Small-sample estimators of the quantiles of the normal,log-normal and Pareto distributions

Estimators of the quantiles of the normal and log-normal distributions are derived. They are more efficient than the established estimators by a wide margin for small samples and high quantiles of the log-normal distribution. Although their evaluation is iterative, it requires only moderate amount of computing, which is not related to the sample size. The method is also applied to the quantiles of the Pareto distribution, but the resulting estimator is more efficient only in some settings. An application to financial statistics, estimating the return on a unit investment in equity markets over a long term, is presented.     

More efficient logistic analysis using moving extreme ranked set sampling

Logistic regression is the most popular technique available for modeling dichotomous-dependent variables. It has intensive application in the field of social, medical, behavioral and public health sciences. In this paper we propose a more efficient logistic regression analysis based on moving extreme ranked set sampling (MERSS_min) scheme with ranking based on an easy-to-available auxiliary variable known to be associated with the variable of interest (response variable). The paper demonstrates that this approach will provide more powerful testing procedure as well as more efficient odds ratio and parameter estimation than using simple random sample (SRS). Theoretical derivation and simulation studies will be provided. Real data from 2011 Youth Risk Behavior Surveillance System (YRBSS) data are used to illustrate the procedures developed in this paper.     

More efficient local polynomial regression with random-effects panel data models

We propose a modification on the local polynomial estimation procedure to account for the “within-subject” correlation presented in panel data. The proposed procedure is rather simple to compute and has a closed-form expression. We study the asymptotic bias and variance of the proposed procedure and show that it outperforms the working independence estimator uniformly up to the first order. Simulation study shows that the gains in efficiency with the proposed method in the presence of “within-subject” correlation can be significant in small samples. For illustration purposes, the procedure is applied to explore the impact of market concentration on airfare.     

On a further generalization of the savage test

A statistic based on the frequencies within the k+1 intervals specified by k arbitrary quantiles is proposed for a LMP test against Lehmann alternatives generalizing the Savage test for the two-sample problem. The maximum efficiency relative to the Savage test for optimally chosen k quantiles is also provided for k=l(2)l5. The asymptotic normality of the statistic follows from the asymptotic multinomial distribution of the frequencies in the classes determined by the k quantiles.     

Constructing quantile confidence intervals using extended simple random sample in finite populations

This paper constructs quantile confidence intervals based on extended simple random sample (SRS) from a finite population, where ranks of population units are all known. Extended simple random sample borrows additional information from unmeasured observations in the population by conditioning on the population ranks of the measured units in SRS. The confidence intervals are improved using Rao-Blackwell theorem over the conditional distribution of sample ranks given the measured sample units. Empirical evidence shows that the proposed confidence intervals have shorter lengths than confidence intervals constructed from an SRS sample.     

A generalized semiparametric regression and its efficient estimation

We investigate a generalized semiparametric regression. Such a model can avoid the risk of wrongly choosing the base measure function. We propose a profile likelihood to efficiently estimate both parameter and nonparametric function. The main difference from the classical profile likelihood is that the profile likelihood proposed is a functional of the base measure function, instead of a function of a real variable. By making the most of the structure information of the semiparametric exponential family, we get an explicit expression of the estimator of the least favorable curve. It ensures that the new profile likelihood is computationally simple. Due to the use of the least favorable curve, the semiparametric efficiency is achieved successfully and the estimation bias is reduced significantly. Simulation studies can illustrate that our proposal is much better than the existing methodologies for most cases under study, and is robust to the different model conditions.     

On nonparametric regression estimators based on regression quantiles

In the ciassical regression model Y_i=h(x_i) + ? _i, i=1,…,n, Cheng (1984) introduced linear combinations of regression quantiles as a new class of estimators for the unknown regression function h(x). The asymptotic properties studied in Cheng (1984) are reconsidered. We obtain a sharper scrong consistency rate and we improve on the conditions for asymptotic normality by proving a new result on the remainder term in the Bahadur representation for regression quantiles.     

Graphical Display of Means

A graphical procedure for the display of treatment means that enables one to determine the statistical significance of the observed differences is presented. It is shown that the widely used least significant difference and honestly significant difference statistics can be used to construct plots in which any two means whose uncertainty intervals do not overlap are significantly different at the assigned probability level. It is argued that these plots, because of their straightforward decision rules, are more effective than those that show the observed means with standard errors or confidence limits. Several examples of the proposed displays are included to illustrate the procedure.     

Generating random points in a ball

A fast method is presented for generating pseudorandom uniform points in a ball of dimension 3, 4, or 5.     

The asymptotic behaviour of a class of L.-estimators under long-range dependence

This paper obtains asymptotic representations of a class of L-estimators in a linear regression model when the errors are a function of long-range-dependent Gaussian random variables. These representations are then used to address some of the efficiency robustness properties of L-estimators compared to the least-squares estimator. It is observed that under the Gaussian error distribution, each member of the class has the same asymptotic efficiency as that of the least-squares estimator. The results are obtained as a consequence of the asymptotic uniform linearity of some weighted empirical processes based on long-range-dependent random variables.     

Some comments on sample quantiles for dependent observations

Sample quantiles obtained from a linear process are examined and their asymptotic properties are derived by using simple elementary arguments.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

Solving For the parameters of a beta a distribution under two quantile constraints

It will be shown that a solution exists for the parameters of a beta distribution given any combination of a lower quantile and upper quantile constraint. A numerical procedure is developed to solve for the parameters of the beta distribution given these quantile constraints. Example solutions are provided.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

Confidence bands for bivariate quantiles

The problem of calculating joint confidence bands for the quantiles of the bivariate normal distribution is solved. Numerical implementation of the results is discussed and calculations are presented for selected cases.     

Expressions for the distribution and percentiles of the sums and products of chi-squares

Linear combinations of central or non-central chi-squares occur naturally in a variety of contexts. The products of chi-squares occur when a variance has a chi-square prior and in electrical engineering. Here, we give expansions for their distribution and quantiles and also for the products of the powers of chi-squares, including ratios. These provide much more accurate approximations than those based on asymptotic normality. The larger the degrees of freedom or the larger the non-centrality parameters, the better the approximations. We give the first four terms of these expansions. These provide approximations with errors smaller by five magnitudes than those based on asymptotic normality or on Satterthwaite's approximation. His method matched the first two moments of the target and a multiple of a chi-square and is only a first-order approximation like that based on the central limit theorem. We show that it can be made second order by matching the first three moments. The appendices show how to obtain analytical expressions for the distribution of weighted sums of chi-squares.     

Sensitivity of some posterior summaries when the prior is unimodal with specified quantiles

Prior information is modeled either by (1) a class of unimodal distributions with specified quantiles, or by (2) an ∈-contamination class with contaminations subject to unimodal and quantile restrictions. We show how the ranges of certain posterior summaries can be found. Using numerical examples, we compare the robustness behaviour under these classes, and assess the effect, in terms of robustness, of specifying a base prior.