Tukey's Lambda Research Articles

Small‐sample properties of robust willingness‐to‐pay estimators

AbstractThe small‐sample properties of robust, binary choice willingness‐to‐pay (WTP) estimators are analyzed. A Monte Carlo simulation compares the bias and mean‐squared error of marginal and expected WTP estimates from probit, zero‐centered (“robit”), logit, and a generalized Tukey lambda distribution called a “pregibit” regression under normal and nonnormal distributional assumptions. Robust binary choice estimators allow for variation in tail thickness (the robit and pregibit) or tail asymmetry (the pregibit). No previous studies have compared the performance of these WTP estimators. The findings will interest researchers who use contingent validation methods to estimate WTP for nonmarket or hypothetical goods.

Sparse simulation-based estimator built on quantiles

The method of simulated quantiles is extended to a general multivariate framework and to provide sparse estimation of the scaling matrix. The method is based on the minimisation of a distance between appropriate statistics evaluated on the true and synthetic data simulated from the postulated model. Those statistics are functions of the quantiles providing an effective way to deal with distributions that do not admit moments of any order like the α–Stable or the Tukey lambda distribution. The lack of a natural ordering represents the major challenge for the extension of the method to the multivariate framework, which is addressed by considering the notion of projectional quantile. The SCAD ℓ1–penalty is then introduced in order to achieve sparse estimation of the scaling matrix which is mostly responsible for the curse of dimensionality. The asymptotic properties of the proposed estimator have been discussed and the method is illustrated and tested on several synthetic datasets simulated from the Elliptical Stable distribution for which alternative methods are recognised to perform poorly.

GTL regression: a linear model with skewed and thick-tailed disturbances

A maximum likelihood estimator of a linear regression model is efficient relative to the customary Ordinary Least Squares (OLS) estimator when disturbances are skewed and/or thick-tailed. In order to model skewed and thick-tailed disturbances, we specify a highly flexible Generalized Tukey Lambda (GTL) distribution that can closely mimic many other unimodal distributions. The GTL-based maximum likelihood regression estimator is consistent and asymptotically normal. A Monte Carlo study demonstrates the potential gains of this GTL-based estimator over the OLS estimator, and as a real-life application, an analysis of speeding tickets illustrates how GTL regression might modify standard OLS estimation results. For the applied data analyst, an LM test statistic is suggested as a straightforward post-estimation diagnostic of whether the standard OLS regression approach is suitable for the data at hand. Stata do-files are provided to perform the OLS post-estimation LM test and to implement GTL regression models.

Mean Difference and Mean Deviation of Tukey Lambda Distribution

The purpose of this paper is to broaden the knowledge of mean difference and, in particular, of an important distribution model known as Tukey lambda, which is generally used to choose a model to fit data. We have obtained compact formulas, which are not yet reported in literature, of mean deviation and mean difference related to the said distribution model. These results made it possible to analyze the relationships among variability indexes, namely standard deviation, mean deviation and mean difference, regarding Tukey lambda model.

Linking Tukey’s legacy to financial risk measurement

Financial data are often thick-tailed and exhibit skewness. The versatile Generalized Tukey Lambda (GTL) distribution is able to capture varying degrees of skewness in thin- or thick-tailed data. Such versatility makes the GTL distribution potentially useful in the area of financial risk measurement. Moreover, for GTL-distributed random variables, the familiar risk measures of Value at Risk (VaR) and Expected Shortfall (ES) may be expressed in simple analytical forms. It turns out that, both analytically and through Monte Carlo simulations, GTL’s VaR and ES differ significantly from other flexible distributions. The asymptotic properties of the maximum likelihood estimator of the GTL parameters are also examined. In order to study risk in financial data, the GTL distribution is inserted into a GARCH model. This GTL-GARCH model is estimated with data on daily returns of GE stock, demonstrating that, for certain data sets, GTL may capture risk measurements better than other distributions.11Online supplementary materials consist of appendices with proofs and additional Monte Carlo results, data used in this study, an R script for fitting GTL densities by maximum likelihood, and an R script for estimation of the GTL-GARCH model (see Appendix A).

GTL Regression: A Linear Model with Skewed and Thick-Tailed Disturbances

If the disturbances of a linear regression model are skewed and/or thick-tailed, a maximum likelihood estimator is efficient relative to the customary Ordinary Least Squares (OLS) estimator. In this paper, we specify a highly flexible Generalized Tukey Lambda (GTL) distribution to model skewed and thick-tailed disturbances. The GTL-regression estimator is consistent and asymptotically normal. We demonstrate the potential gains of the GTL estimator over the OLS estimator in a Monte Carlo study and in five applications that are typical of applied economics research problems: log-wage equations, hedonic housing price equations, an analysis of speeding tickets, the issue of trade creation and trade diversion that result from preferential trade agreements, and the familiar CAPM model in financial economics.

