Sample Quantiles Research Articles

Simulation Estimation of Quantiles From a Distribution With Known Mean

It is common in practice to estimate the quantiles of a complicated distribution by using the order statistics of a simulated sample. If the distribution of interest has known population mean, then it is often possible to improve the mean square error of the standard quantile estimator substantially through the simple device of mean-correction: subtract off the sample mean and add on the known population mean. Asymptotic results for the meancorrected quantile estimator are derived and compared to the standard sample quantile. Simulation results for a variety of distributions and processes illustrate the asymptotic theory. Application to Markov chain Monte Carlo and to simulation-based uncertainty analysis is described.

Coeficientes de variação de algumas características do meloeiro: uma proposta de classificação

Uma proposta para classificação dos coeficientes de variação das características: produtividade, peso do fruto, número de frutos, teor de sólidos solúveis, firmeza da polpa, açúcares totais e acidez total titulável foi estabelecida em dados de trabalhos com meloeiro (Cucumis melo L.) conduzidos na cidade de Mossoró. Os valores de CVs foram obtidos de 98 trabalhos, em sua maioria dissertações e monografias defendidas na ESAM. Foram utilizados os métodos de Garcia (1989) e dos Quantis Amostrais. Houve discordância nas freqüências observadas entre as duas metodologias propostas neste trabalho e a de Pimentel Gomes (1985) para as características açúcar total, teor de sólidos solúveis, peso do fruto, produtividade e número de frutos. Para a acidez total titulável, observaram-se reduzidas diferenças quanto aos limites baixo, médio e alto, porém, segundo a classificação de Pimentel Gomes (1985), nenhum dos dados foi classificado como muito alto. Apenas para a característica firmeza da polpa houve alta concordância entre as metodologias de Garcia (1985) e dos Quantis Amostrais e a de Pimentel Gomes (1985). Os métodos de Garcia (1985) e dos Quantis Amostrais permitiram a classificação mais adequada dos coeficientes de variação para o meloeiro.

Derivative Estimation with Finite Differences

This article discusses the implementation of using finite differences to construct a confidence interval for a simulation estimator of the derivative of the steady-state distribution of a stochastic process. The quasi-independent procedure increases the simulation run length progressively until a certain number of essentially independent and identically distributed systematic samples are obtained. The author computes sample quantiles at certain grid points and constructs a histogram from those grid points. The derivative estimate is then computed from the histogram (i.e., the empirical distribution). An experimental performance evaluation demonstrates the validity of using this procedure to estimate the derivatives.

Scale invariance of regional wet and dry durations of rain fields: A diagnostic study

Most of the recent work on rainfall data analyses and modeling has focused on either spatial or temporal variability. In this paper the structure of rainfall intermittence in space and time is investigated. Using a series of TOGA‐COARE radar scans converted to maps of pixel rain rate over a tropical oceanic region of size 240 × 240 km2, regionally scale‐invariant behavior of the probability distributions of wet and dry epoch durations is explored. Durations of wet and dry epochs are estimated by lengths of wet and dry spells, respectively, in time series of spatially averaged rain rate over sampled square subregions of spatial scales ranging from 120 km to 2 km. The investigation is based on sample quantiles and sample moments of the underlying marginal probability distributions, focusing on the behavior of their tails and their variation with respect to spatial scale. We find that sample tail quantiles and sample moments of wet durations exhibit power law multiscaling, while sample tail quantiles and sample moments of dry durations exhibit exponential multiscaling across the above range of scales. These findings provide new statistical diagnostic tools for validation of spatiotemporal models for rain fields.

A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of Rd. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.

