Location-scale Research Articles

Stochastic Comparisons for Finite Mixtures from Location-scale Family of Distributions

In this study, we consider two finite mixture models (FMMs) with location-scale family distributed components, in which ordering results are established in various stochastic senses. For heterogeneity in one parameter, the comparisons are obtained with respect to usual stochastic order, hazard rate order, reversed hazard rate order and likelihood ratio order. Further, for heterogeneity in two parameters, we derive sufficient conditions for the stochastic comparison of FMMs with respect to usual stochastic order and hazard rate order. Various examples and counterexamples are presented to illustrate the proposed results.

Symmetry, Asymmetry and Studentized Statistics

Inferences on the location parameter λ in location-scale families can be carried out using Studentized statistics, i.e., considering estimators λ˜ of λ and δ˜ of the nuisance scale parameter δ, in a statistic T=g(λ˜,δ˜) with a sampling distribution that does not depend on (λ,δ). If both estimators are independent, then T is an externally Studentized statistic; otherwise, it is an internally Studentized statistic. For the Gaussian and for the exponential location-scale families, there are externally Studentized statistics with sampling distributions that are easy to obtain: in the Gaussian case, Student’s classic t statistic, since the sample mean λ˜=X¯ and the sample standard deviation δ˜=S are independent; in the exponential case, the sample minimum λ˜=X1:n and the sample range δ˜=Xn:n−X1:n, where the latter is a dispersion estimator, which are independent due to the independence of spacings. However, obtaining the exact distribution of Student’s statistic in non-Gaussian populations is hard, but the consequences of assuming symmetry for the parent distribution to obtain approximations allow us to determine if Student’s statistic is conservative or liberal. Moreover, examples of external and internal Studentizations in the asymmetric exponential population are given, and an ANalysis Of Spacings (ANOSp) similar to an ANOVA in Gaussian populations is also presented.

On optimal joint prediction of order statistics

In this paper, we discuss the joint estimation and prediction of unobserved order statistics based on a Type-II censored sample from a location-scale family. Using the concept of Loewner order, we simplify the derivations made earlier, and also strengthen in the process some of the existing results. We then study the efficiency of the methods and finally examine the determination of optimal number of order statistics to be observed as well as the performance of non-linear predictors.

Model checks for two-sample location-scale

Two-sample location-scale refers to a model that permits a pair of standardised random variables to have a common base distribution. Function-based hypothesis testing in these models refers to formal tests based on distributions functions, or direct transformations thereof, that would help decide whether or not two samples come from some location-scale family of distributions. For uncensored data, an approach of testing based on plug-in empirical likelihood (PEL) is carried out with sample means and standard deviations as the plug-ins. The method extends to censored data, where censoring adjusted moment estimators provide the requisite plug-ins. The large sample null distribution of the PEL statistic is derived. Since it is not distribution free, a two-sample location-scale appropriate resampling is employed to obtain thresholds needed for the testing. Numerical studies are carried out to investigate the performance of the proposed method. Real examples are presented for both the uncensored and censored cases.

On f-Divergences Between Cauchy Distributions

We prove that the $f$-divergences between univariate Cauchy distributions are all symmetric, and can be expressed as strictly increasing scalar functions of the symmetric chi-squared divergence. We report the corresponding scalar functions for the total variation distance, the Kullback-Leibler divergence, the squared Hellinger divergence, and the Jensen-Shannon divergence among others. Next, we give conditions to expand the $f$-divergences as converging infinite series of higher-order power chi divergences, and illustrate the criterion for converging Taylor series expressing the $f$-divergences between Cauchy distributions. We then show that the symmetric property of $f$-divergences holds for multivariate location-scale families with prescribed matrix scales provided that the standard density is even which includes the cases of the multivariate normal and Cauchy families. However, the $f$-divergences between multivariate Cauchy densities with different scale matrices are shown asymmetric. Finally, we present several metrizations of $f$-divergences between univariate Cauchy distributions and further report geometric embedding properties of the Kullback-Leibler divergence.

A new family of quantile regression models applied to nutritional data

This paper introduces a new family of quantile regression models whose response variable follows a reparameterized Marshall-Olkin distribution indexed by quantile, scale, and asymmetry parameters. The family has arisen by applying the Marshall-Olkin approach to distributions belonging to the location-scale family. Models of higher flexibility and whose structure is similar to generalized linear models were generated by quantile reparameterization. The maximum likelihood (ML) method is presented for the estimation of the model parameters, and simulation studies evaluated the performance of the ML estimators. The advantages of the family are illustrated through an application to a set of nutritional data, whose results indicate it is a good alternative for modeling slightly asymmetric response variables with support on the real line.

