Inference with Bivariate Truncated Data

Inthis paper we build on previous work for estimation of the bivariatedistribution of the time variables T ₁ and T ₂when they are observable only on the condition that one of thetime variables, say T ₁, is greater than (left-truncation)or less than (right truncation) some observed time variable C ₁.In this paper, we introduce several results based on the InfluenceCurve (which we derive in this paper) of the NPMLE of the distributionF of (T ₁,T ₂) developed by van derLaan (van der Laan, 1996). Specifically we will: prove that theNPMLE is asymptotically equivalent to an estimator developedby Gürler (Gürler, 1997), derive the asymptotic distributionof the NPMLE based on its Influence Curve, present tests to determinethe amount of dependence between T ₁ and T ₂,present the results of simulation studies that compare the NPMLEand Gürler's estimator and evaluate the performance of boththe above mentioned tests and confidence intervals of Fbased on the asymptotic distribution of the NPMLE, and finallywe will apply the methods in a data analysis in which we alsopoint out practical issues that arise in the implementation ofthe estimator.     

Correcting the bias due to dependent censoring of the survival estimator by conditioning

We consider the conditional estimation of the survival function of the time T₂ to a second event as a function of the time T₁ to a first event when there is a censoring mechanism acting on their sum T₁+T₂. The problem has been motivated by a treatment interruption study aimed at improving the quality of life of HIV-infected patients. We base the analysis on the survival function of T₂ given that T₁∈I, where I represents a period of scientific interest (1 trimester, 1 year, 2 years, etc.) and propose a non-parametric estimator for the survival function of T₂ given that T₁∈I, which takes into account both the selection bias and the heterogeneity due to the dependent censoring. The proposed estimator for the survival function uses the risk group of T₂ conditioned on the categories of T₁ and corrects for the dependent censoring using weights defined by the observed values of T₁. The estimator, properly normalized, converges weakly to a zero-mean Gaussian process. We estimate the variance of the limiting process via a bootstrap methodology. Properties of the proposed estimator are illustrated by an extensive simulation study. The motivating data set is analysed by means of this new methodology.     

A note on estimation in a minimum-type model with an exponential factor

The problem of estimating the parameter Q appearing in the distribution function of a continuous random variable T=min{T₁,T₂} is considered, when T₁,T₂ are non-negative independent random variables such that T₁ has an exponential distribution with scale parameter θ and T₂ has a possibly defective distribution function G₂(t) that is G₂(∞) < 1. It is shown that the estimator proposed by Gertsbach [(1967) Theory of Probability and Its Applications, 12 is weekly consistent and asymptotically normal. The merit of this estimator is that the above properties do not depend even on the form of G₂(t) except that G₂(t) and its derivative vanish at zero.     

Asymptotic properties of Bayes risk for one-sided tests

Consider a given sequence {T_n} of estimators for a real-valued parameter θ. This paper studies asymptotic properties of restricted Bayes tests of the following form: reject H₀:θ ≤ θ₀ in favour of the alternative θ > θ₀ if T_n ≤ C_n, where the critical point C_n is determined to minimize among all tests of this form the expected probability of error with respect to the prior distribution. Such tests may or may not be fully Bayes tests, and so are called T_n-Bayes. Under fairly broad conditions it is shown that and the T_n-Bayes risk where a_n is the order of the standard error of T_n, - is the prior density, and μ is the median of F, the limit distribution of (T_n – θ)/a_nb(θ). Several examples are given.     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

Nonparametric location-scale models for censored successive survival times

Let (T₁,T₂) be gap times corresponding to two consecutive events, which are observed subject to (univariate) random right-censoring. The censoring variable corresponding to the second gap time T₂ will in general depend on this gap time. Suppose the vector (T₁,T₂) satisfies the nonparametric location-scale regression model T₂=m(T₁)+σ(T₁)?, where the functions m and σ are ‘smooth’, and ? is independent of T₁. The aim of this paper is twofold. First, we propose a nonparametric estimator of the distribution of the error variable under this model. This problem differs from others considered in the recent related literature in that the censoring acts not only on the response but also on the covariate, having no obvious solution. On the basis of the idea of transfer of tail information (Van Keilegom and Akritas, 1999), we then use the proposed estimator of the error distribution to introduce nonparametric estimators for important targets such as: (a) the conditional distribution of T₂ given T₁; (b) the bivariate distribution of the gap times; and (c) the so-called transition probabilities. The asymptotic properties of these estimators are obtained. We also illustrate through simulations, that the new estimators based on the location-scale model may behave much better than existing ones.     

