Pitman closeness as a criterion for the determination of the optimal progressive censoring scheme

Selecting the optimal progressive censoring scheme for the exponential distribution according to Pitman closeness criterion is discussed. For small sample sizes the Pitman closeness probabilities are calculated explicitly, and it is shown that the optimal progressive censoring scheme is the usual Type-II right censoring case. It is conjectured that this to be the case for all sample sizes. A general algorithm is also presented for the numerical computation of the Pitman closeness probabilities between any two progressive censoring schemes of the same size.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Two-sample Pitman closeness comparison under progressive Type-II censoring

In this paper, we consider two problems concerning two independent progressively Type-II censored samples. We first consider the Pitman closeness (PC) of order statistics from two independent progressively censored samples to a specific population quantile. We then consider the point prediction of a future progressively censored order statistic and discuss the determination of the closest progressively censored order statistic from the current sample according to the simultaneous closeness probabilities. For both these problems, explicit expressions are derived for the pertinent PC probabilities, and then special cases are given as examples. For various censoring schemes, we also present numerical results for the standard uniform, standard exponential, and standard normal distributions. Finally, a distribution-free result for the median is obtained.     

On simultaneous closeness probabilities of order statistics from odd sample sizes to the population median

In this note, we present alternative derivations for the probability that an individual order statistic is closest to the target parameter among all order statistics from a complete random sample. This approach is simpler than the geometric arguments used earlier. We also provide a simple direct proof for the symmetry property of the simultaneous closeness probabilities among order statistics for the estimation of percentiles from a symmetric family. Finally, we offer an alternative simpler proof for the result that sample medians from larger odd sample sizes are Pitman closer to the population median than sample medians from smaller odd sample sizes.     

Optimal Progressive Type-II Censoring Schemes for Nonparametric Confidence Intervals of Quantiles

In this article, optimal progressive censoring schemes are examined for the nonparametric confidence intervals of population quantiles. The results obtained can be universally applied to any continuous probability distribution. By using the interval mass as an optimality criterion, the optimization process is free of the actual observed values from the sample and needs only the initial sample size n and the number of complete failures m. Using several sample sizes combined with various degrees of censoring, the results of the optimization are presented here for the population median at selected levels of confidence (99, 95, and 90%). With the optimality criterion under consideration, the efficiencies of the worst progressive Type-II censoring scheme and ordinary Type-II censoring scheme are also examined in comparison to the best censoring scheme obtained for fixed n and m.     

Simultaneous Pitman closeness of progressively type-II right-censored order statistics to population quantiles

In this work, we extend prior results concerning the simultaneous Pitman closeness of order statistics (OS) to population quantiles. By considering progressively type-II right-censored samples, we derive expressions for the simultaneous closeness probabilities of the progressively censored OS to population quantiles. Explicit expressions are deduced for the cases when the underlying distribution has bounded and unbounded supports. Illustrations are provided for the cases of exponential, uniform and normal distributions for various progressive type-II right-censoring schemes and different quantiles. Finally, an extension to the case of generalized OS is outlined.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.     

On designing an acceptance sampling plan for the Pareto lifetime model

In this paper, the design of reliability sampling plans for the Pareto lifetime model under progressive Type-II right censoring is considered. Sampling plans are derived using the decision theoretic approach with a suitable loss or cost function that consists of sampling cost, rejection cost, and acceptance cost. The decision rule is based on the estimated reliability function. Plans are constructed within the Bayesian context using the natural conjugate prior. Simulations for evaluating the Bayes risk are carried out and the optimal sampling plans are reported for various sample sizes, observed number of failures and removal probabilities.     

Pitman Closeness of kth Records Based on a Two-Sequence Case

Suppose upper kth records were observed from an X-sequence of iid continuous random variables, and kth upper records from another independent Y-sequence of iid variables from the same distribution are to be observed. The Pitman closeness probabilities of these statistics are derived. For symmetric distribution, the Pitman closeness probabilities of kth record statistics to the population median, are also examined and it is shown that these probabilities are distribution free. Numerical computations are conducted to illustrate the results developed here.     

Optimal experimental plan for multi-level stress testing with Weibull regression under progressive Type-II extremal censoring

In the design of constant-stress life-testing experiments, the optimal allocation in a multi-level stress test with Type-I or Type-II censoring based on the Weibull regression model has been studied in the literature. Conventional Type-I and Type-II censoring schemes restrict our ability to observe extreme failures in the experiment and these extreme failures are important in the estimation of upper quantiles and understanding of the tail behaviors of the lifetime distribution. For this reason, we propose the use of progressive extremal censoring at each stress level, whereas the conventional Type-II censoring is a special case. The proposed experimental scheme allows some extreme failures to be observed. The maximum likelihood estimators of the model parameters, the Fisher information, and asymptotic variance–covariance matrices of the maximum likelihood estimates are derived. We consider the optimal experimental planning problem by looking at four different optimality criteria. To avoid the computational burden in searching for the optimal allocation, a simple search procedure is suggested. Optimal allocation of units for two- and four-stress-level situations is determined numerically. The asymptotic Fisher information matrix and the asymptotic optimal allocation problem are also studied and the results are compared with optimal allocations with specified sample sizes. Finally, conclusions and some practical recommendations are provided.     

Asymptotically optimal progressive censoring plans based on Fisher information

Considering the Fisher information about a single parameter contained in a progressively Type-II censored sample, the problem of optimal progressive censoring plans arises. By introducing the notion of asymptotically optimal designs, we show that right censoring is ‘almost’ optimal for many important distributions including scale families of extreme value, normal, logistic, and Laplace distributions. Moreover, it turns out that for the extreme value distribution right censoring is the only asymptotically optimal one-step censoring scheme.     

