Estimation of parameters in the beta binomial model

This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970,Appl. Statist.,19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of ones has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.     

Testing the hypothesis on the “drift” of parameters in the moving average model

Test statistics for checking the conformity of the moving average model to observations are constructed. The statistics are based on sequential processes constructed using residuals and compatible with the equation for estimation of the unknown parameter of the model postulated. In the stationary case and in the case when the parameters of the model permit a drift with respect to time the limiting distributions of the test statistics are obtained.     

Time-fractional telegraph equations and telegraph processes with brownian time

We study the fundamental solutions to time-fractional telegraph equations of order 2. We are able to obtain the Fourier transform of the solutions for any and to give a representation of their inverse, in terms of stable densities. For the special case =1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.This research has been partially supported by the NATO grant No. SA (PST.CLG.976361) 5437.     

Estimation of the drift coefficient parameter in diffusion processes

The article considers the bias and the mean-square error of the maximum likelihood estimate of the drift coefficient parameter in diffusion processes. Weaker conditions than those in [1] are derived.     

Estimation of the parameters of an infectious disease model using neural networks

In this paper, we propose a realistic mathematical model taking into account the mutual interference among the interacting populations. This model attempts to describe the control (vaccination) function as a function of the number of infective individuals, which is an improvement over the existing susceptible–infective epidemic models. Regarding the growth of the epidemic as a nonlinear phenomenon we have developed a neural network architecture to estimate the vital parameters associated with this model. This architecture is based on a recently developed new class of neural networks known as co-operative and supportive neural networks. The application of this architecture to the present study involves preprocessing of the input data, and this renders an efficient estimation of the rate of spread of the epidemic. It is observed that the proposed new neural network outperforms a simple feed-forward neural network and polynomial regression.     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

An option pricing model based on jump telegraph processes

A new class of financial market models is developed. These models are based on generalized telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black-Scholes fundamental differential equation is derived, but, in contrast with the Black-Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters

We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution.     

Estimation of error variance in ANOVA model and order restricted scale parameters

We consider the estimation of error variance in the analysis of experiments using two level orthogonal arrays. We address the estimator which is the minimum of all the estimators which we obtain by pooling some sums of squares for factorial effects. Under squared error loss, we discuss whether or not this estimator uniformly improves upon the best positive multiple of error sum of squares. We show that when we have two factorial effects, we obtain uniform improvement. However, we show that when we have more than two factorial effects, we cannot necessarily obtain uniform improvement. Further, the above results are applied to the problem of estimating the smallest scale parameter of chi-square distributions.     

Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean

We construct a least squares estimator for the drift parameters of a fractional Ornstein Uhlenbeck process with periodic mean function and long range dependence. For this estimator we prove consistency and asymptotic normality. In contrast to the classical fractional Ornstein Uhlenbeck process without periodic mean function the rate of convergence is slower depending on the Hurst parameter H, namely \(n^{1-H}\).     

Estimation of an unobservable parameters in the type II constraints in singular linear regression model

In regular linear regression model with the type II constraints the unobservable parameters in the constraints can be estimated by the BLUE of the observable parameters. The BLUE respects the constraints. If the estimator of the observable parameters which is the BLUE in the model without constraints is used, then the estimator of the unobservable parameters is the same.     

Periodic solutions of the nonlinear telegraph equations with bounded nonlinearities

In this article, using the Leray-Schauder degree theory, we discuss existence, nonexistence and multiplicity for the periodic solutions of the nonlinear telegraph equation

utt−uxx+cut+Φ(u)=f(t,x)+s,     

Series solution of the telegraph equation

Polynomial solutions and other sets of particulr solutions of the telegrph equation are constructed, and their relations with the heat and wave polynomials are established. Regions of convergence for series in terms of these functions are obtained and certian intial value problems for the telegraph equation can be solved in terms of these functions.     

Estimation of the parameters for unstable AR models

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