Reliability interpretations of some concepts from economics

In a recent article, Chandra and Singpurwalla have pointed out the close relationship between the Lorenz curve, which is frequently used when illustrating income distributions in economics, and the total-time-on-test transform (TTT transform), which has proved to be a very useful tool in reliability. They also presented some characterizations of aging properties by using the Lorenz curve. The purpose of this article is to point out some further results in the same area and to give reliability interpretations of some common measures of income inequality.     

Testing goodness of fit to the increasing failure rate family with censored data

One important thrust in the reliability literature is the development of statistical procedures under various “restricted family” model assumptions such as the increasing failure rate (IFR) and decreasing failure rate (DFR) distributions. However, relatively little work has been done on the problem of testing fit to such families as a null hypothesis. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model in single-sample problems. For the same problem with complete data, Tenga and Santner studied several analytic tests of the null hypothesis that the common underlying distribution is IFR versus the alternative that it is not IFR for complete data. This article considers the same problem for the case of four types of censored data: (i) Type I (time) censoring, (ii) Type I1 (order statistic) censoring, (iii) a hybrid of Type I and Type I1 censoring, and (iv) random censorship. The least favorable distributions of several intuitive test statistics are derived for each of the four types of censoring so that valid small-sample-size α tests can be constructed from them. Properties of these tests are investigated.     

Bivariate models from univariate life distributions: A characterization cum modeling approach

Bivariate life distribution models are of importance for studying interdependent components. We present a generic approach by introducing a new concept of characterized model in stead of a characterized distribution. It strikes a balance between characterization and modeling approaches to eliminate their individual limitations and incorporate their respective strengths. The proposed model, being a characterized one, admits many important properties irrespective of the choice of marginal distributions. The retention of univariate IFR, DFR, IFRA, DFRA, NBU, and NWU class properties in the bivariate setup has been ensured along with some results on series combinations and convolution. No other models, available in the literature, can ensure simultaneous retention of these fundamental and extremely important class properties. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Closure properties of the relevation transform*

Sufficient conditions under which the relevation of two probability distributions is (i) NBU, (ii) IFRA, (iii) IFR are derived. The result for case (iii) corrects an error in a previous article by Baxter.     

A note on two IFR systems

We present probabilistic proofs for the following two facts: (i) A k out of n system of i.i.d (independent identically distributed). IFR (increasing failure rate) components has an IFR life distribution. (ii) A compound Poisson process with nonnegative i.i.d jumps with PF₂ distribution is IFR.     

Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.     

Log‐concave and concave distributions in reliability

Nonparametric classes of life distributions are usually based on the pattern of aging in some sense. The common parametric families of life distributions also feature monotone aging. In this paper we consider the class of log‐concave distributions and the subclass of concave distributions. The work is motivated by the fact that most of the common parametric models of life distributions (including Weibull, Gamma, log‐normal, Pareto, and Gompertz distributions) are log‐concave, while the remaining life of maintained and old units tend to have a concave distribution. The classes of concave and log‐concave distributions do not feature monotone aging. Nevertheless, these two classes are shown to have several interesting and useful properties. We examine the closure of these classes under a number of reliability operations, and provide sharp reliability bounds for nonmaintained and maintained units having life distribution belonging to these classes. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 419–433, 1999     

Remarks on a univariate shock model with some bivariate generalizations

In a 1973 paper J. D. Esary, A. W. Marshall, and F. Proschan [5] considered a shock model giving rise to various nonparametric classes of life distributions of interest in reliability theory. A number of authors have extended these results in a variety of directions. In this paper, alternative proofs of the increasing failure rate (IFR) and decreasing mean residual life (DMRL) results are given which do not make use of the theory of total positivity. Some bivariate extensions are then obtained using a shock model similar to that originally used by H. W. Block, A. S. Paulson, and R. C. Kohberger [2] to unify various bivariate exponential distributions.     

A note on first passage times in birth and death and nonnegative diffusion processes

Consider a birth and death process starting in state 0. Keilson has shown by analytical arguments that the time of first passage into state n has an increasing failure rate (IFR) distribution. We present a probabilistic proof for this. In addition, our proof shows that for a nonnegative diffusion process, the first passage time from state 0 to any state x is IFR.     

IFR results for repairable systems

Consider a k-out-of-n system with independent repairable components. Assume that the repair and failure distributions are exponential with parameters {μ₁, ?,μ_n} and {λ₁, ?,λ_n}, respectively. In this article we show that if λ_i – μ_i = Δ for all i then the life distribution of the system is increasing failure rate (IFR).     

Discriminating between the Weibull and log‐normal distributions

Log‐normal and Weibull distributions are the most popular distributions for modeling skewed data. In this paper, we consider the ratio of the maximized likelihood in choosing between the two distributions. The asymptotic distribution of the logarithm of the maximized likelihood ratio has been obtained. It is observed that the asymptotic distribution is independent of the unknown parameters. The asymptotic distribution has been used to determine the minimum sample size required to discriminate between two families of distributions for a user specified probability of correct selection. We perform some numerical experiments to observe how the asymptotic methods work for different sample sizes. It is observed that the asymptotic results work quite well even for small samples also. Two real data sets have been analyzed. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

时频分析方法及其在自导信号检测中的应用

首先引入时频分析的概念,介绍了短时傅里叶变换、Wigner—Ville分布和连续小波变换的数学表示和性质.其次基于经典检测理论,讨论并比较了它们在高斯白噪声背景下对确知信号的检测问题.最后对它们在自导信号检测应用中的有关问题进行了分析,并指出了需进一步研究的方向.     

