For a long time there were a lot of unsuccessful efforts directed toward the solution of many nonparametric problems with the help of the Bayes approach. This can be explained mainly by difficulties a researcher encounter, when he attempts to find a suitable prior distribution, determined on a sample space. Such a distribution in nonparametric problems is chosen in the form of a set of probability distributions on the given sample space. The first work in this field where some progress has been achieved belongs to Ferguson. Ferguson formulated the requirements which must be imposed on a prior distribution

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X_t = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.

Zusammenfassung

Im Mittelpunkt bei einem Zufallsvorgang steht das Interesse, welches der möglichen Elementarereignisse bzw. welches Ereignis eintreten wird. Für das Treffen von Entscheidungen oder das Verhalten in Situationen ist es oft von erheblicher Bedeutung, Kenntnisse über die Chancen oder Risiken für den Eintsitt der Ereignisse zu besitzen.

In this paper, we consider the test for constancy of parameters in long memory fractional autoregressive integrated moving average (FARIMA or ARFIMA) models. We construct the cusum test on the basis of the quasi-log-likelihood estimator (QMLE) and derive its limiting null distribution. Simulation results are provided for illustration.

Let U₁, U₂, . . . , U_n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as $${G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}$$ with notation U₀ = 0, U_n = 1, where $${N^\prime=\left\lfloor n/s\right\rfloor}$$ is the integer part of n/s. Let $${ N=\left\lceil n/s\right\rceil}$$ be the smallest integer greater than or equal to n/s, f_m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic $${f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}$$ is proved.

Zusammenfassung

In Abschnitt 15.5 wurde der Zweistichproben-t-Test vorgestellt. Mit diesem lassen sich für zwei normalverteilte Stichproben die in (15.26) formulierten Hypothesen überprüfen, ob es bei den beiden Gruppen Unterschiede bezüglich der Erwartungswerte gibt. Für die Stichproben wurde in Abschnitt 15.5 vorausgesetzt, dass sie unabhängig sind.