On the admissibility of c[Xbar] + d with respect to the linex loss function

Let X₁, …,X_n be a random sample from a normal distribution with mean θ and variance σ² The problem is to estimate θ with loss function L(θ,e) = v(e-θ) where v(x) = b(exp(ax)-ax-l) and where a, b are constants with b>0, a¦0. Zellner (1986), showed that [Xbar] ? σ²a/2n dominates [Xbar] and hence [Xbar] is inadmissible. The question of what values of c and d render c[Xbar]+ d admissible is studied here.     

Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S^-1 (x-[Xbar]) = c_i centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x² (M-1) and x² (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.     

Smooth nonparametric quantile estimation under censoring: simulations and bootstrap methods

Based on right-censored data from a lifetime distribution F , a smooth nonparametric estimator of the quantile function Q (p) is given by Qn(p)=h 1jjQn(t)K((t-p)/h)dt, where QR(p) denotes the product-limit quantile function. Extensive Monte Carlo simulations indicate that at fixed p this kernel-type quantile estimator has smaller mean squared error than (L(p) for a range of values of the bandwidth h. A method of selecting an "optimal" bandwidth (in the sense of small estimated mean squared error) based on the bootstrap is investigated yielding results consistent with the simulation study. The bootstrap is also used to obtain interval estimates for Q (p) after determining the optimal value of h.     

NONPARAMETRIC QUANTILE ESTIMATION FROM RECORD-BREAKING DATA

Sometimes, in industrial quality control experiments and destructive stress testing, only values smaller than all previous ones are observed. Here we consider nonparametric quantile estimation, both the ‘sample quantile function’ and kernel-type estimators, from such record-breaking data. For a single record-breaking sample, consistent estimation is not possible except in the extreme tails of the distribution. Hence replication is required, and for m. such independent record-breaking samples the quantile estimators are shown to be strongly consistent and asymptotically normal as m-→∞. Also, for small m, the mean-squared errors, biases and smoothing parameters (for the smoothed estimators) are investigated through computer simulations.     

Exact Non-Parametric Confidence, Prediction and Tolerance Intervals with Progressive Type-II Censoring

Abstract.  This article extends recent results [Scand. J. Statist. 28 (2001) 699] about exact non-parametric inferences based on order statistics with progressive type-II censoring. The extension lies in that non-parametric inferences are now covered where the dependence between involved order statistics cannot be circumvented. These inferences include: (a) tolerance intervals containing at least a specified proportion of the parent distribution, (b) prediction intervals containing at least a specified number of observations in a future sample, and (c) outer and/or inner confidence intervals for a quantile interval of the parent distribution. The inferences are valid for any parent distribution with continuous distribution function. The key result shows how the probability of an event involving k dependent order statistics that are observable/uncensored with progressive type-II censoring can be represented as a mixture with known weights of corresponding probabilities involving k dependent ordinary order statistics. Further applications/developments concerning exact Kolmogorov-type confidence regions are indicated.     

Sensitivity analysis of the t-distribution under truncated normal populations

In recent literature, the truncated normal distribution has been used to model the stochastic structure for a variety of random structures. In this paper, the sensitivity of the t-random variable under a left-truncated normal population is explored. Simulation results are used to assess the errors associated when applying the student t-distribution to the case of an underlying left-truncated normal population. The maximum errors are modelled as a linear function of the magnitude of the truncation and sample size. In the case of a left-truncated normal population, adjustments to standard inferences for the mean, namely confidence intervals and observed significance levels, based on the t-random variable are introduced.     

Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.     

Simultaneous Inferences on the Cumulative Distribution Function of a Normal Distribution

This paper addresses the problem of constructing simultaneous confidence intervals for the cumulative distribution function of a normal distribution at several specified points. The procedure is based upon the observation of a random sample of independent observations from a normal distribution with an unknown mean and variance. A new methodology is proposed for obtaining confidence intervals with a specified overall simultaneous confidence level through the inversion of acceptance sets. Both one-sided and two-sided confidence intervals are considered. Some illustrations of the new method are provided, and comparisons are made with other approaches to the problem.     