On the quantile past lifetime of the components of the parallel systems

ABSTRACT

Let T_{1: n} ? T_{2: n} ? ??? ? T_{n: n} be ordered lifetimes of components of a parallel system. In this article, the α-quantile past lifetime from the failure of the component with lifetime T_{r: n} provided that the system has failed at or before time t has been introduced. Then, some properties of this measure have been studied.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

MODELING THE LIFETIME OF LONGITUDINAL ELEMENTS

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function, which denotes the probability that an element of length x survives till time t, will be given by S_T (t, x) = P(T > t/α(x), where α(x) is a monotonically decreasing function. In particular, it is often assumed that T has a Weibull distribution. In this paper, we propose a generalization of this Weibull model by assuming that the distribution of T is Generalized gamma (GG). Since the GG model contains the Weibull, Gamma and Lognormal models as special and limiting cases, a GG regression model is an appropriate tool for describing the size effect on the lifetime and for selecting among the embedded models. Maximum likelihood estimates are obtained for the GG regression model with α(x) = cx^b . As a special case this provide an alternative to the usual approach to estimation for the GG distribution which involves reparametrization. Related parametric inference issues are addressed and illustrated using two experimental data sets. Some discussion of censored data is also provided.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.     

Regression analysis using order statistics

In this article, we study the joint distribution of X and two linear combinations of order statistics, a ^T Y ₍₂₎ and b ^T Y ₍₂₎, where a = (a ₁, a ₂) ^T and b = (b ₁, b ₂) ^T are arbitrary vectors in R ² and Y ₍₂₎ = (Y ₍₁₎, Y ₍₂₎) ^T is a vector of ordered statistics obtained from (Y ₁, Y ₂) ^T when (X, Y ₁, Y ₂) ^T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).     

Optimal component test plans for a parallel system based on Type-II censoring

Consider a parallel system with n independent components. Assume that the lifetime of the jth component follows an exponential distribution with a constant but unknown parameter λ_j, 1≤j≤n. We test r_j components of type-j for failure and compute the total time T_j of r_j failures for the jth component. Based on T=(T₁,T₂,…,T_n) and r=(r₁,r₂,…,r_n), we derive optimal reliability test plans which ensure the usual probability requirements on system reliability. Further, we solve the associated nonlinear integer programming problem by a simple enumeration of integers over the feasible range. An algorithm is developed to obtain integer solutions with minimum cost. Finally, some examples have been discussed for various levels of producer’s and consumer’s risk to illustrate the approach. Our optimal plans lead to considerable savings in costs over the available plans in the literature.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

Thermal,viscoelastic and mechanical properties' optimization of polyphenylene sulfide via optimal processing parameters using the Taguchi method

Thermal, viscoelastic and mechanical properties of polyphenylene sulfide (PPS) were optimized as a function of extrusion and injection molding parameters. For this purpose, design of experiments approach utilizing Taguchi's L₂₇ (3⁷) orthogonal arrays was used. Effect of the parameters on desired properties was determined using the analysis of variance. Differential scanning calorimeter (DSC) tests were performed for the analysis of thermal properties such as melting temperature (T_m) and melting enthalpy (ΔH_M). Dynamic mechanical analysis (DMA) tests were performed for the analysis of viscoelastic properties such as damping factor (tan?δ) and glass transition temperature (T_g). Tensile tests were performed for the analysis of mechanical properties such as tensile strength and modulus. With optimized process parameters, verification DSC, DMA and tensile tests were performed for thermal, viscoelastic and mechanical properties, respectively. The Taguchi method showed that ‘barrel temperature’ and its level of ‘340°C’ were seen to be the most effective parameter and its level; respectively. It was suggested that PPS can be reinforced for further improvement after optimized thermal, viscoelastic and mechanical properties.     

Hotelling’s T2 control chart with double warning lines

Recent studies have shown that the T ² control chart with variable sampling intervals (VSI) and/or variable sample sizes (VSS) detects process shifts faster than the traditional T ² chart. This article extends these studies for processes that are monitored with VSI and VSS using double warning lines (T ² —DWL). It is assumed that the length of time the process remains in control has exponential distribution. The properties of T ² —DWL chart are obtained using Markov chains. The results show that the T ² —DWL chart is quicker than VSI and/or VSS charts in detecting almost all shifts in the process mean.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Jackknifing L-estimates

Linear functions of order statistics (“L-estimates”) of the form T_n =under jackknifing are investigated. This paper proves that with suitable conditions on the function J, the jackknifed version T_n of the L-estimate T_n has the same limit distribution as T_n. It is also shown that the jackknife estimate of the asymptotic variance of n^1/2 is consistent. Furthermore, the Berry-Esséen rate associated with asymptotic normality, and a law of the iterated logarithm of a class of jackknife L-estimates, are characterized.     

FIRST-EXIT TIMES FOR POISSON SHOT NOISE

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T _b at which a given level b > 0 is exceeded. An integral equation for the joint density of T _b and X(T _b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T _b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T _b > t, X(t) < u), that is the joint distribution function of sup_{0 ≤ s ≤ t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).