A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.     

Bayesian pitman closeness

The present article brings in the notion of Posterior Pitman Closeness (PPC) in contrast to the usual Pitman Closeness (PC) criterion. Unlike the PC criterion, the PPC criterion does not suffer from lack of transitivity. Also, a posterior median is usually a posterior Pitman closest estimator of the parameter of interest. Examples are provided to illustrate these ideas. Also, some multivariate analogs of these results are presented.     

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.     

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.     

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.     

Simultaneous closeness among order statistics to population quantiles

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.     

Pitman closeness,monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring

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 than the BLIE. Next, we introduce the notions of Pitman monotonicity and Pitman consistency, and then establish that both BLUE and BLIE possess these two properties.     

Comparison of order statistics in two-sample problem in the sense of Pitman closeness

In this paper, we study the Pitman measure of closeness of order statistics of two independent samples from the same distribution to population quantiles. We then derive various exact expressions of the probability closeness of order statistics from the X and Y samples. Some distribution-free results for the median of the sampling distribution are obtained. Exact and explicit expressions are presented for Uniform(?1, 1) and exponential distributions. Numerical results for illustrative purposes are also provided.     

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.     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

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.     

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.     

Pitman closeness for hierarchical bayes predictors in mixed linear models

The present article considers the Pitman Closeness (PC) criterion of certain hierarchical Bayes (HB) predictors derived under a normal mixed linear models for known ratios of variance components using a uniform prior for the vector of fixed effects and some proper or improper prior on the error variance. For a generalized Euclidean error, simultaneous HB predictors of several linear combinations of vector of effects are shown to be the Pitman-closest in the frequentist sense in the class of equivariant predictors for location group of transformations. The normality assumption can be relaxed to show that these HB predictors are the Pitman-closest for location-scale group of transformations for a wider family of elliptically symmetric distributions. Also for this family, the HB predictors turn out to be Pitman-closest in the class of all linear unbiased predictors (LUPs). All these results are extended for the HB predictor of finite population mean vector in the context of finite population sampling.     

A Note on Comparison of (Non) Linear Estimators Using Pitman's Measure of Closeness

We present some sufficient and necessary conditions under which some linear (or nonlinear) estimators (see Sections 2 and 3) dominate (are better than) others in the sense of PMC. Its applications in linear regressions are also discussed. Furthermore, we obtain results about the eigenvalues of two matrices, which seem to be hard to be prove through pure matrix theory.     

Distribution-free confidence intervals for quantiles and tolerance intervals in terms of k-records

In this paper, we consider the problem of determining non-parametric confidence intervals for quantiles when available data are in the form of k-records. Distribution-free confidence intervals as well as lower and upper confidence limits are derived for fixed quantiles of an arbitrary unknown distribution based on k-records of an independent and identically distributed sequence from that distribution. The construction of tolerance intervals and limits based on k-records is also discussed. An exact expression for the confidence coefficient of these intervals are derived. Some tables are also provided to assist in choosing the appropriate k-records for the construction of these confidence intervals and tolerance intervals. Some simulation results are presented to point out some of the features and properties of these intervals. Finally, the data, representing the records of the amount of annual rainfall in inches recorded at Los Angeles Civic Center, are used to illustrate all the results developed in this paper and also to demonstrate the improvements that they provide on those based on either the usual records or the current records.     

Closeness of k-records to Progressive Type-II Censored Order Statistics for Location-scale Families

In this article, the Pitman closeness of upper and lower k-records to progressive Type-II censored order statistics for location-scale families is investigated. In each case, the special properties of the probability of Pitman closeness are obtained and the corresponding monotonicity properties are discussed. Moreover, the closest k-record to a specific progressive Type-II censored data is obtained. Finally, for the standard exponential and standard uniform distributions, explicit expressions for the probability of Pitman closeness are derived. For various censoring schemes, the results of the numerical computations are displayed in tables. Most of the results in Ahmadi and Balakrishnan (2013) Ahmadi, J., Balakrishnan, N. (2013). On the nearness of record values to order statistics from Pitman measure of closeness. Metrika 76:521–541.[Crossref], [Web of Science ®] , [Google Scholar] can be achieved as special cases.     

Minimal–maximal correlation-type goodness-of-fit tests

ABSTRACT

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.     

Equivariant estimation under the pitman closeness criterion

For the point estimation in models with group structures, an invariance approach to deriving superior estimators is discussed in the Pitman closeness (PC) criterion. When the maximal invariant statistic is parameter-free, that is, ancillary, the closest equivariant estimator to the true value in the PC criterion is presented. On the other hand, as an example where a distribution of the maximalinvariant statistic depends on unknown parameters, the paper treats the Stein problem in estimation of a variance and obtains an improved estimator in the PC criterion by Stein's invariance approach. Also the Stein problem in simultaneous estimation of a location vector of a spherical symmetric distribution is studied.