A. N. V. Rao

Estimation of failure intensity for the Weibull process

A Weibull process/non-homogeneous Poisson process is commonly used to analyze the failure behavior of repairable systems. The object of the present study is to obtain exact estimates of the failure intensity of this model at the time of n failures. The resulting MLE estimate is biased and the corrected version for biasedness along with some approximate estimates is given. An analytical and numerical comparison of the relative efficiencies of the MLE of the exact biased, approximated biased, exact unbiased and approximated unbiased of the intensity function is presented. It will be shown that for small n (n < 30) there is quite a large relative difference between the mean squared errors of the exact and approximate estimates of the intensity function. Real failure data are used to illustrate the difference between the exact and approximate estimates of the intensity function.

Solution of a Stochastic Integral Equation Using Integral Contractors

A general stochastic integral equation of the form x(t; ω) = h(t; ω) + ∫0tƒ(t, s, x(s; ω); ω) ds + ∫0tg(t, s, x(s; ω); ω) dβ(s; ω) is studied, where t ⩾ 0, ω e Ω, ( Ω , A , P ) is a complete probability space, and β(t; ω) is a Brownian motion process. The concept of a bounded integral vector contractor is utilized to obtain very general conditions for the existence of solutions to the stochastic integral equation. The existence theorems are then applied to give stability results.

On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation

The purpose of the present paper is to study a perturbed nonlinear stochastic integro-differential equation of the form x ′ ( t ; ω ) = h ( t , x ( t ; ω ) ) − ∫ 0 t k 1 ( t , τ ; ω ) f 1 ( τ , x ( τ ; ω ) ) d τ + ∫ 0 t k 2 ( t , τ ; ω ) f 2 x ( τ ; ω ) ) d β ( τ ; ω ) , t ⩾ 0 , , where ′ denotes the sample path derivitive; w , the sample point of a complete probability measure space ( ω, A, P ) and β ( t ; ω ), a martingale adapted to an increasing family of sub- σ -algebras A t in A . Conditions which guarantee the existence and uniqueness of a random solution are obtained using functional analytic techniques. Also, the stability and boundedness of the second moments of the random solution are discussed, along with an application to a stochastic feedback system.

On the existence of a random solution to a nonlinear perturbed stochastic integral equation

There are two basic classes of random or stochastic integral equations cur ren t ly under s tudy by probabilists and mathemat ical statisticians. Those integral equations involving Ito or Ito-Doob type stochastic integrals and those which can be considered as probabilistic analogues of classical determinist ic integral equations whose formulation involves only the Lebesque integral. I t has been shown [1]-[4], [6]-[11], [13], [14], [16], [17], among others tha t the theory of random integral equations of the l a t t e r ca tegory are ex t remely important in stochastically character izing many physical situations in life sciences and engineering. In this paper we shall be concerned with a stochastic integral equation of the Ito-Doob type. We shall be concerned with a nonlinear stochastic integral equation of the form given by

Statistical properties of a linear stochastic system

In this paper a random linear system of the form of y(t; ω = ∫ t ∁ K(t, τ; ω)x(τ; ω)dr is studied, where the kernel is a stochastic process defined on a probability space. The concept of the modified characteristic function for the output process is introduced. These characteristic functions are used to identify the distribution of the output process over certain subsets of the probability space, Ω , in order to study the statistical properties of the process. Several examples are given to illustrate the usefulness of the resulting theory. These results extend the previous theory of random linear systems, in that until now, the kernel was deterministic in nature.

Maximum likelihood estimators for bivariate distributions with monotone failure rates

The object of the present paper is to develop maximum likelihood estimates of a vector hazard rate under the characterization of a bivariate piecewise exponential distribution for a given bivariate failure density function.In addition, the consistency of these estimators have been investigated

Stochastic systems and integral inequalities

A stochastic control system of the form dx=F (t,x) dt+G (t,x) dβ (t) +ζ1(t) dt+ζ2(t) dβ (t), t ∈ R+≡ (0,∞), x,F,ζ1 e Rn, G and ζ2 are n×p matrix‐valued functions, and β (t) is a p‐dimensional Brownian motion process, is investigated. Utilizing stochastic Lyapunov functions and the theory of stochastic integral inequalities, conditions are stated under which the above stochastic control is integrally stable in the mean and asymptotically integrally stable in the mean. Conditions are also developed for the stochastic differential system in finite time to be mean stable, uniformly stable in mean, quasiexpansively stable in the mean, and quasicontractively stable in the mean.