Location-scale Research Articles

Generalized confidence interval and p-value in location and scale family

The concept of generalized pivotal quantity (GPQ) and generalized p-value (GPV) have recently become popular in small-sample inferences for complex problems such the Behrens–Fisher problem. These techniques have been shown to be efficient in specific distributions by using MLEs. In this paper, we extend the current literature in two directions; first, we propose a unified approach for constructing GPQ and GPV for the parameters of location-scale families and secondly, we use such approach along with Pitman estimators of location and scale parameters to make inferences based on GPQ and GPV. The Pitman estimators are known to be Minimum Risk equivariant (MRE) estimators which may exist even when the MLEs may not. Thus, this paper extends the generalized inference methodology that has so far been applied to specific distributions and by use of MLEs and conditional inference, to general location-scale families and to cases where the MLE may not exist. The performance of the approach is illustrated by Monte Carlo simulations and by application to air lead levels data set, collected by the National Institute of Occupational Safety and Health (NIOSH).

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 the choice of a noninformative prior for Bayesian inference of discretized normal observations

We consider the task of Bayesian inference of the mean of normal observations when the available data have been discretized and when no prior knowledge about the mean and the variance exists. An application is presented which illustrates that the discretization of the data should not be ignored when their variability is of the order of the discretization step. We show that the standard (noninformative) prior for location-scale family distributions is no longer appropriate. We work out the reference prior of Berger and Bernardo, which leads to different and more reasonable results. However, for this prior the posterior also shows some non-desirable properties. We argue that this is due to the inherent difficulty of the considered problem, which also affects other methods of inference. We therefore complement our analysis by an empirical Bayes approach. While such proceeding overcomes the disadvantages of the standard and reference priors and appears to provide a reasonable inference, it may raise conceptual concerns. We conclude that it is difficult to provide a widely accepted prior for the considered problem.

Fisher Information in Record Values and Their Concomitants: A Comparison of Two Sampling Schemes

Two sampling designs via inverse sampling for generating record data and their concomitants are considered: single sample and multisample. The purpose here is to compare the Fisher information in these two sampling schemes. It is shown that the comparison criterion depends on the underlying distribution. Several general results are established for some parametric families and their well known subclasses such as location-scale and shape families, exponential family and proportional (reversed) hazard model. Farlie-Gumbel-Morgenstern (FGM) family, bivariate normal distribution, and some other common bivariate distributions are considered as examples for illustrations and are classified according to this criterion.

Failure Probability Under Parameter Uncertainty

In many problems of risk analysis, failure is equivalent to the event of a random risk factor exceeding a given threshold. Failure probabilities can be controlled if a decisionmaker is able to set the threshold at an appropriate level. This abstract situation applies, for example, to environmental risks with infrastructure controls; to supply chain risks with inventory controls; and to insurance solvency risks with capital controls. However, uncertainty around the distribution of the risk factor implies that parameter error will be present and the measures taken to control failure probabilities may not be effective. We show that parameter uncertainty increases the probability (understood as expected frequency) of failures. For a large class of loss distributions, arising from increasing transformations of location-scale families (including the log-normal, Weibull, and Pareto distributions), the article shows that failure probabilities can be exactly calculated, as they are independent of the true (but unknown) parameters. Hence it is possible to obtain an explicit measure of the effect of parameter uncertainty on failure probability. Failure probability can be controlled in two different ways: (1) by reducing the nominal required failure probability, depending on the size of the available data set, and (2) by modifying of the distribution itself that is used to calculate the risk control. Approach (1) corresponds to a frequentist/regulatory view of probability, while approach (2) is consistent with a Bayesian/personalistic view. We furthermore show that the two approaches are consistent in achieving the required failure probability. Finally, we briefly discuss the effects of data pooling and its systemic risk implications.

Parametric estimation of location and scale parameters in ranked set sampling

This paper develops parametric inference for the parameters of location-scale family of distributions based on a ranked set sample. Likelihood function incorporates within-set ranking errors into the model through a missing data mechanism. The maximum likelihood estimators of the location-scale and missing data model parameters are constructed and an EM-algorithm is provided. It is shown that the proposed estimator is robust against imperfect ranking error and provides higher efficiency over its competitors.

Bayesian Analysis of Location-Scale Family of Distributions Using S-PLUS and R Software

The Normal and Laplace’s methods of approximation for posterior density based on the location-scale family of distributions in terms of the numerical and graphical simulation are examined using S-PLUS and R Software.

Linear moments study under ranked set sampling

A ranked set sample consists of independently distributed order statistics and can occur naturally in many experimental settings. The weighted least squares method is used to find linear estimators of unknown parameters from location-scale family of distributions which required the full matrix of all variances and covariances of order statistics of the sample size which is difficult to obtain in large samples, finding the inverse of this matrix and the estimators can be computed numerically only for small sample sizes. Also, the weighted least squares can not be used when we have distribution which has more than two parameters, for example, generalized Pareto distribution. In this article, we are looking for method in the class of linear estimation which can be applied for any distribution under ranked set sample regardless of the number of the parameters and easy to use. The linear moment-L-moments- method does not require the full matrix of order statistics and easy to use. Also, we derive unbiased estimators of population L-moments using sample linear moments based on independent ranked set sample. We obtain distribution-free estimate for the sample mean from any distribution under ranked set sample in terms of sample variance and sample L-moments. We illustrate our method on the generalized Pareto distribution.

