In this paper we study the local behaviour of a characteristic of two types of shock models. In many physical systems, a failure occurs when the stress or the fatigue, represented by $\epsilon(n)$, reaches a critical level $x$. We are interested in the time $\tau(x)$ for which this happens for...

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h 6? 0 and g 0 such that f(xt) ? f(t) = h(x)g(t) + o(g(t)) as t ? ? for all x ? (0,?). Zooming in on the remainder term o(g(t))...

Let fX;Xi; i = 1; 2; :::g denote independent positive random variables having a common distribution function F(x) and, independent of X, let N denote an integer valued random variable. Using S(0) = 0 and S(n) = S(n ?? 1) + Xn, the random sum S(N) has distribution function G(x) = 1Xi=0 P(N =...

In this paper, we introduce a two state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is...

Let fXi; i _ 1g denote a sequence of f0; 1g-variables and suppose that the sequence forms a Markov Chain. In the paper we study the number of successes Sn = X1 + X2 + ::: + Xn and we study the number of experiments Y (r) up to the rth success. In the i.i.d. case Sn has a binomial distribution and...

Wang et al. (2007) elaborate on an interesting approach to estimate the equilibrium distribution for the number of customers in the M[x]/M/1 queueing model with multiple vacations and server breakdowns. Their approach consists of maximizing an entropy function subject to constraints, where the...

Let fXi; i _ 1g denote a sequence of variables that take values in f0; 1g and suppose that the sequence forms a Markov chain with transition matrix P and with initial distribution (q; p) = (P(X1 = 0); P(X1 = 1)). Several authors have studied the quantities Sn, Y (r) and AR(n), where Sn = Pn i=1 Xi...

In the Bayesian modelling the data and the prior information concerning a certain parameter of interest may conflict, in the sense that the information carried by them disagree. The most common form of conflict is the presence of outlying information in the data, which may potentially lead to...