Central Limit Theorem for ISE of Kernel Density Estimators in Censored Dependent Model

In some long-term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the failure times and its kernel estimate f _n is the integrated square error(ISE). In this article, we derive a central limit theorem for the integrated square error of the kernel density estimators under a censored dependent model.     

Simple bounds on availability in a model with unknown life and repair distributions

We take a fresh look at the classic model of a device supported by a single statistically identical spare and provision for repairs, with system failure resulting whenever the currently operating unit fails before the repair of the previously failed unit is completed to allow it to become a spare. The limiting availability A(F,G) of this system depends on the life distribution F and repair time distribution G through α=∫GdF and the expected downtime. In this paper we derive several computable and sharp bounds on A(F,G) when F,G have suitable life distribution characteristics in the sense of reliability theory but are otherwise unknown except for at most two moments. Among other results, we find a sharp bound which involves the MTBF, MTTR and the second moment of the life-distribution of the device through its coefficient of variation. This leads to a maximin result for DFR repairs and DMRL lives.     

New estimators of distribution functions

ABSTRACT

This article considers the estimation of a distribution function F_X(x) based on a random sample X₁, X₂, …, X_n when the sample is suspected to come from a close-by distribution F₀(x). The new estimators, namely the preliminary test (PTE) and Stein-type estimator (SE) are defined and compared with the “empirical distribution function” (edf) under local departure. In this case, we show that Stein-type estimators are superior to edf and PTE is superior to edf when it is close to F₀(x). As a by-product similar estimators are proposed for population quantiles.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

Deficiency of the mle of a smooth survival function under the proportional hazard model

The problem of estimating a smooth distribution function F at a point t is treated under the proportional hazard model of random censorship. It is shown that a certain class of properly chosen kernel type estimator of F asymptotically perform better than the maximum likelihood estimator. It is shown that the relative deficiency of the maximum likelihood estimator of F under the proportional hazard model with respect to the properly chosen kernel type estimator tends to infinity as the sample size tends to infinity.     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.     

A product-limit estimator for use with length-biased data

The following life-testing situation is considered. At some time in the distant past, n objects, from a population with life distribution F, were put in use; whenever an object failed, it was promptly replaced. At some time τ, long after the start of the process, a statistician starts observing the n objects in use at that time; he knows the age of each of those n objects, and observes each of them for a fixed length of time? ∞, or until failure, whichever occurs first. In the case where T is finite, some of the observations may be censored; in the case where T =∞, there is no censoring. The total life of an object in use at time ∞ is a length-biased observation from F. A nonparametric estimator of the (cumulative) hazard function is proposed, and is used to construct an estimator of F which is of the product-limit type. Strong uniform consistency results (for n → ∞) are obtained. An “Aalen-Johansen” identity, satisfied by any pair of life distributions and their (cumulative) hazard functions, is used in obtaining rate-of-convergence results.     

A modified kernel quantile estimator under censoring

Based on right-censored data from a lifetime distribution F₀, a modification of the kernel quantile estimator is proposed. The advantage of this estimator is that the data play a role in the degree of smoothing of the estimator while retaining the desirable features of the kernel estimator. Convergence in probability and almost sure convergence of the estimator are discussed. Also, asymptotic normality and confidence bands are presented and some examples are given.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

On Estimating The Quantile Function For Distributions Constrained by Star-Shaped Ordering

Consider distributions F and G such that G ^-1 F is star-shaped. In the problem of estimating the quantile functions for lifetime distributions, the estimators developed by Rojo (1998) are compared with the commonly used empirical quantile function. Both the one-sample and the two-sample methods of estimation are considered for a wide class of lifetime distributions. In addition, the behavior of the estimators is examined for star-shaped ordered lifetime distributions of the important class of coherent k- out-of-n reliability systems. Results of a Monte Carlo study are presented which compare the behavior of the new estimators with that of the empirical quantile function interms of bias and mean-squared error. As the behavior of these estimators typically depends on the tail behavior of the underlying distributions, the examples presented here include distributions with short, medium and long tails. A formula for the inverse of the Kaplan-Meier estimator is provided and used to generate the simulations in the case of censored data.     

