Reliability Properties of Reversed Residual Lifetime

Abstract

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable X _t  = (t ? X|X ≤ t), for a fixed t > 0, is known as `time since failure', which is analogous to the residual lifetime random variable used in reliability and survival analysis. The reversed hazard rate function, which is related to the random variable X _t , has received the attention of many researchers in the recent past [(cf. Shaked, M., Shanthikumar, J. G., 1994 Shaked, M. and Shanthikumar, J. G. 1994. Stochastic Orders and Their Applications New York: Academic Press.  [Google Scholar]). Stochastic Orders and Their Applications. New York: Academic Press]. In this paper, we define some new classes of distributions based on the random variable X _t and study their interrelations. We also define a new ordering based on the mean of the random variable X_t and establish its relationship with the reversed hazard rate ordering.     

Response

The aim of the paper is to consider the implicit restrictions imposed when adopting an AR(1) error term in the context of the linear regression model. It is shown that these restrictions amount to assuming a largely identical temporal structure for all the variables involved in the specification. Implicit in this is the assumption that these variables are mutually Granger non-causal. The main implication of this result is that in most cases when residual autocorrelation is detected boththe OLS and GLS estimators are biased and inconsistent.     

Basic structure of the asymptotic theory in dynamic nonlinear econometric models

This is the second of two papers that provide an expository discussion of the basic structure of the asymptotic theory of M-estimators in dynamic nonlinear models and a review of the literature. The first paper, Pötscher and Prucha(1991), deals with consistency. In the present paper we discuss asymptotic normality. As an important ingredient to the asymptotic normality proof in dynamic nonlinear models we consider central limit theorems for dependent random variables. We also discuss the estimation of the variance covariance matrix of m-estimators under heteroscedasticity and autocorrelation.     

On the Role of Risk Premia in Volatility Forecasting

I explain why at-the-money implied volatility is a biased and inefficient forecast of future realized volatility using the insights from the empirical option-pricing literature. First, I explain how the risk premia, which manifest themselves through disparity between objective and risk-neutral probability measures, lead to the disparity between realized and implied volatilities. Second, I show that this disparity is a function of the latent spot volatility, which I estimate using the historical volatility and high–low range. An empirical exercise that is based on at-the-money implied volatility series of foreign currencies and stock market indexes, is supportive of my risk premia-based explanation of the bias.     

Checking Normality and Homoscedasticity in the General Linear Model Using Diagnostic Plots

Inference for the general linear model makes several assumptions, including independence of errors, normality, and homogeneity of variance. Departure from the latter two of these assumptions may indicate the need for data transformation or removal of outlying observations. Informal procedures such as diagnostic plots of residuals are frequently used to assess the validity of these assumptions or to identify possible outliers. A simulation-based approach is proposed, which facilitates the interpretation of various diagnostic plots by adding simultaneous tolerance bounds. Several tests exist for normality or homoscedasticity in simple random samples. These tests are often applied to residuals from a linear model fit. The resulting procedures are approximate in that correlation among residuals is ignored. The simulation-based approach accounts for the correlation structure of residuals in the linear model and allows simultaneously checking for possible outliers, non normality, and heteroscedasticity, and it does not rely on formal testing.

[Supplementary materials are available for this article. Go to the publisher's online edition of Communications in Statistics—Simulation and Computation® for the following three supplemental resource: a word file containing figures illustrating the mode of operation for the bisectional algorithm, QQ-plots, and a residual plot for the mussels data.]     

MRL and MPL Functions of Parallel Systems with INID Components under Double Monitoring

This article investigates parallel systems with n independent but not identically distributed (INID) components. Under the condition that, at time t ₁ (t ₁ > 0) the total number of failures of the components is r (r < n), and at time t ₂ (t ₂ ≥ t ₁) the sys-tem is still working or it has failed, the mean residual life (MRL) function of the parallel system and the mean past lifetime (MPL) function of the components are conducted. Some representations and corresponding properties of the MRL function and the MPL function under several specific conditions are obtained.     

Nonlinear programming solutions to the nonstationary markov model

A nonstationary Markov process model with embedded explanatory variables offers a means to account for underlying causal factors while retaining unrestrictive assumptions and the predictive ability of a stochastic framework. We find that a direct search algorithm requiring minimal user preparation is a feasible computational procedure for estimating such a model. We compare this method with several others using factorially designed Monte Carlo simulations and find evidence that a small state space and a long time series lead to better algorithmic performance.     

A relationship between a family of waiting time distributions and the asymptotic residual waiting time distribution

For a class of renewal process waiting time distributions defined herein, one may describe the distribution of asymptotic residual waiting times. The relationship between the two distributions characterizes the class, which includes the gamma distribution. Possible consequences for hypothesis testing are discussed.     

Asymptotic equivalence between two empirical mean residual life processes

There are two mean residual life estimates for right censored data. One is based on the Kaplan-Meier estimate, the other, based on the Susarla-Van Ryzin estimate for survival function. In this paper, we define the empirical mean residual life process for right censored data and show that the two empirical mean residual life processes based on the Kaplan-Meier and Susarla-Van Ryzin estimates are asymptotically equivalent uniformly on an interval under some conditions. Also we discuss the case which the asymptotic equivalence might fail.     

A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.