The Combination Test for Multivariate Normality

A basic graphical approach for checking normality is the Q - Q plot that compares sample quantiles against the population quantiles. In the univariate analysis, the probability plot correlation coefficient test for normality has been studied extensively. We consider testing the multivariate normality by using the correlation coefficient of the Q - Q plot. When multivariate normality holds, the sample squared distance should follow a chi-square distribution for large samples. The plot should resemble a straight line. A correlation coefficient test can be constructed by using the pairs of points in the probability plot. When the correlation coefficient test does not reject the null hypothesis, the sample data may come from a multivariate normal distribution or some other distributions. So, we use the following two steps to test multivariate normality. First, we check the multivariate normality by using the probability plot correction coefficient test. If the test does not reject the null hypothesis, then we test symmetry of the distribution and determine whether multivariate normality holds. This test procedure is called the combination test. The size and power of this test are studied, and it is found that the combination test, in general, is more powerful than other tests for multivariate normality.

Minimum Disparity Estimators for Discrete and Continuous Models

Disparities of discrete distributions are introduced as a natural and useful extension of the information-theoretic divergences. The minimum disparity point estimators are studied in regular discrete models with i.i.d. observations and their asymptotic efficiency of the first order, in the sense of Rao, is proved. These estimators are applied to continuous models with i.i.d. observations when the observation space is quantized by fixed points, or at random, by the sample quantiles of fixed orders. It is shown that the random quantization leads to estimators which are robust in the sense of Lindsay [9], and which can achieve the efficiency in the underlying continuous models provided these are regular enough.

LEAST ABSOLUTE DEVIATIONS REGRESSION UNDER NONSTANDARD CONDITIONS

Most work on the asymptotic properties of least absolute deviations (LAD) estimators makes use of the assumption that the common distribution of the disturbances has a density that is both positive and finite at zero. We consider the implications of weakening this assumption in a number of regression settings, primarily with a time series orientation. These models include ones with deterministic and stochastic trends, and we pay particular attention to the case of a simple unit root model. The way in which the conventional assumption on the error distribution is modified is motivated in part by N.V. Smirnov's work on domains of attraction in the asymptotic theory of sample quantiles. The approach adopted usually allows for simple characterizations (often featuring a single parameter, γ), of both the shapes of the limiting distributions of the LAD estimators and their convergence rates. The present paper complements the closely related recent work of K. Knight.

Spherical harmonics in quadratic forms for testing multivariate normality

We study two statistics for testing goodness of fit to the null hypothesis of multivariate normality, based on averages over the standardized sample of multivariate spherical harmonics, radial functions and their products. These statistics (of which one was studied in the two-dimensional case in Quiroz and Dudley, 1991) have, as limiting distributions, linear combinations of chi-squares. In arbitrary dimension, we obtain closed form expressions for the coefficients that describe the limiting distributions, which allow us to produce Monte Carlo approximate limiting quantiles. We also obtain Monte Carlo approximate finite sample size quantiles and evaluate the power of the statistics presented against several alternatives of interest. A power comparison with other relevant statistics is included. The statistics proposed are easy to compute (with Fortran code available from the authors) and their finite sample quantiles converge relatively rapidly, with increasing sample size, to their limiting values, a behaviour that could be explained by the large number of orthogonal functions used in the quadratic forms involved.

An exact iterated bootstrap algorithm for small-sample bias reduction

It is well known that bootstrap accuracy can be theoretically enhanced by iterating the bootstrap procedure. Monte Carlo approximation to bootstrap iteration incurs prohibitively expensive computational cost, especially when higher levels of resampling are involved. The theoretical gain promised by high-level bootstrap iteration can thus hardly be materialized in practice. By considering bootstrap iteration as a Markov process, we propose an algorithm for its implementation in the context of small-sample bias reduction. The algorithm caters for any number of bootstrap iterations and computes exact bootstrap bias-corrected estimates without the need for extensive Monte Carlo resampling. We discuss the practical value of our algorithm in situations where infinite-level bootstrap iteration yields an unbiased estimate irrespective of the sample size and where our algorithm converges rapidly. Numerical examples are given to illustrate applications to estimates such as functions of sample means, sample quantiles and the Nadaraya–Watson estimate.