On progressive hybrid censored exponential distribution

In this paper, we introduce a new adaptive Type-I progressive hybrid censoring scheme, which has some advantages over the progressive hybrid censoring schemes already discussed in the literature. Based on an adaptive Type-I progressively hybrid censored sample, we derive the exact distribution of the maximum-likelihood estimator (MLE) of the mean lifetime of an exponential distribution as well as confidence intervals for the failure rate using exact distribution, asymptotic distribution, and three parametric bootstrap resampling methods. Furthermore, we provide computational formula for the expected number of failures and investigate the performance of the point and interval estimation for the failure rate in this case. An alternative simple form for the distribution of the MLE under adaptive Type-II progressive hybrid censoring scheme proposed by Ng et al. [Statistical analysis of exponential lifetimes under an adaptive Type-II progressive censoring scheme, Naval Res. Logist. 56 (2009), pp. 687–698] is obtained. Finally, from the exact distribution of the MLE, we establish the explicit expression for the Bayes risk of a sampling plan under adaptive Type-II progressive hybrid censoring scheme when a general loss function is used, and present some optimal Bayes solutions under four different progressive hybrid censoring schemes to illustrate the effectiveness of the proposed method.     

Generalized inverted exponential distribution under progressive first-failure censoring

This article deals with progressive first-failure censoring, which is a generalization of progressive censoring. We derive maximum likelihood estimators of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using progressive first-failure censored samples. The asymptotic confidence intervals and coverage probabilities for the parameters are obtained based on the observed Fisher's information matrix. Bayes estimators of the parameters and reliability characteristics under squared error loss function are obtained using the Lindley approximation and importance sampling methods. Also, highest posterior density credible intervals for the parameters are computed using importance sampling procedure. A Monte Carlo simulation study is conducted to analyse the performance of the estimators derived in the article. A real data set is discussed for illustration purposes. Finally, an optimal censoring scheme has been suggested using different optimality criteria.     

Estimation for the Parameters of the Pareto Distribution Under Progressive Censoring

In this article, progressive Type-II right censored sample from Pareto distribution is considered. Exact confidence region is derived for the parameters of the corresponding distribution under progressive censoring. Simulation study is performed to investigate the coverage probabilities of the proposed confidence region. Illustrative example is also given.     

Pitman comparisons of predictors of censored observations from progressively censored samples for exponential distribution

On the basis of a progressively censored sample, Basak et al. [On some predictors of times to failure of censored items in progressively censored samples. Comput Statist Data Anal. 2006;50:1313 –1337] considered the problem of predicting the unobserved censored units at various stages of progressive censoring. They then discussed several different point predictors of these censored units and compared them with respect to mean square prediction error. In this work, we use the Pitman closeness (PC) criterion to compare the maximum likelihood, best linear unbiased, best linear equivariant, and conditional median predictors (CMPs) of these progressively censored units. Next, we compare all these with respect to the median unbiased predictor in terms of PC. Numerical computations are then performed to compare all these predictors. By comparing our results to those of Basak et al. (2006), we note that our findings in the sense of PC are similar to theirs in which the CMP competes well when compared to all other predictors.     

Point estimation of lower quantiles based on two-sampling scheme

ABSTRACT

Consider a two-sampling scheme in which an initial sample is first taken from the underlying population and then by assuming a suitable restriction on this sample, some more data points are observed as a new restricted sample. This sampling scheme is used to do inference about the lower quantiles of the underlying distribution. The results are compared with those of simple random sampling in view of mean squared error and Pitman’s measure of closeness criteria for exponential and uniform distributions. It will be shown that the proposed sampling scheme would improve the performance of the point estimators of the lower quantiles of the population.     

On the Nearness of Records to Population Quantiles with Respect to Order Statistics in the Sense of Pitman Closeness

In this paper, Pitman closeness criterion is used to compare the nearness of record values and order statistics from two independent samples to a specific population quantile of the parent distribution while the underlying distributions are the same. General expressions for the associated Pitman closeness probability are obtained when the support of the parent distribution is bounded and also unbounded. Some distribution-free results are achieved for symmetric distributions. The exponential and uniform distributions are considered for illustrative proposes and exact expressions are obtained in each case.     

Pitman closeness of record values from two sequences to population quantiles

Suppose upper records from two independent sequences from iid continuous random variables from the same distribution are observed. Pitman's measure of closeness of these statistics to population quantiles of the parent distribution is studied and various exact expressions are derived. For symmetric distributions, Pitman closeness probabilities of records to median are also obtained. Examples including exponential and uniform distributions are discussed. Numerical evaluations are presented to illustrate all the results developed here.     

Pitman Closeness of Order Statistics to Population Quantiles

In this article, Pitman closeness of sample order statistics to population quantiles of a location-scale family of distributions is discussed. Explicit expressions are derived for some specific families such as uniform, exponential, and power function. Numerical results are then presented for these families for sample sizes n = 10,15, and for the choices of p = 0.10, 0.25, 0.75, 0.90. The Pitman-closest order statistic is also determined in these cases and presented.     

Pitman closeness comparison of linear estimators:a canonical form

A general form is presented for the comparison of two linear estimators of a common parameter by means of the Pitman measure of closeness. Several asymptotic results are given. The case in which the estimators are linear combinations of the order statistics is discussed. The asymptotic comparison of the sample mean versus the sample median is derived for the Laplace distribution, and two other examples are given.