namely a mixed Poisson process with an inverse gamma mixing distribution. In particular, we analyze (1) the extreme shock model, (2) the δ-shock model, and the (3) cumulative shock model. For the latter, we assume a constant and an exponentially distributed random threshold and consider different choices for the distribution of the amount of damage caused by a single shock. For all the treated cases, we obtain the survival function, together with the expected value and the variance of the failure time. Some properties of the inverse gamma mixed Poisson process are also disclosed.

Scale Mixtures of Frechet Distributions as Asymptotic Approximations of Extreme Precipitation

Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.

I-Delaporte process and applications

In this paper we introduce a mixed Pólya–Aeppli process with shifted gamma mixing distribution and call it an Inflated-parameter Delaporte process (I-Delaporte process). We derive the probability mass function, moments and some basic properties. Then we define the process as a pure birth process and derive differential equations for the probabilities. As application, we consider a risk model in which the claim counting process is the defined I-Delaporte process. For the defined risk model we derive the joint distribution of the time to ruin and the deficit at ruin as well as the ruin probability. We discuss in detail the particular case of exponentially distributed claims.

I-Delaporte Process and Applications

In this paper we introduce a mixed Polya-Aeppli process with shifted gamma mixing distribution and call it an Inated-parameter Delaporte process (I-Delaporte). We derive the pro bability mass function, recursion formulas and some basic properties. Then we de ne the process as a pure birth process and derive differential equations for the probabilities. As application, we consider a risk model in which the claim counting process is the de ned I-Delaporte process. For the defined risk model we derive the joint distribution of the time to ruin and the de cit at ruin as well as the ruin probability. We discuss in detail the particular case of exponentially distributed claims.

Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein–Uhlenbeck processes

Continuous superpositions of Ornstein–Uhlenbeck processes are proposed as a model for asset return volatility. An interesting class of continuous superpositions is defined by a Gamma mixing distribution which can define long memory processes. In contrast, previously studied discrete superpositions cannot generate this behaviour. Efficient Markov chain Monte Carlo methods for Bayesian inference are developed which allow the estimation of such models with leverage effects. The continuous superposition model is applied to both stock index and exchange rate data. The continuous superposition model is compared with a two-component superposition on the daily Standard and Poor’s 500 index from 1980 to 2000.

Negative binomial loglinear mixed models

The Poisson loglinear model is a common choice for explaining variability in counts. However, in many practical circumstances the restriction that the mean and variance are equal is not realistic. Overdispersion with respect to the Poisson distribution can be modeled explicitly by integrating with respect to a mixture distribution, and use of the conjugate gamma mixing distribution leads to a negative binomial loglinear model. This paper extends the negative binomial loglinear model to the case of dependent counts, where dependence among the counts is handled by including linear combinations of random effects in the linear predictor. If we assume that the vector of random effects is multivariate normal, then complex forms of dependence can be modelled by appropriate specification of the covariance structure. Although the likelihood function for the resulting model is not tractable, maximum likelihood estimates (and standard errors) can be found using the NLMIXED procedure in SAS or, in more complicated examples, using a Monte Carlo EM algorithm. An alternate approach is to leave the random effects completely unspecified and attempt to estimate them using nonparametric maximum likelihood. The methodologies are illustrated with several examples.

A Bayesian Approach to Modeling Purchase Frequency

Direct marketers are often faced with the task of ranking, or scoring individual customers in terms of their expected value to the firm. A critical element of their scoring systems is expected frequency of customer interaction. In this paper the authors develop a hierarchical Bayes model of purchase frequency that combines a Poisson likelihood with a gamma mixing distribution, where the mixing distribution is a function of covariates. The proposed model is evaluated with two direct marketing datasets, and is shown to provide improved estimates of purchase frequency, particularly for customers with short purchase histories or who have infrequent interaction with the firm.

Bayesian analysis of volcanic eruptions

The simple Poisson model generally gives a good fit to many volcanoes for volcanic eruption forecasting. Nonetheless, empirical evidence suggests that volcanic activity in successive equal time-periods tends to be more variable than a simple Poisson with constant eruptive rate. An alternative model is therefore examined in which eruptive rate(λ) for a given volcano or cluster(s) of volcanoes is described by a gamma distribution (prior) rather than treated as a constant value as in the assumptions of a simple Poisson model. Bayesian analysis is performed to link two distributions together to give the aggregate behavior of the volcanic activity. When the Poisson process is expanded to accomodate a gamma mixing distribution on λ, a consequence of this mixed (or compound) Poisson model is that the frequency distribution of eruptions in any given time-period of equal length follows the negative binomial distribution (NBD). Applications of the proposed model and comparisons between the generalized model and simple Poisson model are discussed based on the historical eruptive count data of volcanoes Mauna Loa (Hawaii) and Etna (Italy). Several relevant facts lead to the conclusion that the generalized model is preferable for practical use both in space and time.

A note on a limiting behaviour of the occurrence times of a mixed Poisson process

In a recent publication Pfeifer (1982) shows that for a P61ya-Lundberg birth process the limiting distribution of the sequence {n/T,}, where T, denotes the nth occurrence time of the process, is a gamma distribution. As the P61ya-Lundberg birth process is a mixed Poisson process with a gamma mixing distribution, Pfeifer's result implies that the limiting distribution of {n/T,} is just the mixing distribution of the process. A stronger result of this type is valid for every mixed Poisson process, which can be seen as follows. Given a random variable A whose distribution is the mixing distribution of the process, we notice, following Grandell ((1976), p. 12), that the process N(t) = N(tA), where N(t) is a Poisson process with unit intensity, is a version of the desired mixed Poisson process. Hence ( N(t) N(At) P(im = A t= = P lim -= At) = 11.

The analysis of residence histories and other longitudinal panel data: A continuous time mixed markov renewal model incorporating exogeneous variables

The analysis of residence histories and other longitudinal panel data is fraught with methodological problems. Much recent progress has been made in methods of analysis within discrete time. This paper extends the development of empirically tractable mixed continuous time stochastic models. Analysis of a sample of intra-urban residential histories identifies the effect of tenure type, age of household head, size of household and duration of stay on movement probabilities. Surprisingly, no further variation, as represented by a gamma mixing distribution over a hazard rate parameter, may be identified.