APPROXIMATIONS TO POWERS OF φ-DISPARITY GOODNESS-OF-FIT TESTS1*

The paper studies a class of tests based on disparities between the real-valued data and theoretical models resulting either from fixed partitions of the observation space, or from the partitions by the sample quantiles of fixed orders. In both cases there are considered the goodness-of-fit tests of simple and composite hypotheses. All tests are shown to be consistent, and their power is evaluated at the nonlocal as well as local alternatives.

Minimum Divergence Estimators Based on Grouped Data

The paper considers statistical models with real-valued observations i.i.d. by F(x, θ0) from a family of distribution functions (F(x, θ); θ e Θ), Θ ⊂ R s , s ≥ 1. For random quantizations defined by sample quantiles (F n −1 (λ1),θ, F n −1 (λ m−1)) of arbitrary fixed orders 0 < λ1 θ < λm-1 < 1, there are studied estimators θφ,n of θ0 which minimize φ-divergences of the theoretical and empirical probabilities. Under an appropriate regularity, all these estimators are shown to be as efficient (first order, in the sense of Rao) as the MLE in the model quantified nonrandomly by (F −1 (λ1,θ0),θ, F −1 (λ m−1, θ0)). Moreover, the Fisher information matrix I m (θ0, λ) of the latter model with the equidistant orders λ = (λ j = j/m : 1 ≤ j ≤ m − 1) arbitrarily closely approximates the Fisher information J(θ0) of the original model when m is appropriately large. Thus the random binning by a large number of quantiles of equidistant orders leads to appropriate estimates of the above considered type.

Quantile Regression via an MM Algorithm

Quantile regression is an increasingly popular method for estimating the quantiles of a distribution conditional on the values of covariates. Regression quantiles are robust against the influence of outliers and, taken several at a time, they give a more complete picture of the conditional distribution than a single estimate of the center. This article first presents an iterative algorithm for finding sample quantiles without sorting and then explores a generalization of the algorithm to nonlinear quantile regression. Our quantile regression algorithm is termed an MM, or majorize—minimize, algorithm because it entails majorizing the objective function by a quadratic function followed by minimizing that quadratic. The algorithm is conceptually simple and easy to code, and our numerical tests suggest that it is computationally competitive with a recent interior point algorithm for most problems.

On ranked-set sample quantiles and their applications

The properties of the sample quantiles of ranked-set samples are dealt with in this article. For any fixed set size in the ranked-set sampling the strong consistency and the asymptotic normality of the ranked-set sample quantiles for large samples are established. Bahadur representation of the ranked-set sample quantiles are also obtained. These properties are used to develop procedures for the inference on population quantiles such as confidence intervals and hypotheses testings. The efficiency of these procedures relative to their counterpart in simple random sampling is investigated. The ranked-set sampling is generally more efficient than the simple random sampling in this context. The gain in efficiency by using ranked-set sampling is quite substantial for the inference on middle quantiles and is the largest on the median. However, the relative efficiency damps away as the quantiles move away from the median on both directions. The gain in efficiency becomes negligible for extreme quantiles.

Asymptotic Properties of Sample Quantiles Constructed from Samples with Random Sizes

This paper gives necessary and sufficient conditions for the weak convergence of joint distributions of sample quantiles constructed from samples of random sizes. The limit distributions are described.

Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles

Summary For a sample from a given distribution the difference of two order statistics and the Studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences. This paper gives saddlepoint approximations for densities and saddlepoint approximations of the Lugannani–Rice form for tail probabilities of these statistics. The relative errors of the approximations are n −1 uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log-concave.

Theory &amp; Methods: Saddlepoint Approximation for Sample and Bootstrap Quantiles

The paper gives the saddlepoint approximation for the distribution function of the sample quantile. A comparison of the saddlepoint approximations for the distribution functions of the sample quantile and the bootstrap quantile shows that the error of the bootstrap approximation to the distribution of the sample quantile obtained by Singh (1981) as an absolute error is actually a relative error.

