Location-scale Research Articles

Sample sizes for comparing means of two lifetime distributions with type II censored data:: application in an aging intervention study

Sample size determination is a very important part of planning for clinical trials. Most clinical trials do not follow all their subjects to the terminal event, resulting in censored observations. This article presents a method of computing sample sizes required to achieve adequate statistical power to compare the means of two lifetime distributions when both samples are subject to type II censoring. Our approach is based on the location-scale family of log-transformed lifetime distributions as compared to that based on the log-rank test and the family of proportional hazards. Specific applications to log-normal distribution and Weibull distribution are also discussed.

The Statistical Basis of Laboratory Data Normalization

The problem of laboratory data normalization in clinical trials is considered in the statistical framework. Leading principles for laboratory data normalization are proposed and practical normalization formulas are derived using as general assumptions as possible. The location-scale family and the scale family are used to model the assay value distributions. These assumptions lead to straightforward normalization formulas that have been proposed as ad hoc solutions. The preferred normalization is found to depend on the distribution of an assay value. We also consider the case where some normalized values become negative and propose that the negative values indicate that an unsuitable normalization is applied.

Minimum Wage Effects on Labor Market Outcomes Under Search with Bargaining

Building upon a continuous-time model of search with Nash bargaining in a stationary environment, we analyze the effect of changes in minimum wages on labor market outcomes and welfare. While minimum wage increases invariably lead to employment losses in our model, they may be welfare-improving to labor market participants using any one of a number of welfare criteria. A key determinant of the welfare impact of a minimum wage increase is the Nash bargaining power parameter. We discuss identification of this model using Current Population Survey data on accepted wages and unemployment durations, and demonstrate that key parameters are not identified when the distribution of match values belongs to a location-scale family. By incorporating a limited amount of information from the demand side of the market, we are able to obtain credible and precise estimates of all primitive parameters, including bargaining power. Direct estimates of the welfare impact of the minimum wage increase from $4.25 to $4.75 in 1996 provide limited evidence of a small improvement. Using estimates of the primitive parameters we show that more substantial welfare gains for labor market participants could have been obtained by doubling the minimum wage rate in 1996, though at the cost of a perhaps unacceptably high unemployment rate.

Goodnes-of-fit tests for location and scale families based on a weighted L2-Wasserstein distance measure

In two recent papers del Barrio et al. (1999) and del Barrio et al. (2000) consider a new class of goodness-of-fit statistics based on the L2-Wasserstein distance. They derive the limiting distribution of these statistics and show that the normal distribution is the only location-scale family for which this limiting distribution has the "loss of degrees of freedom" property, due to the estimation of the unknown parameters. In this paper a weighted L2-Wasserstein distance is considered and it is proven that these statistics retain the loss of degrees of freedom property for general classes of distributions, if applied separately to the location family and to the scale family and if the "right" weight function is used. These weight functions are such that the corresponding minimum distance estimators for the location parameter and the scale parameter are asymptotically efficient. Examples are discussed for both location and scale families.

A comparison of the maximum likelihood estimators under ranked set sapling some of its modifications

This paper compares the maximum likelihood estimators (MLEs) of the parameters of the location-scale parameter family of distributions, under the methods of ranked set sampling (S.L. Stokes, Annals of the Institute of Statistical Mathematics 47 (3) (1995) 465), median ranked set sampling and extreme ranked set sampling (A.-B Shaibu, H.A. Muttlak, Annals of the Institute of Statistical Mathematics, submitted), and percentile ranked set sampling. Several distributions have been considered and the estimators are compared based on their asymptotic relative precision with respect to the corresponding simple random sample estimators.

The use of spacings in the estimation of a scale parameter

Linear functions on spacings—instead of linear functions on order statistics—are considered, in order to simplify the form of best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the scale parameter in the classical location-scale family. Also, a sufficient condition for the non-negativity of the scale estimator is presented and, moreover, necessary and sufficient conditions for the BLUE (and the BLIE) to be a constant multiple of the sample range are derived. Finally, a modification of this approach is applied in order to simplify the derivations of both the location and the scale estimators in the Uniform Type-II Censored model.

Estimation of location extremes within general families of scale mixtures

In this paper, we study the estimation of the minimum and maximum location parameters, respectively, representing the minimum guaranteed lifetime of series and parallel systems of components, within a general class of scale mixtures. The conditional or underlying distribution has only the primary restriction of being a location-scale family with positive support. The mixing distribution is also quite general in that we only assume that it has positive support and finite second moment. For demonstrative purposes several special cases are highlighted such as the gamma, inverse-Gaussian, and discrete mixture. Various estimators, including bootstrap bias corrected estimators, are compared with respect to both mean-squared-error and Pitman's measure of closeness.

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.

BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.

The Natural Sets of Wang's Premium Principle

AbstractRecently, Wang's premium principle (Wang, 1995, 1996) has been discussed by many authors. Considerable attention has been given to the conditions under which Wang's premium principle can be reduced to the standard deviation premium principle. In this paper, we have got two results on this problem. One is that the natural set is a location-scale family if Wang's premium principle can be reduced to the SD premium principle for all surjective distortions. The other is that the natural set is a location-scale family for all power distortions.