Comparación de los procedimientos de Fleishman y Ramberg et al. para generar datos no normales en estudios de simulación

Simulation techniques must be able to generate the types of distributions most commonly encountered in real data, for example, non-normal distributions. Two recognized procedures for generating non-normal data are Fleishman’s linear transformation method and the method proposed by Ramberg et al. that is based on generalization of the Tukey lambda distribution. This study compares these procedures in terms of the extent to which the distributions they generate fit their respective theoretical models, and it also examines the number of simulations needed to achieve this fit. To this end, the paper considers, in addition to the normal distribution, a series of non-normal distributions that are commonly found in real data, and then analyses fit according to the extent to which normality is violated and the number of simulations performed. The results show that the two data generation procedures behave similarly. As the degree of contamination of the theoretical distribution increases, so does the number of simulations required to ensure a good fit to the generated data. The two procedures generate more accurate normal and non-normal distributions when at least 7000 simulations are performed, although when the degree of contamination is severe (with values of skewness and kurtosis of 2 and 6, respectively) it is advisable to perform 15000 simulations.

GT&#923: A More Flexible Distribution for Garch-Type Models

We introduce the Generalized Tukey Lambda distribution in estimation of GARCH-type models. To assess its performance against other well-known distributions, we implement two methods. First, with an extensive Monte Carlo study, we check for bias in finite samples. Second, we carry out an experiment to evaluate out-of-sample forecasting performance of the new estimator on two different data sets, and test for robustness.

A Flexible Sample Selection Model: A Gtl-Copula Approach

In this paper, we propose a new approach to estimating sample selection models that combines Generalized Tukey Lambda (GTL) distributions with copulas. The GTL distribution is a versatile univariate distribution that permits a wide range of skewness and thick- or thin-tailed behavior in the data that it represents. Copulas help create versatile representations of bivariate distribution. The versatility arising from inserting GTL marginal distributions into copula-constructed bivariate distributions reduces the dependence of estimated parameters on distributional assumptions in applied research. A thorough Monte Carlo study illustrates that our proposed estimator performs well under normal and nonnormal settings, both with and without an instrument in the selection equation that fulfills the exclusion restriction that is often considered to be a requisite for implementation of sample selection models in empirical research. Five applications ranging from wages and health expenditures to speeding tickets and international disputes illustrate the value of the proposed GTL-copula estimator.

Selection and Ranking Procedures for Type I Extreme Value Populations and a Related Homogeneity Test

Consider k (≥2) independent Type I extreme value populations with unknown location parameters and common known scale parameter. With samples of same size, we study procedures based on the sample means for (1) selecting the population having the largest location parameter, (2) selecting the population having the smallest location parameter, and (3) testing for equality of all the location parameters. We use Bechhofer's indifference-zone and Gupta's subset selection formulations. Tables of constants for implemention are provided based on approximation for the distribution of the standardized sample mean by a generalized Tukey's lambda distribution. Examples are provided for all procedures.

A hierarchy of lorenz curves based on the generalized tukey's lambda distribution

A hierarchy of Lorenz curves based on the generalized Tukey's Lambda distribution is proposed. Representations of the corresponding distribution and density function are also provided, together with popular inequality measures. Estimation methods are suggested. Finally, a comparison with other parametric families of Lorenz curves is established.

A Bivariate Generalized Lambda Distribution (GLD-2) using Plackett's Method of Construction: Distribution, Examples and Applications

SYNOPTIC ABSTRACTThe Generalized Lambda Distribution (GLD), an extension of Tukey's Lambda Distribution, has proved to be a useful family for fitting probability distributions. However, the specification of the distribution via its inverse function would seem to preclude direct bivariate or multivariate extensions (at least in a reasonably straightforward form), since any bivariate F(x,y) is a function of the marginals F1(x) and F2(y) called a copula, F(x,y) = C(F1(x),F2(y)). Johnson and Kotz (1973) proposed a non-closed-form multivariate extension of Tukey's original (ungeneralized) distribution, but given the lack of a closed form for the univariate GLD it seems likely that a multivariate (even bivariate) extension of the four-parameter GLD would be even more difficult to give a closed-form distribution function for. However, R.L. Plackett (1965) proposed a method for generating bivariate distributions from specified marginals, and his method is employed in this paper to furnish a bivariate GLD-2. Altho...