The distribution of the product of two noncentral beta variates

In this paper the exact distribution of the product of two noncentral beta variates is derived using Mellin integral transform. The density function of the product is represented as a mixture of Beta distributions and the distribution function as a mixture of Incomplete Beta Functions.     

Signal‐to‐interference‐plus‐noise ratio estimation for wireless communication systems: Methods and analysis

The Signal‐to‐Interference‐plus‐Noise Ratio (SINR) is an important metric of wireless communication link quality. SINR estimates have several important applications. These include optimizing the transmit power level for a target quality of service, assisting with handoff decisions and dynamically adapting the data rate for wireless Internet applications. Accurate SINR estimation provides for both a more efficient system and a higher user‐perceived quality of service. In this paper, we develop new SINR estimators and compare their mean squared error (MSE) performance. We show that our new estimators dominate estimators that have previously appeared in the literature with respect to MSE. The sequence of transmitted bits in wireless communication systems consists of both pilot bits (which are known both to the transmitter and receiver) and user bits (which are known only by the transmitter). The SINR estimators we consider alternatively depend exclusively on pilot bits, exclusively on user bits, or simultaneously use both pilot and user bits. In addition, we consider estimators that utilize smoothing and feedback mechanisms. Smoothed estimators are motivated by the fact that the interference component of the SINR changes relatively slowly with time, typically with the addition or departure of a user to the system. Feedback estimators are motivated by the fact that receivers typically decode bits correctly with a very high probability, and therefore user bits can be thought of as quasipilot bits. For each estimator discussed, we derive an exact or approximate formula for its MSE. Satterthwaite approximations, noncentral F distributions (singly and doubly) and distribution theory of quadratic forms are the key statistical tools used in developing the MSE formulas. In the case of approximate MSE formulas, we validate their accuracy using simulation techniques. The approximate MSE formulas, of interest in their own right for comparing the quality of the estimators, are also used for optimally combining estimators. In particular, we derive optimal weights for linearly combining an estimator based on pilot bits with an estimator based on user bits. The optimal weights depend on the MSE of the two estimators being combined, and thus the accurate approximate MSE formulas can conveniently be used. The optimal weights also depend on the unknown SINR, and therefore need to be estimated in order to construct a useable combined estimator. The impact on the MSE of the combined estimator due to estimating the weights is examined. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

The error in the normal approximation to the multinomial with an increasing number of classes

In an earlier paper, it was shown that under certain conditions, if the number of classes in a multinomial distribution increases as the number of trials increases, the probabilities assigned to arbitrary regions by the multinomial distribution are close to the probabilities assigned by the distribution of slightly rounded-off normal random variables. A different method of studying the approximation of the multinomial distribution by a normal distribution is to use the multivariate Berry-Esseen bound. In this paper, these two methods are compared, particularly with respect to the class of multinomial distributions for which the bounds on the error remain useful.     

A finite-range distribution of failure times

Properties of a finite-range failure-time distribution, that includes the exponential as a particular case, have been studied. The distribution is IFR when the shape parameter exceeds unity, but is IFRA always. For a given value of the shape parameter, the distribution is NBUE over a segment of its range. Estimators of parameters have been derived. Distributions of two-component series, parallel, and standby system lives have also been worked out.     

Weak aging properties for coherent systems with statistically dependent component lifetimes

As a relevant topic in reliability theory, the preservation of aging properties under the formation of various coherent structures contributes to improving system performance through better structure design and more effective system maintenance. The classical research in this line usually focuses upon coherent systems with independent component lifetimes. Recently, some authors discussed the preservation of IFR, NBU, and DMRL in the setting of dependent component lifetimes. This paper further investigates sufficient conditions for coherent systems with dependent component lifetimes to preserve aging properties including NBUC, NBU (2), DMRL, and their dual versions. Some examples are presented to illustrate coherent structures and typical copula functions fulfilling the present sufficient conditions as well.     

一种寿命分布类——剩余寿命按凸序递减类

在前人研究的寿命分布类的基础上 ,给出一种新的寿命分布类的定义 ,并研究了其有关的性质 (主要是封闭性质 )。     

Some test functions for the parameters of the weibull distributions

Test functions, based on various types of censored and noncensored data, for testing several hypotheses about the location, the scale, and the shape parameters of the Weibull distributions are proposed. The exact sampling distributions of these test statistics are derived and their properties in special cases are discussed. A numerical example is considered to illustrate the application of the test functions. The results of this paper possess good possibility of wide application in view of the fact that hosts of real data arising from diverse fields of human endeavor are adequately described by the Weibull distribution.     

Periodic replacement when minimal repair costs vary with time

A policy of periodic replacement with minimal repair at failure is considered for a complex system. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening system failures. The cost of a minimal repair to the system is assumed to be a nonde-creasing function of its age. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the system. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited in the case where the system life distribution is strictly increasing failure rate (IFR).