Comparison of the estimators of the intra-cluster correlation for the nested error regression model

The intra-cluster correlation is insisted on nested error regression model that, in practice, is rarely known. This article demonstrates the size in generalized least squares (GLS) F-test using Fuller–Battese transformation and modification F-test. For the balanced case, the former using strictly positive, analysis of covariance (ANCOVA) and analysis of variance (ANOVA) estimators of intra-cluster correlation can control the size for moderate intra-cluster correlations. For small intra-cluster correlation, they perform well when the numbers of cluster are large. The latter using the ANOVA estimator performs well except for small numbers of cluster. When intra-cluster correlation is large, it cannot control the size. For the unbalanced case, the GLS F-test using the Fuller–Battese transformation and the modification F-test using the strictly positive, the ANCOVA and the ANOVA estimators maintain the significance level for small total sample size and small intra-cluster correlations when there is a large variation in cluster sizes, but they perform well in controlling the size for large total sample size and small different variation in cluster sizes. Besides, Henderson’s method 3 estimator maintains the significance level for a few situations.     

Nonparametric estimators for shift in quantal bioassay

Let F(x) and F(x+θ) be log dose-response curves for a standard preparation and a test preparation, respectively, in a parallel quantal bioassay designed to test the relative potency of a drug, toxicant, or some other substance, and suppose the form of F is unknown. Several estimators of the shift parameter θ or relative potency, are compared, including some generalized and trimmed Spearman-Kärber estimators and a non parametric maximum likelihood estimator. Both point and interval estimation are discussed. Some recommendations concerning the choices of estimators are offered.     

Preventive replacement policies with products update announcements

Abstract

When we consider the improvement of the functional performances that are released by the new updates of the products, it is an interesting problem to revisit the existing replacement policies. For such a viewpoint, four replacement models with product update announcements, i.e., PUA for abbreviation, are given in this paper: Model 1, the unit is replaced at time T or at PUA over time T. Model 2, the unit is replaced at the Kth failure or at PUA over the Kth failure. By considering both time T and failure K, Models 3 and 4 are obtained based on the approaches of replacement first and last. We obtain the expected cost rates for four models and discuss analytically their optimal replacement policies Further, numerical examples are given when the time for PUA has an exponential distribution.     

Stress-Strength Reliability of a Two-Parameter Bathtub-shaped Lifetime Distribution Based on Progressively Censored Samples

Based on progressively Type-II censored samples, this article deals with inference for the stress-strength reliability R = P(Y < X) when X and Y are two independent two-parameter bathtub-shape lifetime distributions with different scale parameters, but having the same shape parameter. Different methods for estimating the reliability are applied. The maximum likelihood estimate of R is derived. Also, its asymptotic distribution is used to construct an asymptotic confidence interval for R. Assuming that the shape parameter is known, the maximum likelihood estimator of R is obtained. Based on the exact distribution of the maximum likelihood estimator of R an exact confidence interval of that has been obtained. The uniformly minimum variance unbiased estimator are calculated for R. Bayes estimate of R and the associated credible interval are also got under the assumption of independent gamma priors. Monte Carlo simulations are performed to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose. Finally, we will generalize this distribution to the proportional hazard family with two parameters and derive various estimators in this family.     

A new aging concept derived from the increasing convex ordering

In this paper, a comparison between the life distribution of a new unit with that of a used unit in the increasing convex order is made leading to a new class of life distributions which we call “new better than used in convex ordering of second order”. This class includes as subclasses the NBU and the NBUC and is a subclass of the NBUCA class. Preservation properties under convolution, random maxima, mixing and formation of coherent structures are established. Stochastic comparisons of the excess lifetime when the inter-arrival times belong to the NBUC(2) class are developed. Some applications of Poisson shock models and a test of exponentiality against NBUC(2) alternative are presented.     

On testing more IFRA ordering-II

ABSTRACT

Suppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ? _*G) if G^{? 1}F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H₀: F = _*G against H₁: F ? _*G and F ≠ _*G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.     

The exponentiated exponential–geometric distribution: a distribution with decreasing,increasing and unimodal failure rate

In this paper, we proposed a new family of distributions namely exponentiated exponential–geometric (E2G) distribution. The E2G distribution is a straightforwardly generalization of the exponential–geometric (EG) distribution proposed by Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum-likelihood inference is implemented straightforwardly. From a mis-specification simulation study performed in order to assess the extent of the mis-specification errors when testing the EG distribution against the E2G, and we observed that it is usually possible to discriminate between both distributions even for moderate samples with presence of censoring. The practical importance of the new distribution was demonstrated in three applications where we compare the E2G distribution with several lifetime distributions.