Location-scale Research Articles

Riskiness in gambles that belong to the same location-scale family and with well-defined means and variances

This paper investigates the orderings of gambles with well-defined means and variances that belong to the same location-scale family under: (i) the index of riskiness analyzed by Aumann and Serrano (AS) and (ii) the measure of riskiness proposed by Foster and Hart (FH). For both measures we study the characteristics and properties of the representation on the (mean, variance) plane of the curves that include gambles with the same level of riskiness. Our analysis applies to gambles with truncated normal distributions and beta distributions. We also discuss the relationships between different riskiness measures derived from the AS and FH measures.

Two sample comparisons including zero-inflated continuous data: A parametric approach with applications to microarray experiment

Micro-array experiments are important fields in molecular biology where zero values mixed with a continuous outcome are frequently encountered leading to a mixed distribution with a clump at zero. Comparison of two mixed populations, e.g. of a control and a treated group; of two groups with different types of cancer, to name a few, are often encountered in these contexts. Fairly skewed distribution of the continuous part coupled with small sample sizes are issues of main concern to be attended for the quality of inference in such situations while popularly used nonparametric methods rely on asymptotic distribution of the underlying test statistics which are valid only under large sample sizes. We address the aforementioned issues via a newly proposed exact test for location-scale family distributions and Generalized pivot quantity (GPQ) based parametric test procedures for non-location-scale distributions. Simulation based assessment showed their superior performance with respect to size and power in comparison to the popular two-part tests (Wilcoxon rank sum, t-test, Kolmogrov–Smirnov, Ansari–Bradley and Sigel–Tukey) more prominently for small sample sizes.

On stochastic comparisons of minimum order statistics from the location–scale family of distributions

We consider stochastic comparisons of minimum order statistics from the location–scale family of distributions that contain most of the popular lifetime distributions. Under certain assumptions, we show that the minimum order statistic of one set of random variables dominates that of another set of random variables with respect to different stochastic orders. Furthermore, we illustrate our results using some well-known specific distributions.

The two-moment decision model with additive risks

With multiple additive risks, the mean–variance approach and the expected utility approach of risk preferences are compatible if all attainable distributions belong to the same location–scale family. Under this proviso, we survey existing results on the parallels of the two approaches with respect to risk attitudes, the changes thereof, and the comparative statics for simple, linear choice problems under risks. In mean–variance approach all effects can be couched in terms of the marginal rate of substitution between mean and variance. We provide some simple proofs of some previous results. We apply the theory we stated or developed in our paper to study the behavior of banking firm and study risk-taking behavior with background risk in the mean–variance model.

Fisher information in exponential-weighted location-scale distributions

ABSTRACTIn this article, the comparison between the Fisher information on parameters of the weighted distributions and the parent distributions is done. The most common family of distributions, location–scale family, is considered with the exponential weight function w(x) = eβx where β is a constant. Conditions under which the weighted distributions are more (less) informative than the parent distribution are given. This was done for location, scale, and location–scale families when the scale parameter is considered as a nuisance parameter. Furthermore, using the transformation technique, we show that the results in location–scale family can be generalized to the broader classes of problems that studied the Fisher information of the weighted distributions such as Tzavelas and Economou (2014). As the exponential weight function can include some other weight functions, the obtained results in this article can be generalized for some other weight functions.

Between First- and Second-Order Stochastic Dominance

We develop a continuum of stochastic dominance rules, covering preferences from first- to second-order stochastic dominance. The motivation for such a continuum is that while decision makers have a preference for “more is better,” they are mostly risk averse but cannot assert that they would dislike any risk. For example, situations with targets, aspiration levels, and local convexities in induced utility functions in sequential decision problems may lead to preferences for some risks. We relate our continuum of stochastic dominance rules to utility classes, the corresponding integral conditions, and probability transfers and discuss the usefulness of these interpretations. Several examples involving, e.g., finite-crossing cumulative distribution functions, location-scale families, and induced utility, illustrate the implementation of the framework developed here. Finally, we extend our results to a combined order including convex (risk-taking) stochastic dominance. This paper was accepted by Manel Baucells, decision analysis.

On Outliers Detection for Location-Scale and Shape-Scale Families

The problem of multiple upper outliers detection in samples from location-scale and shape-scale families is considered. A new test statistic is proposed. The critical values of the new test statistic are tabulated by simulation. The power of the new test and other available tests are compared by simulation.