Distribution tails of sample quantiles and subexponentiality

We show that subexponentiality is not sufficient to guarantee that the distribution tail of a sample quantile of an infinitely divisible process is equivalent to the “tail” of the same sample quantile under the corresponding Lévy measure. However, such an equivalence result is shown to hold under either an assumption of an appropriately slow tail decay or an assumption on the structure of the process.

LQ-moments: Analogs of L-moments

The L-moments, certain linear functions of the expectations of order statistics, were introduced by Sillitto (1951) and comprehensively reviewed by Hosking (1990). L-moments have found wide applications in such fields of applied research as civil engineering, meteorology, and hydrology. We present a class of their analogs, labeled LQ-moments, obtained by replacing the expectations by functionals inducing quick estimators such as the median, Gastwirth estimator and the trimean. The LQ-moments always exist and are simpler to evaluate and estimate. The measures of skewness and kurtosis based on the LQ-moments can be used as more convenient and effective alternatives to the traditional beta coefficients. Simple explicit schemes for their estimation using readily obtainable sample quantiles are presented. The simplicity of their analyses is demonstrated in terms of the asymptotic distributions of the estimators of the LQ-moments, LQ-skewness and LQ-kurtosis. An application to the extreme value problem in flood data analysis in hydrology is discussed, and several potential applications are outlined.

Optimal spacing of the selected sample quantiles for the joint estimation of the location and scale parameters of a symmetric distribution

We are considering the ABLUE’s – asymptotic best linear unbiased estimators – of the location parameter μ and the scale parameter σ of the population jointly based on a set of selected k sample quantiles, when the population distribution has the density of the form g(x)= 1 σ f x−μ σ , −∞<μ<∞, σ>0, where the standardized function f( u) being of a known functional form. A set of selected sample quantiles with a designated spacing λ : λ 1<λ 2<⋯<λ k, or in terms of u=( x− μ)/ σ u : −∞<u 1<u 2<⋯<u k<+∞, where λ i=∫ −∞ u i f(t) dt, i=1,2,…,k are given by x(n 1)<x(n 2)<⋯<x(n k), where n i= nλ i if nλ i is an integer , i=1,2,…,k, [nλ i]+1 otherwise. Asymptotic distribution of the k sample quantiles when n is very large is given by h(x(n 1),x(n 2),…,x(n k);μ,σ)=(2 πσ 2) −k/2[λ 1(λ 2−λ 1)⋯(λ k−λ k−1)(1−λ k)] −1/2n k/2 exp(−nS/2σ 2), where S= ∑ i=1 k λ i+1−λ i−1 (λ i+1−λ i)(λ i−λ i−1) f i 2(x(n i)−μ−u iσ) 2−2 ∑ i=2 k f if i−1 λ i−λ i−1 (x(n i)−μ−u iσ)(x(n i−1)−μ−u i−1σ), f i=f(u i), i=0,1,…,k,k+1, f 0=f k+1=0, λ 0=0, λ k+1=1. The relative efficiency of the joint estimation is given by η(μ,σ)= 1 κ Δ Δ=K 1K 2−K 3 2, where K 1= ∑ i=1 k+1 (f i−f i−1) 2 λ i−λ i−1 , K 2= ∑ i=1 k+1 (u if i−u i−1f i−1) 2 λ i−λ i−1 , K 3= ∑ i=1 k+1 (f i−f i−1) (u if i−u i−1f i−1) λ i−λ i−1 and κ being independent of the spacing λ . The optimal spacing is the spacing which maximizes the relative efficiency η( μ, σ). We will prove the following rather remarkable theorem. Theorem. The optimal spacing for the joint estimation is symmetric, i.e. λ i+λ k−i+1=1, or u i+u k−i+1=0, i=1,2,…,k, if the standardized density f( u) of the population is differentiable infinitely many times and symmetric f(−u)=f(u), f′(−u)=−f′(u).