On using the jackknife to estimate quantile variance

We show that the jackknife technique fails badly when applied to the problem of estimating the variance of a sample quantile. When viewed as a point estimator, the jackknife estimator is known to be inconsistent. We show that the ratio of the jackknife variance estimate to the true variance has an asymptotic Weibull distribution with parameters 1 and 1/2. We also show that if the jackknife variance estimate is used to Studentize the sample quantile, the asymptotic distribution of the resulting Studentized statistic is markedly nonnormal, having infinite mean. This result is in stark contrast with that obtained in simpler problems, such as that of constructing confidence intervals for a mean, where the jackknife-Studentized statistic has an asymptotic standard normal distribution.     

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.     

Distributions of certain variables in samples from two-parameter exponential distribution

Cumulative distribution function of the variable Y=(U+c)/(Z/2ν)) is given. Here U and Z are independent random variables, U has the exponential distribution (1.1) with θ=0, σ=1, Z has the distribution χ² (2ν) and c is a real quantity. The variable Y with U and Z given by (2.2) and (2.3) is used for inference about the parametric functions ?=θ?kσ of a two-parameter exponential distribution (1.1) with k or ? known. Special cases of ? or k are: the parameter θ, the Pth quantile Xp, the mean θ+σ and the value of the cumulative distribution function or of the reliability function at given point a. Also one-sided tolerance limits for a two-parameter exponential distribution can be derived from the distribution of the variable Y. The results are also applied to the Pareto distribution.     

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.     

A multi sample test for the equality of coefficients of variation in normal populations

Two tests are derived for the hypothesis that the coefficients of variation of k normal populations are equal. The k samples may be of unequal size. The first test is the likelihood ratio test with the usual X²-approximation. A simulation study shows that the small sample behaviour under the null hypothesis is unsatisfactory. An alternative test, based on the sample coefficients of variation, appears to have somewhat better properties.     

Tables of two-sided tolerance factors for a normal distribution

Given a random sample of size N from a normal distribution, we consider tolerance intervals of the form X ? ks to X + ks, where X is the sample mean and s is the sample standard deviation. The value of k is chosen so that the interval covers a given proportion P of the population with confidence γ. Exact values of k, computed from numerical integration, are given for N = 2(1)100; P = 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.999; and γ = 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995. The exact values are compared with the values obtained from an approximation developed by Wald and Wolfowitz (1946).     

Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics

We find the distribution that has maximum entropy conditional on having specified values of its first r  L$L$-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics.     

Inference based on taguchi's saturated designs

The present paper studies the validity of inferential procedures which follow the Taguchi method, under saturated designs. The distribution of the signal to noise (S/N) ratio Y [ILM0001] is investigated,for normal parent distributions. We further investigate the distribution of orthonormal contrasts of such S/N variables. Finally, we discuss and provide critical values for mod-F tests of significance of parameters, when the k smallest SS values are pooled to serve as error variance estimate     

On the robust two wided tolerance limits based on MML estimators

Recently, Balakrishnan and Kocherlakota (1985) have proposed robust two sided tolerance limits based on the MML estimators. They have simulated the actual γ values attained by their procedure and the classical procedure based on [Xbar] and s. However, there seems to have been an error in their simulation. Here, we present the corrected table of the simulated values of γ which also reverses their recommendations.     

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number k _n of largest order statistics can be represented as a statistical tail function al, that is a functional T applied to the empirical tail quantile function Q _n. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F . For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T ( Q _n) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k _pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour     

A fractional order statistic towards defining a smooth quantile function for discrete data

This work is motivated in part by a recent publication by Ma et al. (2011) who resolved the asymptotic non-normality problem of the classical sample quantiles for discrete data through defining a new mid-distribution based quantile function. This work is the motivation for defining a new and improved smooth population quantile function given discrete data. Our definition is based on the theory of fractional order statistics. The main advantage of our definition as compared to its competitors is the capability to distinguish the uth quantile across different discrete distributions over the whole interval, u∈(0,1). In addition, we define the corresponding estimator of the smooth population quantiles and demonstrate the convergence and asymptotic normal distribution of the corresponding sample quantiles. We verify our theoretical results through a Monte Carlo simulation, and illustrate the utilization of our quantile function in a Q-Q plot for discrete data.