Kappa ratios and (higher-order) stochastic dominance

This paper first shows the sufficient relationship between the $$(n+1)$$ -order SD and the n-order Kappa ratio. In fact, we clarify the restrictions on necessary beating of the target for the higher-order SD consistency of the Kappa ratios. Thereafter, we show that, in general, the necessary relationship between SD/RSD and the Kappa ratio cannot be established. We find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationship between the $$(n+1)$$ -order SD with the n-order Kappa ratio after imposing some conditions on the means. Our findings enable academics and practitioners to draw better decision in their analysis.

On stochastic comparisons of maximum order statistics from the location-scale family of distributions

We consider the location-scale family of distributions, which contains many standard lifetime distributions. We give conditions under which the largest order statistic of a set of random variables with different/the same location as well as different/the same scale parameters dominates that of another set of random variables with respect to various stochastic orders. Along with general results, we consider important special cases, namely, the Feller–Pareto, generalized Pareto, Burr, exponentiated Weibull, Power generalized Weibull, generalized gamma, Half-normal and Fréchet distributions.

The Preferences of Omega Ratio for Risk Averters and Risk Seekers

It is well-known that under some conditions, the mean-variance rule is equivalent to stochastic dominance rule. Some academics hypothesize that there could exist mean-Omega ratio rule that could be equivalent to stochastic dominance rule under certain conditions. To explore this possible, in this paper, we aim to establish the necessary conditions between Omega ratio and stochastic dominance that leads to the preferences of risk averters/seekers. We find that it is possible to establish the necessary conditions between and Omega ratio and the preferences of risk averters/seekers under the condition that the variables being compared belong to the location-scale family or the same linear combination of location-scale families.

From Volatility Smile to Risk Neutral Probability and Closed Form Solution of Local Volatility Function

The risk neutral measure is identified as a symmetric location-scale family of distribution in the local regime of the λ model. A partial differential equation is derived as the transformation between the implied volatility surface and such risk neutral probability. Given a well-interpolated volatility surface from market data, the risk neutral probability and the implied rate of growth can be extracted in a model-free manner. On the other hand, assuming a priori knowledge on the rate of growth and the risk neutral probability being an symmetric λ distribution, the closed form solution of the local volatility function be derived from Fokker-Planck equation. Based on such solution, I discuss possible forms of diffusion process, implement a leptokurtic extension of Weiner process, and discover a mean-reverting process that bridges between the Ornstein-Uhlenbeck process and Bessel process.

A general algorithm for computing simultaneous prediction intervals for the (log)-location-scale family of distributions

ABSTRACTMaking predictions of future realized values of random variables based on currently available data is a frequent task in statistical applications. In some applications, the interest is to obtain a two-sided simultaneous prediction interval (SPI) to contain at least k out of m future observations with a certain confidence level based on n previous observations from the same distribution. A closely related problem is to obtain a one-sided upper (or lower) simultaneous prediction bound (SPB) to exceed (or be exceeded) by at least k out of m future observations. In this paper, we provide a general approach for computing SPIs and SPBs based on data from a particular member of the (log)-location-scale family of distributions with complete or right censored data. The proposed simulation-based procedure can provide exact coverage probability for complete and Type II censored data. For Type I censored data, our simulation results show that our procedure provides satisfactory results in small samples. We use three applications to illustrate the proposed simultaneous prediction intervals and bounds.

The shapes of things to come: probability density quantiles

ABSTRACTFor every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQs) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback–Leibler divergences. Convergent empirical estimates of these pdQs are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQs from the symmetric class. Further, a precise classification of shapes by tail behaviour can be defined simply in terms of pdQ boundary derivatives.

Two-sided tolerance intervals for members of the (log)-location-scale family of distributions

In this paper, we propose methods to calculate exact factors for two-sided control-the-centre and control-both-tails tolerance intervals for the (log)-location-scale family of distributions, based on complete or Type II censored data. With Type I censored data, exact factors do not exist. For this case, we developed an algorithm to compute approximate factors. Our approaches are based on Monte Carlo simulations. We also provide algorithms for computing TIs that control the probability in both tails of a distribution. A simulation study for Type I censored data shows that the estimated coverage probability is close to the nominal confidence level when the expected number of uncensored observations is moderate to large. We illustrate the methods with applications using different combinations of distributions and types of censoring.

