Estimation from incomplete data in longitudinal surveys

In longitudinal surveys where a number of observations have to be made on the same sampling unit at specified time intervals, it is not uncommon that observations for some of the time stages for some of the sampled units are found missing. In the present investigation an estimation procedure for estimating the population total based on such incomplete data from multiple observations is suggested which makes use of all the available information and is seen to be more efficient than the one based on only completely observed units. Estimators are also proposed for two other situations; firstly when data is collected only for a sample of time stages and secondly when data is observed for only one time stage per sampled unit.     

LSE method using the CHSS model

Testing the reliability at a nominal stress level may lead to extensive test time. Estimations of reliability parameters can be obtained faster thanks to step-stress accelerated life tests (ALT). Usually, a transfer functional defined among a given class of parametric functions is required, but Bagdonavi?ius and Nikulin showed that ALT tests are still possible without any assumption about this functional. When shape and scale parameters of the lifetime distribution change with the stress level, they suggested an ALT method using a model called CHanging Shape and Scale (CHSS). They estimated the lifetime parameters at the nominal stress with maximum likelihood estimation (MLE). However, this method usually requires an initialization of lifetime parameters, which may be difficult when no similar product has been tested before. This paper aims to face this issue by using an iterating least square estimation (LSE) method. It will enable one to initialize the optimization required to carry out the MLE and it will give estimations that can sometimes be better than those given by MLE.     

Sequential testing with interval censored data ∗

Let X ₁,…,X _n be the lifetimes of n items put on testing at the same time. It is not possible to observe the actual lifetimes. However, it is possible to inspect the items at a finite number of time intervals. At each time of inspection, the number of failures can be recorded. Only these numbers of failures at times of inspections will be available to make decision on the distribution of the lifetimes. Decision can be made at the time of inspection. A “sequential” statistical test is developed to test the mean levels of the lifetimes when the probability distribution is assumed to be exponential. Some numerical results will be presented. The power and the expected time for the decision are compared with those for the idealized situation when each and every actual lifetime is recorded. They are also compared with those for the case when one and only one inspection is allowed to make the decision.     

A maximum likelihood estimator for left-truncated lifetimes based on probabilistic prior information about time of occurrence

In forestry, many processes of interest are binary and they can be modeled using lifetime analysis. However, available data are often incomplete, being interval- and right-censored as well as left-truncated, which may lead to biased parameter estimates. While censoring can be easily considered in lifetime analysis, left truncation is more complicated when individual age at selection is unknown. In this study, we designed and tested a maximum likelihood estimator that deals with left truncation by taking advantage of prior knowledge about the time when the individuals enter the experiment. Whenever a model is available for predicting the time of selection, the distribution of the delayed entries can be obtained using Bayes' theorem. It is then possible to marginalize the likelihood function over the distribution of the delayed entries in the experiment to assess the joint distribution of time of selection and time to event. This estimator was tested with continuous and discrete Gompertz-distributed lifetimes. It was then compared with two other estimators: a standard one in which left truncation was not considered and a second estimator that implemented an analytical correction. Our new estimator yielded unbiased parameter estimates with empirical coverage of confidence intervals close to their nominal value. The standard estimator leaded to an overestimation of the long-term probability of survival.     

Statistical Analysis of Coins Lost in Circulation

The loss rate of coins in the Israeli pound (IL) series issued in Israel in 1960–1979 is estimated, and the connection between it and the characteristics of the coins is examined. A lifetime model with time-dependent hazard rate and an estimation procedure for the model parameters are presented. The estimates are then used to forecast the number of coins surviving in circulation and the median lifetime of different denominations. The data are random samples drawn from the coins in circulation at a point in time, the sampled coins being sorted by date of issue.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Estimation on dependent right censoring scheme in an ordinary bivariate geometric distribution

Discrete lifetime data are very common in engineering and medical researches. In many cases the lifetime is censored at a random or predetermined time and we do not know the complete survival time. There are many situations that the lifetime variable could be dependent on the time of censoring. In this paper we propose the dependent right censoring scheme in discrete setup when the lifetime and censoring variables have a bivariate geometric distribution. We obtain the maximum likelihood estimators of the unknown parameters with their risks in closed forms. The Bayes estimators as well as the constrained Bayes estimates of the unknown parameters under the squared error loss function are also obtained. We considered an extension to the case where covariates are present along with the data. Finally we provided a simulation study and an illustrative example with a real data.     

