Representing the mean residual life in terms of the failure rate

In survival or reliability studies, the mean residual life or life expectancy is an important characteristic of the model. Whereas the failure rate can be expressed quite simply in terms of the mean residual life and its derivative, the inverse problem—namely that of expressing the mean residual life in terms of the failure rate—typically involves an integral of a complicated expression. In this paper, we obtain simple expressions for the mean residual life in terms of the failure rate for certain classes of distributions which subsume many of the standard cases. Several results in the literature can be obtained using our approach. Additionally, we develop an expansion for the mean residual life in terms of Gaussian probability functions for a broad class of ultimately increasing failure rate distributions. Some examples are provided to illustrate the procedure.     

Testing goodness of fit for the distribution of errors in multivariate linear models

In this paper, to test goodness of fit to any fixed distribution of errors in multivariate linear models, we consider a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis. We study the limiting behaviour of this test statistic under the null hypothesis and under alternatives. In the asymptotics, the rank of the design matrix is allowed to grow with the sample size.     

MgO films grown on yttria-stabilized zirconia by molecular beam epitaxy

MgO films were grown on (0 0 1) yttria-stabilized zirconia (YSZ) substrates by molecular beam epitaxy (MBE). The crystalline structures of these films were investigated using X-ray diffraction and transmission electron microscopy. Growth temperature was varied from 350 to 550 °C, with crystalline quality being improved at higher temperatures. The MgO films had a domain structure: (1 1 1)[1 1 2¯]_MgO(0 0 1)[1 0 0]_YSZ with four twin variants related by a 90° in-plane rotation about the [1 1 1]_MgO axis. The observed epitaxial orientation was compared to previous reports of films grown by pulsed laser deposition and sputtering and explained as resulting in the lowest interface energy.     

Inviscid,laminar and turbulent opposed flows

This paper attempts to reproduce numerically previous experimental findings with opposed flows and extends their range to quantify the effects of upstream pipes and nozzles with inviscid, laminar and turbulent flows. The choice of conservation equations, boundary conditions, algorithms for their solution, the degree of grid dependence, numerical diffusion and the validity of numerical approximations are justified with supporting calculations where necessary. The results of all calculations on the stagnation plane show maximum strain rates close to the annular exit from the nozzles and pipes for lower separations and it can be expected that corresponding reacting flows will tend to extinguish in this region with the extinction moving towards the axis. With laminar flows, the maximum strain rate increased with Reynolds number and the maximum values were generally greater than with inviscid flows and smaller than with turbulent flows. With large separations, the strain rates varied less and this explains some results with reacting flows where the extinction appeared to begin on the axis. The turbulent‐flow calculations allowed comparison of three common variants of a two‐equation first‐moment closure. They provided reasonable and useful indications of strain rates but none correctly represented the rms of velocity fluctuations on the axis and close to the stagnation plane. As expected, those designed to deal with this problem produced results in better agreement with experiment but were still imperfect. Copyright © 2004 John Wiley & Sons, Ltd.     

Tail probabilities of the limiting null distributions of the Anderson–Stephens statistics

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) z_i, i=1,…,n, Anderson and Stephens (Biometrika 59 (1972) 613–621) proposed testing procedures based on the statistics S_max=max_u S(u) and S_min=min_u S(u), where u is a unit vector and nS(u) is the sum of squares of u′z_i's. In this paper, we also consider another test statistic S_range=S_max−S_min. We provide formulas for the P-values of S_max, S_min, S_range by approximating tail probabilities of the limiting null distributions by means of the tube method, an integral-geometric approach for evaluating tail probability of the maximum of a Gaussian random field. Monte Carlo simulations for examining the accuracy of the approximation and for the power comparison of the statistics are given.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

Quantum Lax Operators and Discrete 2+1-Dimensional Integrable Models

A specific quantum Lax operator, forming a quantum U _q (sl n) chain, is suggested. The intertwining R L L relation is proved. The invariant local 2+1-dimensional reformulation of the auxiliary linear problem for the L-operator is discussed.     

Probabilistic and Average Linear Widths of Sobolev Space with Gaussian Measure in L \infty-Norm

In this paper we investigate the probabilistic linear $(n,\delta)$-widths and $p$-average linear $n$-widths of the Sobolev space $W^r_2$ equipped with the Gaussian measure $\mu$ in the $L_{\infty}$-norm, and determine the asymptotic equalities \begin{eqnarray*} \lambda_{n,\delta}(W^r_2,\mu,L_{\infty}) &\asymp&\frac{\sqrt{\ln (n/\delta)}}{n^{r+(s-1)/2}},\\[3pt] \lambda^{(a)}_n(W^r_2,\mu,L_{\infty})_p &\asymp&\frac{\sqrt{\ln n}}{n^{r+(s-1)/2}}, \qquad 0 < p < \infty. \end{eqnarray*}     

The K–nucleus cross sections in relativistic non-linear models

The K⁺−nucleus cross sections have been calculated in relativistic non-linear models taking into account the in-medium modifications of nucleons and mesons properties. More precisely, we have taken into account both the dressing of the nucleon effective mass by the scalar nuclear field in the target and the coupling of the mesons exchanged between the K⁺ and the nucleons to the polarization of the medium. First, using the most recent relativistic non-linear models, the in-medium σ, ω and ρ meson masses have been obtained in the nuclear matter rest frame. The influence of the non-linear contributions on the K⁺−nucleus cross sections has then been studied. Finally, we have compared our results with the experimental data.     

Regression quantiles for unstable autoregressive models

This paper investigates regression quantiles (RQ) for unstable autoregressive models. The uniform Bahadur representation of the RQ process is obtained. The joint asymptotic distribution of the RQ process is derived in a unified manner for all types of characteristic roots on or outside the unit circle. It involves stochastic integrals in terms of a sequence of independent and identically distributed multivariate Brownian motions with correlated components. The related L-estimator is also discussed. The asymptotic distributions of the RQ and the L-estimator corresponding to the nonstationary componentwise arguments can be transformed into a function of a normal random variable and a sequence of i.i.d. univariate Brownian motions. This is different from the analysis based on the LSE in the literature. As an auxiliary theorem, a weak convergence of a randomly weighted residual empirical process to the stochastic integral of a Kiefer process is established. The results obtained in this paper provide an asymptotic theory for nonstationary time series processes, which can be used to construct robust unit root tests.