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## Statistics of random processes I: General theory

### Metrika (1983-12-01) 30: 100 , December 01, 1983

## Bounds for the mean residual life function of a k-out-of-n system

### Metrika (2011-11-01) 74: 361-380 , November 01, 2011

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*.

## On axiomatic characterization of some non-additive measures of information

### Metrika (1977-12-01) 24: 23-34 , December 01, 1977

An axiomatic characterization of non-additive measures of information associated with a pair of probability distributions having the same number of elements has been given. This quantity under additional suitable postulates leads to the non-additive Entropy, Directed-Divergence and Inaccuracy of one or more parameters.

## The lattice structure of nonlinear congruential pseudorandom numbers

### Metrika (1993-12-01) 40: 115-120 , December 01, 1993

Several known deficiencies of the classical linear congruential method for generating uniform pseudorandom numbers led to the development of nonlinear congruential pseudorandom number generators. In the present paper a general class of nonlinear congruential methods with prime power modulus is considered. It is proved that these generators show certain undesirable linear structures, too, which stem from the composite nature of the modulus.

## The multiresolution histogram

### Metrika (1997-01-01) 46: 41-57 , January 01, 1997

We introduce a new method for locally adaptive histogram construction that doesn’t resort to a standard distribution and is easy to implement: the multiresolution histogram. It is based on a*L*_{2} analysis of the mean integrated squared error with Haar wavelets and hence can be associated with a multiresolution analysis of the sample space.

## Dreistufige klienste Quadrate —einige numerische Ergebnisse

### Metrika (1973-12-01) 20: 193-195 , December 01, 1973

## A Cramér-type large deviation theorem for sums of functions of higher order non-overlapping spacings

### Metrika (2011-07-01) 74: 33-54 , July 01, 2011

Let *U*_{1}, *U*_{2}, . . . , *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 *U*_{0} = 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.

## On kalman filtering, posterior mode estimation and fisher scoring in dynamic exponential family regression

### Metrika (1991-12-01) 38: 37-60 , December 01, 1991

### Summary

Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parameters*β*_{0},*β*_{1}, …,*β*_{t}, …, dim*β*_{t}=*p*. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived from posterior mode estimation of the whole parameter vector (*β*′_{0}, …,*β*′_{t}) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms can be given in a forward-backward recursive form where only inverses of “small”*p×p*-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which is explicit up to*p×p*-matrices.