Robust generalized confidence intervals

ABSTRACTMost interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues.

A New Estimation Method Based on Order Statistics in the Families of Symmetric Location-scale Distributions

In this study, new unbiased and nonlinear estimators based on order statistics are proposed for the family of symmetric location-scale distributions and these estimators can be computed from both uncensored and symmetric doubly Type II censored samples. In addition, other relevant unbiased estimators are proposed to estimate standard deviations of these new estimators. A simulation study has been performed to evaluate the performance of the new estimators compared to BLU estimators for small sample sizes. As a result of the simulation study, the new estimators proposed for the location-scale family in general performed nearly as good as BLU estimators. Furthermore, the computational advantage of the proposed estimators over BLU and ML estimators are worthy of notice. In addition, these new estimators have been applied to real data, and the estimation results obtained have been compatible with those of BLUE methods.

OPTIMAL SHRINKAGE ESTIMATION OF MEAN PARAMETERS IN FAMILY OF DISTRIBUTIONS WITH QUADRATIC VARIANCE.

This paper discusses the simultaneous inference of mean parameters in a family of distributions with quadratic variance function. We first introduce a class of semi-parametric/parametric shrinkage estimators and establish their asymptotic optimality properties. Two specific cases, the location-scale family and the natural exponential family with quadratic variance function, are then studied in detail. We conduct a comprehensive simulation study to compare the performance of the proposed methods with existing shrinkage estimators. We also apply the method to real data and obtain encouraging results.

Nonparametric two-sample estimation of location and scale parameters from empirical characteristic functions

ABSTRACTTwo random variables X and Y are said to belong to the same location-scale family when with unknown constants and . Given iid observations , and , satisfying this location-scale assumption, we wish to estimate μ and σ with high efficiency in the absence of knowledge of the functional form of the underlying common family of distributions of X and Y. Here, ‘high efficiency’ means that the estimator is asymptotically unbiased and that its asymptotic variance is close to the asymptotic variance of the maximum likelihood estimator that would be used had the form of the underlying location-scale family of distributions been known. We propose in the present paper two methods for estimating these parameters based on the empirical characteristic function (ECF). The first approach considered minimizes a weighted distance between the ECFs of the X and Y data. The second approach constructs a quadratic form comparing the real and imaginary parts of the X- and Y-sample ECFs at a preselected number of points. In both approaches, the constructed distance metric is minimized to estimate μ and σ. The asymptotic distributions of the estimators are found, and small sample performance is investigated via a Monte Carlo simulation study.

EM algorithm in Gaussian copula with missing data

Rank-based correlation is widely used to measure dependence between variables when their marginal distributions are skewed. Estimation of such correlation is challenged by both the presence of missing data and the need for adjusting for confounding factors. In this paper, we consider a unified framework of Gaussian copula regression that enables us to estimate either Pearson correlation or rank-based correlation (e.g. Kendall’s tau or Spearman’s rho), depending on the types of marginal distributions. To adjust for confounding covariates, we utilize marginal regression models with univariate location-scale family distributions. We establish the EM algorithm for estimation of both correlation and regression parameters with missing values. For implementation, we propose an effective peeling procedure to carry out iterations required by the EM algorithm. We compare the performance of the EM algorithm method to the traditional multiple imputation approach through simulation studies. For structured types of correlations, such as exchangeable or first-order auto-regressive (AR-1) correlation, the EM algorithm outperforms the multiple imputation approach in terms of both estimation bias and efficiency.

Managing Inventory with Limited History of Intermittent Demand

We consider a single-product discrete-time inventory model with intermittent demand. In every period, either zero demand or a positive demand is observed with an unknown probability. The distribution of the positive demand is assumed to be from the location-scale family with unknown mean and variance. The functional form of the optimal inventory target is available but it is a function of the unknown intermittent demand parameters that must be estimated from a limited amount of historical demand data. We first quantify the expected cost associated with implementing the optimal inventory policy using the point estimates of the unknown parameters by ignoring the uncertainty around them. We then minimize this expected cost with respect to a threshold variable that factors the statistical estimation errors of the unknown parameters into the inventory decision. We find that the use of an optimized threshold leads to a significant reduction in the a priori expected cost of the decision maker.