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In this paper, we examine properties of k-out-of-n systems when the component lifetimes are discretely distributed. The primary focus is the random variable which represents the number of failed components upon system failure. For this variate we derive the probability mass function under the most general setting of possibly dependent and heterogeneous components. We also present conditional probabilities for this variable given some information about the time of system failure. In addition to these results, we show that, in the special case of series systems, this variable exhibits some aging properties, and we establish several characterizations of probability distributions based on such properties. For illustration, we provide three numerical examples.

In life-testing and reliability studies, it is commonly assumed that the lifetimes come from a continuous distribution. While this often makes derivations more tractable, under certain circumstances, it may be more realistic to assume a discrete distribution. Recent literature has heavily focused on the former and with this, results have been aimed at deriving properties in the case of independent and identically distributed random variables and associated order statistics. In this paper, we consider the case of discrete order statistics from non-identical distributions. We derive expression for single and product moments, and with this, covariances. We demonstrate the results for the geometric and binomial distributions.

In this paper we consider correlation-type tests based on plotting points which are modifications to the simultaneous closeness probability plotting points as recently introduced in the literature. In particular, we consider a maximal correlation test and a minimal correlation test. Furthermore, we provide two methods to carry out each test, where one method uses plotting points which are data dependent and the other test uses plotting points which are not. Some numerical properties on the associated correlation statistics are provided for various distributions, as well as a comprehensive power study to assess their performance in comparison to correlation-type tests based on more traditional plotting points. Two illustrative examples are also provided to demonstrate the tests. Finally, we make some observations and provide ideas for future work.

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.

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 θˆ1 and θˆ2 and obtain necessary and sufficient conditions for θˆ1 to be Pitman closer to the common median θ than θˆ2. 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 p...

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.

We first compare the ranked set sampling (RSS) and simple random sampling schemes for the estimation of the population median through Pitman’s measure of closeness. Then, by using Banks’ criterion, we determine the frequency with which an individual order statistic is closer to the median than some symmetric estimator in an ϵ -neighbourhood of the median. Finally, we consider the simultaneous Pitman closeness criterion and identify the median RSS as the optimal RSS scheme for estimating the population median.

Prediction of censored order statistics from a Type-II censored sample can be done with trivial bounds having perfect confidence. However, given independent samples from the same absolutely continuous distribution, improved bounds can be attained. In this regard, we develop here point prediction based on L-statistics for predicting order statistics (OS) from a future sample as well as for predicting censored OS from a Type-II censored sample. An example is taken to illustrate these ideas, and the limiting case wherein a single independent sample is arbitrarily large is also discussed.

Plotting points of order statistics are often used in the determination of goodness-of-fit of observed data to theoretical percentiles. Plotting points are usually determined by using nonparametric methods which produce, for example, the mean- and median-ranks. Here, we use a distribution-based approach which selects plotting points (quantiles) based on the simultaneous-closeness of order statistics to population quantiles. We show that the plotting points so determined are robust over a multitude of symmetric distributions and then demonstrate their usefulness by examining the power properties of a correlation goodness-of-fit test for normality.

Best linear unbiased, best linear invariant, and maximum likelihood predictors are commonly used in reliability studies for predicting either censored failure times or lifetimes from a future life-test. In this article, by assuming a Type-II right-censored sample from an exponential distribution, we compare best linear unbiased (BLUP) and best linear invariant (BLIP) predictors of the censored order statistics in the one-sample case and order statistics from a future sample in the two-sample case, in terms of Pitman closeness criterion. Some specific conclusions are drawn and supporting numerical results are presented.

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last 50 years since the ground breaking work of Lloyd [E.H. Lloyd, Least squares estimation of location and scale parameters using order statistics, Biometrika 39 (1952), pp. 88–95]. These comparisons have been made under many different criteria across different parametric families of distributions. A noteworthy one is by Nagaraja [H.N. Nagaraja, Comparison of estimators and predictors from two-parameter exponential distribution, Sankhyā Ser. B 48 (1986), pp. 10–18], who made a comparison of best linear unbiased (BLUE) and best linear invariant (BLIE) estimators in the case of exponential distribution. In this paper, continuing along the same lines by assuming a Type II right censored sample from a scaled-exponential distribution, we first compare BLUE and BLIE of the exponential mean parameter in terms of Pitman closeness (nearness) criterion. We show that the BLUE is always Pitman closer th...

We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.

In this paper, we consider joint distributions of order statistics generated by two simulation methods. By using these distributions, we study the nature of dependence and exceedence probabilities between them.

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.

Comparisons of best linear unbiased estimators with some other prominent estimators have been carried out over the last six decades since the ground breaking work of Lloyd [13]; see Arnold et al. [1] and David and Nagaraja [9] for elaborate details in this regard. Recently, Pitman closeness comparison of order statistics as estimators for population parameters, such as medians and quantiles, and their applications have been carried out by Balakrishnan et al. [3, 4, 5, 7]. In this paper, we discuss the Pitman closest estimators based on convex linear combinations of two contiguous order statistics, which sheds additional insight with regard to the estimation of the population median in the case of even sample sizes. We finally demonstrate the proposed method for the uniform, exponential, power function and Pareto distributions.