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.     

Bayesian sample size determination for estimating binomial parameters from data subject to misclassification

We investigate the sample size problem when a binomial parameter is to be estimated, but some degree of misclassification is possible. The problem is especially challenging when the degree to which misclassification occurs is not exactly known. Motivated by a Canadian survey of the prevalence of toxoplasmosis infection in pregnant women, we examine the situation where it is desired that a marginal posterior credible interval for the prevalence of width w has coverage 1−α, using a Bayesian sample size criterion. The degree to which the misclassification probabilities are known a priori can have a very large effect on sample size requirements, and in some cases achieving a coverage of 1−α is impossible, even with an infinite sample size. Therefore, investigators must carefully evaluate the degree to which misclassification can occur when estimating sample size requirements.     

On the robustness of x-charts

In process control the simple x-charts are widely used. In determining an optimum economic design for such a control procedure the time of satisfactory production is usually assumed to be exponentially distributed. In this paper deviations from this optimistic approach are investigated by comparing E-optimal x-charts (using the assumption of an exponentially distributed lifetime) and M-optimal x-charts which are obtained by tfie pessimistic muumax principle applied to all possible lifetime distributions with the same mean value. The comparison shows that the differences between E- and M-optimal x-charts are only minor with respect to a suitable loss function and therefore, the model under consideration is verv robust against deviations from the assumption about the distribution of the time of satisfactory production.     

Nonparametric inference about service time distribution from indirect measurements

Summary.  In studies of properties of queues, for example in relation to Internet traffic, a subject that is of particular interest is the 'shape' of service time distribution. For example, we might wish to know whether the service time density is unimodal, suggesting that service time distribution is possibly homogeneous, or whether it is multimodal, indicating that there are two or more distinct customer populations. However, even in relatively controlled experiments we may not have access to explicit service time data. Our only information might be the durations of service time clusters, i.e. of busy periods. We wish to 'deconvolve' these concatenations, and to construct empirical approximations to the distribution and, particularly, the density function of service time. Explicit solutions of these problems will be suggested. In particular, a kernel-based 'deconvolution' estimator of service time density will be introduced, admitting conventional approaches to the choice of bandwidth.     

A regularized bridge sampler for sparsely sampled diffusions

Sparsely sampled diffusion processes, in this paper interpreted as data sampled sparsely in time relative to the time constant, is a challenging statistical problem. Most approximations of the transition kernel are derived under the assumption that data is frequently sampled and these approximations are often severely biased for sparsely sampled data. Monte Carlo methods can be used for this problem as the transition density can be estimated with arbitrary accuracy regardless of the sampling frequency, but this is computationally expensive or even prohibited unless effective variance reduction is applied.     

代际研究中终生收入估算偏差问题探讨

在我国代际收入流动性问题研究中,微观数据分析常用父代40岁左右的单年收入替代其终生收入,由此导致代际收入弹性估计偏大问题。文章假定微观个体观测值服从对数正态分布并进行建模,对个体终生收入做出统计预测。模拟得到学历为硕士人群的工资年增长率是本科人群的1.06倍,收入达到峰值的时间则晚了约8年,对全部人群父代的终生收入用其40岁左右收入逼近的思路可依据父代教育水平进行再次细分。     

The influence of load on the lifetime distribution

This paper derives lifetime distributions for components that fail due to overloading. We introduce the notion of energy similarity. This makes it possible to use knowledge about how components behave under different loads in order to establish a probability model. To obtain parametric families, we consider two physically distinct cases: (1) components that function individually and (2) components that function as part of a system. The former leads to univariate and the latter to multivariate models. The results are particularly useful when little or no data is available.     

