Comparison of methods of estimation of parameters of the weibull distribution

The generalised least squares, maximum likelihood, Bain-Antle 1 and 2, and two mixed methods of estimating the parameters of the two-parameter Weibull distribution are compared. The comparison is made using (a) the observed relative efficiency of parameter estimates and (b) themean squared relative error in estimated quantiles, to summarize the results of 1000 simulated samples of sizes 10 and 25. The results are that: generalised least squares is the best method of estimating the shape parameter ß the best method of estimating the scale parameter a depends onthe size of ß for quantile estimation maximum likelihood is best Bain-Antle 2 is uniformly the worst of the methods.     

Nonparametric estimation of the number of components of a superposition of renewal processes

Suppose all events occurring in an unknown number (ν)$(\nu )$ of iid renewal processes, with a common renewal distribution F  , are observed for a fixed time τ$\tau$, where both ν$\nu$ and F   are unknown. The individual processes are not known a priori, but for each event, the process that generated it is identified. For example, in software reliability application, the errors (or bugs) in a piece of software are not known a priori, but whenever the software fails, the error causing the failure is identified. We present a nonparametric method for estimating ν$\nu$ and investigate its properties. Our results show that the proposed estimator performs well in terms of bias and asymptotic normality, while the MLE of ν$\nu$ derived assuming that the common renewal distribution is exponential may be seriously biased if that assumption does not hold.     

Table of percentage points of ratios of linear combinations of order statistics of samples from exponential distributions

A form of the distribution function of ratios of linear combinations of order statistics of samples from an exponential distribution is given. From the distribution, tables of percentage points of the statistic for α = .05, .95, and n = 3(1)50, and for censoring up to five observations are presented. Use of the tables is made to find critical values of the most powerful scale and location invariant test of exponentiality against uniformity, and also to find critical values for a test of outliers in an exponential population.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.     

Comparison of dispersions of mixtures of ordered distributions

In the paper we consider mixtures of unknown stochastically ordered distribution functions according to known mixing distribution functions. We provide optimal lower and upper bounds on ratios of general dispersion measures of such mixtures. The bounds do not depend on the particular form of dispersion measure. We present applications of the results in reliability theory, insurance mathematics, Bayesian statistics, and regression analysis.     

Minutes of Committee of Presidents of Statistical Societies

The general approach to generating random variates through transformations with multiple roots is discussed. Multinomial probabilities are determined for the selection of the different roots. An application of the general result yields a new and simple technique for the generation of variates from the inverse Gaussian distribution.     

Estimate of variance of u-statistics

Let g(x₁,… , x_k) be a symmetric function with k arguments. Let U be a U-statistic based on a random sample of size n with kernel function g . In this paper, the problem of estimating var(U) is considered. Several estimators are compared by computer simulations and we conclude that two estimators, one is constructed as a U-statistic and the other is the bootstrap estimator, give good estimates for many U-statistics.     

Estimation of an indicator of the representativeness of survey response

Nonresponse is a major source of estimation error in sample surveys. The response rate is widely used to measure survey quality associated with nonresponse, but is inadequate as an indicator because of its limited relation with nonresponse bias. Schouten et al. (2009) proposed an alternative indicator, which they refer to as an indicator of representativeness or R-indicator. This indicator measures the variability of the probabilities of response for units in the population. This paper develops methods for the estimation of this R-indicator assuming that values of a set of auxiliary variables are observed for both respondents and nonrespondents. We propose bias adjustments to the point estimator proposed by Schouten et al. (2009) and demonstrate the effectiveness of this adjustment in a simulation study where it is shown that the method is valid, especially for smaller sample sizes. We also propose linearization variance estimators which avoid the need for computer-intensive replication methods and show good coverage in the simulation study even when models are not fully specified. The use of the proposed procedures is also illustrated in an application to two business surveys at Statistics Netherlands.     

SNA关于机构部门分类的修订与中国机构部门的调整研究

 本文对2008SNA关于机构部门分类的修订进行了梳理。对比了2008SNA和1993SNA在机构部门与机构单位内容上的主要区别,分析了中国国民经济核算体系关于机构部门分类所存在的问题,提出了机构部门相关核算方法的调整建议。     

中国第三产业就业效应的实证分析

就业效应是第三产业的一个显著特点。一方面,通过发展第三产业可以促进就业;另一方面,又可通过就业的扩大推动第三产业的发展,实现发展第三产业与扩大就业的良性互动。文章在分析我国第三产业发展和劳动就业现状的基础上,运用计量分析的方法,对第三产业发展中的劳动就业贡献率及其对劳动力的吸纳能力进行了分析,并提出了相关的政策建议。     

Estimation of variance of mean using known coefficient of variation

An optimum unbiased estimator of the variance of mean is given It is defined as a function of the mean and itscustomary unbiased variance estimator, utilizing known coefficient of variation, skewness and kurtosis of the underlying distributions. Exact results are obtained. Normal and large sample cases receive particular treatment. The proposed variance estimator is generally more efficient than the customary variance estimator; its relative efficiency becomes appreciably higher for smaller coefficient of variation, smaller sample (in the normal case at least), higher negative skewness, or higher positive skewness with sufficiently large kurtosis. The empirical findings are reassuring and supportive.     