Tolerance intervals for symmetric location-scale families based on uncensored or censored samples

Exact methods for constructing two-sided tolerance intervals (TIs) and tolerance intervals that control percentages in both tails for a location-scale family of distributions are proposed. The proposed methods are illustrated by constructing TIs for a normal, logistic, and Laplace (double exponential) distributions based on type II singly censored samples. Factors for constructing one-sided and two-sided TIs for a logistic distribution are tabulated for the case of uncensored samples. Factors for constructing TIs based on censored samples for all three distributions are also tabulated. The factors for all cases are estimated by Monte Carlo simulation. An adjustment to the tolerance factors based on type II censored samples is proposed so that they can be used to find approximate TIs based on type I censored samples. Coverage studies of the approximate TIs based on type I censored samples indicate that the approximation is satisfactory as long as the proportion of censored observations is no more than 0.70. The methods are illustrated using some practical examples.

Linear estimation of location and scale parameters using partial maxima

Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.

Pitman closeness of current records for location-scale families

In this paper, the largest and the smallest observations are considered, at the time when a new record of either kind (upper or lower) occurs based on a sequence of independent random variables with identical continuous distributions. These statistics are referred to as current upper and lower records, respectively, in the statistical literature. We derive expressions for the Pitman closeness of current records to a common population parameter and then apply these results to location-scale families of distributions with a special emphasis on the estimation of quantiles. In the case of symmetric distributions, we show that this criterion possesses some symmetry properties. Exact expressions are derived for the Pitman closeness probabilities in the case of Uniform (−1, 1) and exponential distributions. Moreover, for the population median, we show that the Pitman closeness probability is distribution-free.

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.

The concept of risk unbiasedness in statistical prediction

This paper extends the concept of risk unbiasedness for applying to statistical prediction and nonstandard inference problems, by formalizing the idea that a risk unbiased predictor should be at least as close to the “true” predictant as to any “wrong” predictant, on the average. A novel aspect of our approach is measuring closeness between a predicted value and the predictant by a regret function, derived suitably from the given loss function. The general concept is more relevant than mean unbiasedness, especially for asymmetric loss functions. For squared error loss, we present a method for deriving best (minimum risk) risk unbiased predictors when the regression function is linear in a function of the parameters. We derive a Rao–Blackwell type result for a class of loss functions that includes squared error and LINEX losses as special cases. For location-scale families, we prove that if a unique best risk unbiased predictor exists, then it is equivariant. The concepts and results are illustrated with several examples. One interesting finding is that in some problems a best unbiased predictor does not exist, but a best risk unbiased predictor can be obtained. Thus, risk unbiasedness can be a useful tool for selecting a predictor.

Random fuzzy location-scale family with linear fuzzy location function

Coexistence of fuzziness and the randomness has become an intrinsic character in modern manufacturing environments. Accordingly, it is necessary to introduce a new random fuzzy reliability concept and regression models to facilitate the random and fuzzy uncertainties. In this article, we propose the average chance reliability as a measure of quality index under random fuzzy uncertainty, which can be regarded as a counterpart of conventional reliability under probability theory. Furthermore, we extend the probabilistic location-scale family, which is actually a non-normal regression model, into a random fuzzy location-scale family, which is developed for the facilitation of the covariate modeling under random fuzzy environments. We will particularly focus our development on a random fuzzy location-scale family, in which the location function is a fuzzy regression model with each fuzzy exploratory variable being piecewise linear credibility distributed.

Do Economic Variables Follow Scale or Location‐Scale Distributions?

AbstractEconomics researchers often assume that random variables are drawn from distributions that are members of scale or location‐scale families of distributions. This article generalizes earlier results in the literature on the bias in least squares estimates of multiplicative error models, and uses those results to construct a test of the scale and location‐scale hypotheses. A Monte Carlo simulation shows that the test is powerful in large samples. The empirical relevance of these findings is illustrated with estimates of a supply function for U.S. wheat production. Implications for applied economics research are discussed.

Linear Estimators and Predictors Based on Generalized Order Statistics from Generalized Pareto Distributions

Linear estimation and prediction based on several samples of generalized order statistics from generalized Pareto distributions is considered. Representations of best linear unbiased estimators (BLUEs) and best linear equivariant estimators in location-scale families are derived, as well as corresponding optimal linear predictors. Moreover, we study positivity of the linear estimators of the scale parameter. An example illustrates that the BLUE may attain negative values with positive probability in certain situations.

Data-Driven Smooth Tests for a Location-Scale Family Revisited

A new data-driven, smooth goodness of test for a location-scale family is proposed and studied. The new test statistic is a combination of an efficient score statistic and an appropriate selection rule. Some examples are presented and by using extensive simulations the test is shown to have desirable properties.

Pitman closeness of record values to population quantiles

In this paper, we examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of Uniform (−1,1) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.

Percentile estimators in location-scale parameter families under absolute loss

Estimators of percentiles of location-scale parameter families are optimized based on median unbiasedness and absolute risk. Median unbiased estimators and minimum absolute risk estimators are shown to exist within a class of equivariant estimators and depend upon medians of two completely specified distributions. This work extends earlier findings to a larger class of equivariant estimators. These estimators are illustrated in the normal and exponential distributions.

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.