Nonparametric maximum likelihood estimation for artificially truncated absence data

In manpower planning it is cornmoniy tue case tnat employees withuraw from active service for a period of time before returning to take up post at a later date. Such periods of absence are frequently of major concern to employers who are anxious to ensure that employees return as soon as possible. The distribution of duration of such periods of absence are therefore of considerable interest as is the probability that such employees will ever return to active service. In this paper we derive a nonparametric estimator for such a lifetime distribution based on renewal data which are subject to various forms of incompleteness, namely right censoring, left and right truncation, and forward recurrence. Artificial truncation is used to ensure that the data are time homogeneous. A nonparametric maximum likelihood estimator for the lifetime.     

Comparing Nonparametric Estimators for Length-Biased Data

Length-biased data appear when sampling lifetimes by cross-section. Right-censoring may affect the sampled information due to time limitation in following-up, lost to follow-up cases, etc. In this article, we compare by simulations two alternative nonparametric estimators of the lifetime distribution function when the data are length-biased and right-censored. These estimates, recently introduced in the literature, are based on nonparametric maximum-likelihood and moment-based principles. It is shown that the relative benefits associated to each estimator depend on several factors, such as the shape of the underlying distribution, sample size, or censoring level.     

Designing a Degradation Test with a Two-Parameter Exponential Lifetime Distribution

Degradation testing (DT) is a useful approach to assessing the reliability of highly reliable products which are not likely to fail under the traditional life tests or accelerated life tests. There have been a great number of excellent studies investigating the estimation of the failure time distribution and the optimal design (e.g., the optimal setting of the inspection frequency, the number of measurement, and the termination time) for DTs. However, the lifetime distributions considered in the studies mentioned above are all those without failure-free life. Here, failure-free life is characterized by a threshold parameter below which no failure is possible. The main purpose of this article is to deal with the optimal design of a DT with a two-parameter exponential lifetime distribution. More specifically, with respect to a DT where a linearized degradation model is used to model the degradation process and the lifetime is assumed to follow a two-parameter exponential distribution, under the constraint that the total experimental cost does not exceed a predetermined budget, the optimal combination of the inspection frequency, the sample size, and the termination time are determined by minimizing the mean squared error of the estimated 100p-th percentile of the lifetime distribution of the product. An example is provided to illustrate the proposed method and the corresponding sensitivity analysis is also discussed.     

On a generalization of Poisson sampling

In real-time sampling, the units of a population pass a sampler one by one. Alternatively the sampler may successively visit the units of the population. Each unit passes only once and at that time it is decided whether or not it should be included in the sample. The goal is to take a sample and efficiently estimate a population parameter. The list sequential sampling method presented here is called correlated Poisson sampling. The method is an alternative to Poisson sampling, where the units are sampled independently with given inclusion probabilities. Correlated Poisson sampling uses weights to create correlations between the inclusion indicators. In that way it is possible to reduce the variation of the sample size and to make the samples more evenly spread over the population. Simulation shows that correlated Poisson sampling improves the efficiency in many cases.     

The Bayesian approach for highly reliable electro-explosive devices using one-shot device testing

Due to the high reliability and high testing cost of electro-explosive devices, even though an accelerated test is performed, one may observe very few failures or even no failures at all due to censoring. In this paper, we consider modelling the reliability of such devices by an exponential lifetime distribution in which the failure rate is assumed to be a function of some covariates and that the observed data are binary. The Bayesian approach, with three different prior settings, is used to develop inference on the failure rate, lifetime and the reliability under some settings. A Monte Carlo simulation study is carried out to show that this approach is quite useful and suitable for analysing data of the considered form, especially when the failure rates are very small. Finally, illustrative data are analysed using this approach.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.