Wasserstein information matrix

We study information matrices for statistical models by the $$L^2$$ -Wasserstein metric. We call them Wasserstein information matrices (WIMs), which are analogs of classical Fisher information matrices. We introduce Wasserstein score functions and study covariance operators in statistical models. Using them, we establish Wasserstein–Cramer–Rao bounds for estimations and explore their comparisons with classical results. We next consider the asymptotic behaviors and efficiency of estimators. We derive the online asymptotic efficiency for Wasserstein natural gradient. Besides, we establish a Poincaré efficiency for Wasserstein natural gradient of maximal likelihood estimation. Several analytical examples of WIMs are presented, including location-scale families, independent families, rectified linear unit (ReLU) generative models.

An uniformly superior exact multi-sample test procedure for homogeneity of variances under location-scale family of distributions

An exact test procedure based on the generalized variable approach for testing homogeneity of variances of several independent populations belonging to the same location-scale family is proposed. An extensive simulation study within the class of location-scale families reveals the uniformly superior performance of the proposed test over existing popularly used parametric and non-parametric tests, both in terms of the type-I error and power function. In normal populations, [Liu X, Xu X. A new generalized p-value approach for testing the homogeneity of variances. Stat Probab Lett. 2010;80:1486–1491] test designed specifically for normal populations is found equally compatible with the proposed test. The proposed test is found to be robust against moderate deviation from model specification and the location-scale assumption. The proposed test is illustrated with two real life examples.

The Mean-Variance Rule for Investors with Reverse S-Shaped Utility

Our paper contributes to the literature by developing the theory of the mean-variance (MV) rules for investors with reverse S-shaped utility. To do so, we first introduce the definition of the MV rule for investors with reverse S-shaped utility. We then set up the conjecture on the preference for different prospects by using the new MV rule that they could get a higher expected utility for the preferred asset under some conditions. Thereafter, we look for the conditions that the conjecture could hold and construct a theorem for this purpose by showing that when the negative (positive) parts of the assets follow one (another) type of location-scale family or the linear combination of location-scale families, then the preferences of the assets is the same as those by using an expected utility for the investors with reverse S-shaped utility. We then extend the theory by developing some properties of portfolio diversification by using the new MV rule. The theory developed in our paper enables academics and practitioners to apply the theory developed in this paper to analyze some important empirical issues and draw inferences on the preferences of investors with reverse S-shaped utility.

Simultaneous best linear invariant prediction of future order statistics for location-scale and scale families and associated optimality properties

In this article, we first derive an explicit expression for the marginal best linear invariant predictor (BLIP) of an unobserved future order statistic based on a set of early observed ordered statistics. We then derive the joint BLIPs of two future order statistics and prove that the joint predictors are trace-efficient as well as determinant-efficient linear invariant predictors. More generally, the BLIPs are shown to possess complete mean squared predictive error matrix dominance property in the class of all linear invariant predictors of two future unobserved order statistics. Finally, these results are extended to the case of simultaneous BLIPs of any ℓ future order statistics. Both scale and location-scale families of distributions are considered as the parent distribution for the development of results.

Methods to Compute Prediction Intervals: A Review and New Results

The purpose of this paper is to review both classic and modern methods for constructing prediction intervals. We focus, primarily, on model-based non-Bayesian methods for the prediction of a scalar random variable, but we also include Bayesian methods with objective prior distributions. Our review of non-Bayesian methods follows two lines: general methods based on (approximate) pivotal quantities and methods based on non-Bayesian predictive distributions. The connection between these two types of methods is described for distributions in the (log-)location-scale family. We also discuss extending the general prediction methods to data with complicated dependence structures as well as some nonparametric prediction methods (e.g., conformal prediction).

Comments on: “Every variance function ... can be produced by any location-scale family ...”

Comments on: “Every variance function ... can be produced by any location-scale family ...”

An optimal purely sequential strategy with asymptotic second-order properties: Applications from statistical inference and data analysis

We develop a new class of purely sequential methodologies under an assumption that the population distribution belongs to a location-scale family. Both asymptotic first-order and second-order theories are put forward with substantial generality under a big and unified tent that successfully lead to a broad set of illustrations. After we identify an appropriately defined optimal strategy under this unified structure, we introduce applications that handle a variety of interesting inference problems. These are associated with, but not limited to, the following areas: (a) the fixed-width confidence interval (FWCI) estimation, (b) the minimum risk point estimation (MRPE), (c) the fixed-size confidence region (FSCR) estimation, (d) multiple comparisons, and (e) selecting the best normal treatment (StBNT). In illustrations (a)–(d), we have highlighted a number of choices of population distributions. Some illustrations are accompanied with data analyses.