Extremes,extreme spacings and outliers in the tukey and weibull families

In this paper we analyze the tail character of the generalized Tukey lambda and extended Weibull distributions and in the process demonstrate that the details of convergence in law of the extreme spacings are graphically descriptive of the tail-lengths and the associated outlier proneness. In terms of the classical extreme value theory, it is seen that the two shape parameters of the generalized Tukey lambda family separately determine the nature of the two tails individually and that the family is sparse in medium tailed distributions. For the extended Weibull (or generalized extreme value) family on the other hand, the right tail is always medium, while the left tail can be either long or short. An analysis based upon the limiting distribution of extreme spacings especially O p ·, the high probability order of magnitude for large samples, is used to refine and clarify the classical tail classification. The derivations involve elementary enough methods to make the results pedagogically suitable èven at introductory level.

Robustness of mean E(X) and R charts

The effect of nonnormality on E(X) and R charts is reported. The effect of departure from normality can be examined by comparing the probabilities that E(X) and R lie outside their three-standard-deviation and two-standard-deviation control limits. Tukey's lambda -family of symmetric distributions is used because it contains a wide spectrum of distributions with a variety of tail areas. The constants required to construct E(X) and R charts for the lambda -family are computed. Control charts based on the assumption of normality give inaccurate results when the tails of the underlying distribution are thin or thick. The validity of the normality assumption is examined by using a numerical example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A study of the generalized tukey lambda family

The Tukey lambda family of distributions together with its extensions have played an important role in statistical practice. In this paper a con¬tinuously defined two-parameter generalization of this family, which holds promise of a variety of additional applications, is variously studied. The coefficients of skewness and kurtosis and the density shapes of its members are examined and the family is related to the classical Pearsonian system of distributions.

The generalized lambda distribution as a survival model: The generalized single hit model

The generalized single hit (GSH) distribution has been proposed as a survival distribution for modeling populations with decreasing hazard. This distribution is a member of Turner and Pruitt's [15] generic family of survival distributions. The equivalence of the GSH distribution and a member of the generalized Tukey's lambda distribution is demonstrated. In addition, the GSH is shown to be equivalent to other known distributions, such as Hjorth's IDB and Pearson's type VIII. The GSH is the result of compounding a gamma prior distribution on exponential failure times. Thus, from a modeling viewpoint, the GSH can be used when it is assumed that the population with decreasing hazard is made up of individuals with constant hazards. An example is presented fitting the GSH distribution to censored data by the method of maximum likelihood.

Measurement of skewness and kurtosis for the generalized tukey lambda distributions

Generalizations of the symmetric Tukey lambda distributions to asymmetric random variables are considered. These distributions, defined via inverses F-1 of their distribution functions, yield simple algorithms useful in Monte Carlo studies and they are also useful in fitting data. Here a recently recommended measure of skewness is used to determine the skewness of such distributions. This skewness measure is also adapted to the measurement of the kurtosis of members of the symmetric class. Applciations to the simulation of symmetric long-tailed distributions and to the selection of appropriate members of the Tukey families as approximations are considered. Some insight into the interpretation of Monte Carlo studies of tests of normality is presented

Skewness properties of asymmetric foems of tu1cey lambda distributions

Some measures of skewness are the third central moments the mean minus the median, arid measures in terms of percentiles. The signs of these measures are analysed for two asymmetric generalisations of Tukey's lambda distribution. The results expand and explain some results of Ramberg et al.1979, and illustrate the problem of choosing an appropriate measure of skewness.

Robust Estimation of Location Using Optimally Chosen Sample Quantiles

Robust estimation of the location parameter α based on selected order statistics is considered. The distribution function is only known to belong to a subset of a set D of distributions consisting of Tukey's lambda family of symmetric distributions with the inverse distribution function of the form α + β/γ) [λγ – (1 – λ)γ], 0 ≤ λ ≤ 1, and the normal, Cauchy and double exponential distributions. The scale parameter β can be unknown. In the set D, distributions with tail length varying from short to extremely long, when the shape parameter γ varies, are included. The asymptotically best linear estimate (ABLE) based on k (k = 2(1)5) optimally chosen symmetrical sample quantiles is considered. It can be used as a robust estimate and is shown to compete favorably with optimally trimmed and Winsorized means, in the sense of giving a higher guaranteed relative asymptotic efficiency (GRAE) for subsets of D. Tables are provided so that the robust k-ABLE giving the highest GRAE for any subset of D can easily be obtained. An adaptive procedure of trying to identify the prototype by means of estimating the tail length is suggested.

Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form

Distributions whose inverse forms are explicitly defined, such as Tukey's lambda, may present problems in deriving their parameters by more conventional means. Probability weighted moments are introduced and shown to be potentially useful in expressing the parameters of these distributions.