Linear rank tests under general alternatives, with application to summary statistics computed from repeated measures data

Linear rank tests are used extensively for comparing two or more groups of continuous outcomes. Tests in this class retain proper test size with minimal assumptions and can have high efficiency towards an alternative of interest. In recent years, these tests have been increasingly used in settings where an individual's observation is itself a scalar summary of several outcome measures. Here, simple distributional structures on the outcome variables can lead to complex differences between the distributions of summary statistics of the comparison groups. The local asymptotic power of linear rank tests when the groups are assumed to differ by a location or scale alternative has been studied in detail. However, not much is known about their behavior for other types of alternatives. To address this, we derive the asymptotic distribution of linear rank tests under a general contiguous alternative and then investigate the implications for location–scale families and more general settings, including an example drawn from an AIDS clinical trial where the continuous outcome is a summary statistic computed from repeated measures of a biological marker.

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramer-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.

Asymptotic Behavior of Linear Permutation Tests under General Alternatives, with Application to Test Selection and Study Design

Tests based on the permutation of observations are a common and attractive method of comparing two groups of outcomes, in part because they retain proper test size with minimal assumptions and can have high efficiency toward specific alternatives of interest. In addition, permutation tests may be used with discrete or categorical outcomes, for which linear rank tests are not designed. Permutation tests are now increasingly used to analyze discrete or continuous responses that themselves are functions of several statistics. Examples of such summary statistics include the area under the curve generated by repeated measures of a laboratory marker or an overall composite score from a quality of life study. Here even simple structures for the joint distribution of the component statistics can lead to complex differences between the distributions of summary statistics of the comparison groups. Despite their attractive features, surprisingly little is known about the behavior of linear permutation tests when the two groups differ even in simple ways. This lack of knowledge limits an assessment of the relative efficiency of different tests or the planning of the size of a study based on a permutation test. To address these issues, we derive the asymptotic distribution of permutation tests under a general contiguous alternative, and then investigate the implications for test selection and study design for several diverse areas of application. For discrete outcomes, areas of application include permutation tests for ordinal responses and for count data. For continuous outcomes, we explore several applications, including general results for location-scale families, a comparison of different data transformations, and a comparison to linear rank tests.

A Note on Christofides' Conjecture Regarding Wang's Premium Principle

Young (1999) discussed the conjecture proposed by Christofides (1998) regarding the premium principle of Wang (1995, 1996). She shows that this conjecture is true for location-scale families and for certain other families, but false in general. In addition Young (1999) states that it remains an open problem to determine under what circumstances Wang's premium principle reduces to the standard deviation (SD) premium principle.In this paper we will provide further discussion of this problem. We will show that, for a fixed distortion, the natural set on which Wang's premium principle can reduce to the SD premium principle is and only is the union of location-scale families which satisfies some condition. Furthermore, it will be shown that the natural set is and only is a location-scale family if Wang's premium principle can be reduced to the SD premium principle for any distortion.

Distribution-Free Tests for Location-Scale Problem

SYNOPTIC ABSTRACTThe location-scale family in the normal theory set up presents problems regarding inference about the location-scale parameter θ as it does not belong to regular exponential family. In this paper, we present few nonparametric procedures for location-scale parameter θ (θ > 0) in the general set up. A particular solution for the functional equation arising in this problem is also obtained.

Peroxidase kinetic differences and cyclooxygenase catalytic control in PGH synthase-1 and -2

Choosing an appropriate performance measure is important for fund investors, nevertheless, many researchers find empirically that the choice of measures does not matter because those measures generate identical rank ordering, even though the distribution of fund returns is non-normal. In this paper we certify their findings by proving the monotonicity of several widely used performance measures when the distribution is a location-scale family. The mutual fund monthly return data from 1997 to 2015, together with simulation results, collaborate with our proof.

Discussion of Christofides' Conjecture Regarding Wang's Premium Principle

Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

Tests of goodness of fit based on the $L_2$-Wasserstein distance

We consider the Wasserstein distance between a sample distribution and the set of normal distributions as a measure of nonnormality. By considering the standardized version of this distance we obtain a version of Shapiro–Wilk’s test of normality. The asymptotic behavior of the statistic is studied using approximations of the quantile process by Brownian bridges. This method differs from the “ad hoc” method of de Wet and Venter and permits a similar analysis for testing other location scale families.

A characterization of the Behrens—Fisher distribution with applications to Bayesian inference

The Behrens-Fisher distribution is generally defined as the convolution of two Student t distributions and it is well known that it can be represented as a scale mixture of normals. By extending this standardized distribution to a location-scale family in the usual way we prove that this generalised Behrens-Fisher distribution can also be represented as a location mixture of t distributions when the mixing distribution is, in turn, a Student t. This characterization is applied to the computation of certain predictive distributions appearing in the Bayesian analysis of two sample problems.

Equivariant Estimation of the Percentiles Under a Linex Loss Function

The minimum risk equivariant estimation of percentiles in location scale families of distributions under Linex loss function is provided. The theory is illustrated by the exponential, normal and uniform examples.