Power of goodness-of-fit tests and some competitive proposals based on progressively type-II censored data from a location-scale distribution

In this paper, we review some existing methods for testing goodness-of-fit based on progressively type-II censored samples in the location-scale family of distributions. Also, some similar procedures and new modifications are proposed. Using Monte Carlo simulation, the powers of the reviewed and proposed tests are compared for the normal and Gumbel distributions against several alternatives. Then, we present some results based on the simulation studies. Finally, an application to two datasets is presented for numerical illustration.

Robust regression against heavy heterogeneous contamination

The gamma -divergence is well-known for having strong robustness against heavy contamination. By virtue of this property, many applications via the gamma -divergence have been proposed. There are two types of gamma -divergence for the regression problem, in which the base measures are handled differently. In this study, these two gamma -divergences are compared, and a large difference is found between them under heterogeneous contamination, where the outlier ratio depends on the explanatory variable. One gamma -divergence has the strong robustness even under heterogeneous contamination. The other does not have in general; however, it has under homogeneous contamination, where the outlier ratio does not depend on the explanatory variable, or when the parametric model of the response variable belongs to a location-scale family in which the scale does not depend on the explanatory variables. Hung et al. (Biometrics 74(1):145–154, 2018) discussed the strong robustness in a logistic regression model with an additional assumption that the tuning parameter gamma is sufficiently large. The results obtained in this study hold for any parametric model without such an additional assumption.

Parametric estimation of location and scale parameters based on ranked set sampling with unequal set sizes

Ranked set sampling with unequal set sizes (RSSU) are some of the important variants of ranked set sampling with equal set sizes (RSS). The sets that arise naturally in many applications are typically of different set sizes. One may prefer to use the largest sets available naturally along with the smaller sets rather than an attempt to form artificial equal-sized sets. This article develops maximum likelihood estimators for the location-scale family of distributions based on ranked set sampling with unequal set sizes (RSSU). The closed form expressions for MLE under RSSU do not exist, we have proved the existence of MLE for location and scale parameters for some standard distributions for RSSU data. It is proved that MLE based on MedRSSU are more efficient than their counterparts based on SRS for some standard distributions for location-scale parameters. It is also shown that asymptotic efficiencies of the MLE based on MedRSSU are considerably better than those of the estimators based on RSS with the same number of observations. A simulation study is conducted to compare the performances of the MLE’s from RSS and MedRSSU with the corresponding SRS estimators when the underlying distributions are normal and logistic.

Correction to: Every variance function, including Taylor's power law of fluctuation scaling, can be produced by any location-scale family of distributions with positive mean and variance

Correction to: Every variance function, including Taylor's power law of fluctuation scaling, can be produced by any location-scale family of distributions with positive mean and variance

Generalized fiducial methods for testing quantitative trait locus effects in genetic backcross studies

In this paper, we propose generalized fiducial methods and construct four generalized p-values to test the existence of quantitative trait locus effects under phenotype distributions from a location-scale family. Compared with the likelihood ratio test based on simulation studies, our methods perform better at controlling type I errors while retaining comparable power in cases with small or moderate sample sizes. The four generalized fiducial methods support varied scenarios: two of them are more aggressive and powerful, whereas the other two appear more conservative and robust. A real data example involving mouse blood pressure is used to illustrate our proposed methods.

A sequential sampling approach for discriminating log-normal, Weibull, and log-logistic distributions

Log-normal, Weibull, and log-logistic distributions are widely used in modeling nonnegative skewed data. We develop sequential methodologies to discriminate between any two of these three distributions as well as to discriminate among these three distributions. These methods are extended to discriminate distributions from location-scale, log-location-scale and regular families of distributions. Discriminating three or more distributions having similar shapes often requires large sample size. Sequential procedures allow early stopping which in turn reduce the sample size needed for discrimination. Proposed methods yield high probabilities of correct selection that are shown to converge to 1 asymptotically. Asymptotic behavior of expected sample size and error probabilities are studied as stopping boundaries tend to infinity. Extensive simulation study validates finite sample performances of the proposed procedures requiring significantly fewer samples on average. These methods are applied to three benchmark datasets on cancer trials and are shown to select the correct model with high probability.

On simultaneous best linear unbiased prediction of future order statistics and associated properties

In this article, the joint best linear unbiased predictors (BLUPs) of two future unobserved order statistics, based on a set of observed order statistics, are developed explicitly. It is shown that these predictors are trace-efficient as well as determinant-efficient BLUPs. More generally, the BLUPs are shown to possess complete mean squared predictive error matrix dominance in the class of all linear unbiased predictors of two future unobserved order statistics. Finally, these results are extended to the case of simultaneous BLUPs of any ℓ future order statistics. Both scale and location-scale family of distributions are considered as the parent distribution for the underlying random variables.