Dexter O. Cahoy

FRACTIONAL PROCESSES: FROM POISSON TO BRANCHING ONE

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.

A bootstrap test for equality of variances

We introduce a bootstrap procedure to test the hypothesis H o that K + 1 variances are homogeneous. The procedure uses a variance-based statistic, and is derived from a normal-theory test for equality of variances. The test equivalently expressed the hypothesis as H o : ? = ( ? 1 , ? , ? K + 1 ) T = 0 , where ? i s are log contrasts of the population variances. A box-type acceptance region is constructed to test the hypothesis H o . Simulation results indicated that our method is generally superior to the Shoemaker and Levene tests, and the bootstrapped version of the Levene test in controlling the Type I and Type II errors.

Simulation and estimation for the fractional Yule process

In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications.

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.

A statistical reevaluation of the data used in the Lanphear et al. () pooled-analysis that related low levels of blood lead to intellectual deficits in children

Abstract A pooled-analysis by Lanphear et al. (2005) of seven cohort studies of the association between blood lead (BPb) concentrations in children and measures of their intelligence concluded that “environmental lead exposure in children who have maximal blood lead levels <7.5 μg/dL is associated with intellectual deficits.” This study has played a prominent role in shaping the public understanding of the effects upon children’s IQ of low BPb exposures (e.g., BPb ≤ 10 μg/dL). Here we present a reanalysis of the data used by Lanphear et al. to evaluate the robustness of their conclusions. Our analysis differed from that of Lanphear et al. primarily in how we controlled for non-lead variables (allowing a number of them to be site-specific), how we defined summary measures of BPb exposure, and in how we decided which BPb measures and transformations best modeled the data. We also reproduced the Lanphear et al. analysis. Although we found some small errors and questionable decisions by Lanphear et al. that, taken alone, could cause doubt in their conclusions, our reanalysis tended to support their conclusions. We concluded that there was statistical evidence that the exposure-response is non-linear over the full range of BPb evaluated in these studies, which implies that, for a given increase in blood lead, the associated IQ decrement is greater at lower BPb levels. However at BPb below 10 µg/dL, the exposure-response is adequately modeled as linear. We also found statistical evidence for an association with IQ among children who had maximal measured BPb levels ≤7 μg/dL, and concurrent BPb levels as low as ≤5 μg/dL.

Parameter estimation for fractional birth and fractional death processes

The fractional birth and the fractional death processes are more desirable in practice than their classical counterparts as they naturally provide greater flexibility in modeling growing and decreasing systems. In this paper, we propose formal parameter estimation procedures for the fractional Yule, the fractional linear death, and the fractional sublinear death processes. The methods use all available data possible, are computationally simple and asymptotically unbiased. The procedures exploited the natural structure of the random inter-birth and inter-death times that are known to be independent but are not identically distributed. We also showed how these methods can be applied to certain models with more general birth and death rates. The computational tests showed favorable results for our proposed methods even with relatively small sample sizes. The proposed methods are also illustrated using the branching times of the plethodontid salamanders data of (Syst. Zool. 28:579–599, 1979).

Renewal processes based on generalized Mittag–Leffler waiting times

Abstract The fractional Poisson process has recently attracted experts from several fields of study. Its natural generalization of the ordinary Poisson process made the model more appealing for real-world applications. In this paper, we generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag–Leffler function in the kernel. The waiting times of the proposed renewal processes have the generalized Mittag–Leffler and stretched–squashed Mittag–Leffler distributions. Note that the generalizations naturally provide greater flexibility in modeling real-life renewal processes. Algorithms to simulate sample paths and to estimate the model parameters are derived. Note also that these procedures are necessary to make these models more usable in practice. State probabilities and other qualitative or quantitative features of the models are also discussed.

Estimation of Mittag-Leffler Parameters

We propose a procedure for estimating the parameters of the Mittag-Leffler (ML) and the generalized Mittag-Leffler (GML) distributions. The algorithm is less restrictive, computationally simple, and necessary to make these models usable in practice. A comparison with the fractional moment estimator indicated favorable results for the proposed method.

On a Fractional Binomial Process

The classical binomial process has been studied by Jakeman (J. Phys. A 23:2815–2825, 1990) (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart in describing real physical processes. More statistical properties are also derived.

Transient Behavior of Fractional Queues and Related Processes

We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M/1 queue process and the related linear birth-death process are provided. Closed-form expressions of the point and interval estimators of the parameters of the proposed fractional stochastic models are also presented. These methods are necessary to make these models usable in practice. The proposed fractional M/M/1 queue model and the statistical methods are illustrated using financial data.