A comparison of estimators of the degree of insect control

The current estimator of the degree of insect control by an insecticide in a field experiment laid out in randomized blocks is equal to one minus the cross-product ratio of a two way table of total insect counts over blocks. Since much work has been done on estimation of the common odds ratio of a number of strata in medical studies, a series of Monte Carlo studies was performed to investigate the possible use of these estimators and their standard errors in estimating the common degree of inject control of a number of blocks. Maximum likelihood, Mantel-Haenszel, and empirical logit estimators were evaluated and compared with back-transformed means over blocks, of cross-product ratios on the arithmetic, logarithmic, and arcsine scales. Maximum likelihood and Mantel-Haenszel estimators had the smallest mean squared errors, but their standard error estimates were only appropriate when sampling distributions were approximately Poisson and there was little heterogeneity among plots within blocks in the natural rates of population change.     

Efficient computation of the performance of bootstrap and jackknife estimators of the variance of L-statistics

Calculation of the bootstrap and the jackknife estimators of the variance of a statistic often relies on approximation techniques because the exact values are difficult if not impossible to obtain analytically. For the special case where the statistic is a linear combination of order statistics we propose to calculate the exact values combinatorically, thus completely eliminating the second-stage simulation error.     

Simulation of extremes of diffusions

The authors consider the problem of simulating the times of events such as extremes and barrier crossings in diffusion processes. They develop a rejection sampler based on Shepp [Shepp, Journal of Applied Probability 1979; 16:423–427] for simulating an extreme of a Brownian motion and use it in a general recursive scheme for more complex simulations, including simultaneous simulation of the minimum and maximum and application to more general diffusions. They price exotic options that are difficult to price analytically: a knock‐out barrier option with a modified payoff function, a lookback option that includes discounting at the risk‐free interest rate, and a chooser option where the choice is made at the time of a barrier crossing. The Canadian Journal of Statistics 38: 738–755; 2010 © 2010 Statistical Society of Canada     

Analysis of variance of paired data without repetition of measurement

Summary The objective of this analysis of variance of paired data is to estimate positive random error variances for each ofN=2 measurement methods. The two methods measure the same item only once without measurement repetition. The well-known unbiased Grubbs’ estimators are not suitable for practical purpose because they can become negative. With the help of Chebyshev’s inequality the probability was determined that Grubbs’ estimators become negative. Based on the Grubbs’ estimators new estimators were derived. The new estimators are indeed always positive, but they are biased. It is shown that the biases are small. In case the Grubbs’ estimators are positive a bias correction of the new estimators may be envisaged.

---

Zusammenfassung Das Ziel dieser Varianzanalyse von gepaarten Messungen ist die Sch?tzung zuf?lliger Messfehlervarianzen für jede derN=2 Messmethoden. Die beiden Messmethoden messen das gleiche Merkmal eines Elements nur einmal ohne Messwiederholung. Die bekannten unverzerrten Grubbs-Sch?tzer sind für die praktische Anwendung nicht geeignet, weil sie negativ werden k?nnten. Die Tschebyscheffsche Ungleichung wurde genutzt, um die Wahrscheinlichkeit zu ermitteln, dass Grubbs-Sch?tzer negativ werden. Basierend auf Grubbs-Sch?tzern wurden neue Sch?tzer hergeleitet. Diese neuen Sch?tzer sind zwar immer positiv, aber verzerrt. Es wird gezeigt, dass die Verzerrungen klein sind. Für den Fall, dass die Grubbs-Sch?tzer positiv ausfallen, k?nnte eine Korrektur der Verzerrung in Betracht gezogen werden.

     

SAS软件的应用——基于ARMA模型的商品销售额的预测分析

文章运用SAS软件系统中的一些时间序列建模方法及回归分析方法对某商品的月销售额作了预测分析,得到了较高的预测精度,在实际应用中预测值的准确对于指导商家的战略决策起着重要作用.     

Test of equality of normal means in the absence of independent estimator of variance

Let X_l,…,X_n be normally and independently distributed with means θ_l,…,θ_nand a cornmorl variance. Thus there are n observations and n+i unknwon parameters. A test of the null hypothesis that, the θ_i's are all zero and the alternative that the vector (θ_l,…,θ_n) lies in a convex cone with its vertex a.t the origin is connsidered in this paper. It is shown that under a mild condition the likelihood ratio test is possible. The ordinary one sided t - test belongs to the class of tests considered in this paper. The hypothesis of equality of means against the simple order alternative can be tested in certain cases .     

Measures of lack of fit from tests of chi-squared type

If X² is the Pearson chi-squared statistic for testing fit, then $\text{X}{}^2\text{n}$ has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X² as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of $\text{T}\text{n}$, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.     

Automatic repeated-loess decomposition of data consisting of sums of oscillatory curves

Repeated loess is a nonparametric procedure that uses progressive smoothing and differencing to decompose data consisting of sums of curves. Smoothing is by locally weighted polynomial regression. Here the procedure was developed so that the decomposition into components was controlled automatically by the number of maxima in each component. The level of smoothing of each component was chosen to maximize the estimated probability of the observed number of maxima. No assumptions were made about the periodicity of components and only very weak assumptions about their shapes. The automatic procedure was applied to simulated data and to experimental data on human visual sensitivity to line orientation.An erratum to this article can be found at