P. Vellaisamy

Fractional Cauchy problems on bounded domains

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain DR d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordi- nator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brow- nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. 1. Introduction. In this paper, we extend the approach of Meerschaert and Scheffler ( 23) and Meerschaert et al. (24) to fractional Cauchy problems on bounded domains. Our methods involve eigenfunction expansions, killed Markov processes and inverse stable subordinators. In a recent related paper (7), we establish a connection between fractional Cauchy problems with index β = 1/2 on an unbounded domain, and iterated Brownian motion (IBM), defined as Zt = B(|Yt|), where B is a Brownian motion with values in R d and Y is an independent one-dimensional Brownian motion. Since IBM is also the stochastic solution to a Cauchy problem involving a fourth-order derivative in space (2, 14), that paper also establishes a connection between certain higher-order Cauchy problems and their time-fractional analogues. More generally, Baeumer, Meerschaert and Nane (7) shows a connection between fractional Cauchy problems with β = 1/2 and higher-order Cauchy problems that involve the square of the generator. In the present paper, we

Distributed-order fractional diffusions on bounded domains

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. This paper provides explicit strong solutions and stochastic analogues for distributed-order time-fractional diffusion equations on bounded domains, with Dirichlet boundary conditions.

Sequelae of prospective versus retrospective reports of adverse childhood experiences.

Retrospective assessment of adverse childhood experiences is widely used in research, although there are concerns about its validity. In particular, recall bias is assumed to produce significant artifacts. Data from a longitudinal cohort (the British National Child Development Study; N = 7,710) and the retrospective Mainz Adverse Childhood Experiences Study (N = 1,062, Germany) were compared on 10 adverse childhood experiences and psychological adjustment at age 42 yr. Between the two methods, no significant differences in risk effects were detected. Results held for bivariate analyses on all 10 childhood adversities and a multivariate model; the later comprises the childhood adversities which show significant long-term sequelae (not always with natural parent, chronically ill parent, financial hardship, and being firstborn) and three covariates. In conclusion, the present data did not show any bias in the retrospective assessment.

Time-changed Poisson processes

We consider time-changed Poisson processes, and derive the governing difference–differential equations (DDEs) for these processes. In particular, we consider the time-changed Poisson processes where the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDEs. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDEs corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index 0<β<1, when β is a rational number. We then use this result to obtain the governing DDE for the mass function of the Poisson process time-changed by the tempered stable subordinator. Our results extend and complement the results in Baeumer et al. (2009) and Beghin and Orsingher (2009) in several directions.

Estimation of the mean of the selected gamma population

Let л1 and л2 denote two independent gamma populations G(α1, p) and G(α2, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Yi denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Yi is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c1x(1) +c1x(2) and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)-1x(1) is minimax and dominates the UMVUE. Also UMVUE is not minimax.

On compound Poisson approximation for sums of random variables

An upper bound for the total variation distance between the distribution of the sum of a sequence of r.v.s and that of a compound Poisson is derived. Its applications to a general independent sequence and Markov-binomial sequence are demonstrated.

A note on the estimation of the selected scale parameters

Abstract Suppose a subset of the given k gamma populations, with unknown scale parameters and a common known shape parameter, is selected using Guptas subset selection procedure based on unequal sample sizes. The problem of estimating the scale parameters associated with the selected populations is considered. The natural estimators as well as the unbiased estimator are shown to be inadmissible and are improved by the technique of solving differential inequalities on the sample space. As a special case, we obtain an estimator dominating the UMVUE of θ (1) , the scale parameter of the population corresponding to the largest observation X (1) . The inadmissibility of the UMVUE of θ ( k ) , the scale parameter associated with the smallest observation X ( k ) , is also established.

On the number of successes in dependent trials

A new approach to the study of the distributions of sums of n Bernoulli variables by conditional distributions is considered, This leads to a characterization of binomial distribution, and provides a simple approach to the study of generalized binomial distributions. We show that the binomial distribution and Poissons binomial distribution can be derived as the distribution of sum of dependent Bernoulli random variables. Finally the problem of Poisson approximation to the generalized binomial distribution is investigated.

On UMVU estimation following selection

In this paper, we obtain some results concerning the UMVUE (uniformly minimum variance unbiased estimator ) of the selected parameter say, for example, the mean. For the squared error loss, the conditions under which the UMVUE is also a UMMSEUE(uniformly minimum mean squared error unbiased estimator), are Investigated, As an application, the UMVUE of θ

Transient anomalous sub-diffusion on bounded domains

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