几种方差概念的比较

当分布数列中各标志值代表的含义不完全一样时，方差仅仅描述总体的离散程度，而半方差概念则能更准确地描述所关注的标志值的离散程度；相对方差反映的是分布数列中各标志值的相对离散程度，与标准差系数一样，相对方差可用于不同性质、不同单位或具有不同水平的数列的离散程度的比较，而方差却不适合用于这种比较；半相对方差是综合了半方差和相对方差优点的一个新指标。上述几种方差概念，从不同视角描述了数列的离散程度，各有优劣点，而且相互间有着紧密的联系。     

Half-sample variance estimation

This paper studies an alternative to the jackknife variance estimator, the half-sample variance estimator. Both theoretical and Monte Carlo comparisons between the half-sample variance estimator and the jackknife variance estimator indicate that the former is better in some situations.     

On the stability of the jackknife variance estimator in ratio estimation

The stability of a slightly modified version of the usual jackknife variance estimator is evaluated exactly in small samples under a suitable linear regression model and compared with that of two different linearization variance estimators. Depending on the degree of heteroscedasticity of the error variance in the model, the stability of the jackknife variance estimator is found to be somewhat comparable to that of one or the other of the linearization variance estimators under conditions especially favorable to ratio estimation (i.e., regression approximately through the origin with a relatively small coefficient of variation in the x population). When these conditions do not hold, however, the jackknife variance estimator is found to be less stable than either of the linearization variance estimators.     

Estimating the variances of ratio estimates in complex sample surveys with two primary sampling units per stratum—a comparison of balanced replication and jackknife techniques

The balanced half-sample and jackknife variance estimation techniques are used to estimate the variance of the combined ratio estimate. An empirical sampling study is conducted using computer-generated populations to investigate the variance, bias and mean square error of these variance estimators and results are compared to theoretical results derived elsewhere for the linear case. Results indicate that either the balanced half-sample or jackknife method may be used effectively for estimating the variance of the combined ratio estimate.     

Comments on the paper “Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response”

Singh and Arnab (2010) presented a bias adjustment to the jackknife variance estimator of Rao and Sitter (1995) in the presence of non-response. In their paper, they obtained a second-order approximation of the bias of the Rao-Sitter variance estimator and then proposed a bias-adjusted estimator based on this approximation. To compare their proposed variance estimator to various other variance estimators, they performed a simulation study and showed that their variance estimator is superior to the Rao-Sitter variance estimator. In fact they showed that the Rao-Sitter variance estimator suffers from severe underestimation. These results contradict those in the literature, which indicate that the Rao-Sitter variance estimator suffers from a positive bias if the sampling fractions are not negligible; see Rao and Sitter (1995), Lee et al. (1995) and Haziza and Picard (2011). Because of this contradiction, we felt that a further investigation was warranted. In this paper, we attempt to recreate the results of Singh and Arnab (2010) and, in fact, show that their second order approximation to the bias of the Rao-Sitter variance estimator is incorrect and that their simulation results are also questionable.     

R-squared for Bayesian Regression Models

The usual definition of R² (variance of the predicted values divided by the variance of the data) has a problem for Bayesian fits, as the numerator can be larger than the denominator. We propose an alternative definition similar to one that has appeared in the survival analysis literature: the variance of the predicted values divided by the variance of predicted values plus the expected variance of the errors.     

INDEX TO VOLUME 4 (1986)

This article provides an empirical investigation of the risk-neutral variance process and the market price of variance risk implied in the foreign-currency options market. There are three principal contributions. First, the parameters of Heston's mean-reverting square-root stochastic volatility model are estimated using dollar/mark option prices from 1987 to 1992. Second, it is shown that these implied parameters can be combined with historical moments of the dollar/mark exchange rate to deduce an estimate of the market price of variance risk. These estimates are found to be nonzero, time varying, and of sufficient magnitude to imply that the compensation for variance risk is a significant component of the risk premia in the currency market. Finally, the out-of-sample test suggests that the historical variance and the Hull and White implied variance contain no more information than that imbedded in the Heston implied variance.     

Efficient variance reduction approach based on the variation of the input importance

ABSTRACT

To reduce the output variance, the variance-based importance analysis can provide an efficient way by reducing the variance of the ‘important’ inputs. But with the reduction of the variance of those ‘important’ inputs, the input importance will change and it is no longer the most efficient way to reduce the variance of those ‘important’ inputs alone. Thus, analyst needs to consider reducing the variance of other inputs to obtain a more efficient way. This work provides a graphical solution for analyst to decide how to reduce the input variance to achieve the targeted reduction of the output variance efficiently. Furthermore, by the importance sampling-based approach, the graphical solution can be obtained with only a single group of samples, which can decrease the computational cost greatly.     

Testing for abrupt breaks in variance structures with smooth changes

Abstract

We investigate the problem of testing for variance breaks in the case where the variance structure is assumed to be smoothly time-varying under the null. Since the classical tests are aimed to detect any change in the variance, they are not able to distinguish between smooth non constant variance and abrupt breaks. In this paper a new procedure for detecting variance breaks taking into account for smooth changes in the variance under the null is proposed. The finite sample properties of the test we introduce are investigated by Monte Carlo experiments. The theoretical outputs are illustrated using U.S. macroeconomic data.     

On the coefficient of determination for mixed regression models

For mixed regression models, we define a variance decomposition including three terms, explained individual variance, unexplained individual variance and noise variance. In contrast to traditional variance decomposition, it is thus the unexplained  , not the explained, variance that is split. It gives rise to a coefficient of individual determination (CID) defined as the estimated fraction of explained individual variance. We argue that in many applications CID is a valuable complement to R²$R^2$, since it excludes noise variance (which can never be explained) and thus has one as a natural upper bound.     

Second-order variance estimation in poststratified two-stage sampling

The linearization or Taylor series variance estimator and jackknife linearization variance estimator are popular for poststratified point estimators. In this note we propose a simple second-order linearization variance estimator for the poststratified estimator of the population total in two-stage sampling, using the second-order Taylor series expansion. We investigate the properties of the proposed variance estimator and its modified version and their empirical performance through some simulation studies in comparison to the standard and jackknife linearization variance estimators. Simulation studies are carried out on both artificially generated data and real data.     

Finite Population Variance Estimation Under LSS with Multiple Random Starts

In this article, an unbiased estimator for finite population variance is developed under linear systematic sampling with two random starts and an explicit expression for its variance is also obtained. The study is supported by two real life situations. A detailed numerical comparative study has been carried out to compare its average variance with the average variance of the conventional unbiased estimator for finite population variance under simple random sampling for a wide variety of populations. Results based on the study strongly favor the use of the developed estimator for such populations.     

Variance reduction for Markov chains with application to MCMC

In this paper, we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non-asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular, we apply our method to various MCMC Bayesian estimation problems where it favorably compares to the existing variance reduction approaches.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

The effects of the proxy information on the iterative Stein-rule estimator of the disturbance variance

In a regression model with proxy variables, we consider the iterative estimator of the disturbance variance to obtain more precise estimates. In the formula of the estimator of the disturbance variance, the estimator is obtained by using Stein-rule (SR) estimator instead of OLS (ordinary least squares) estimator is called Iterative estimator of the disturbance variance. It is shown that, in a regression model with proxy variables the mean square error (MSE) of the iterative estimator of the disturbance variance is greater than the MSE of the disturbance variance related to the OLS estimator under certain conditions.     

Analysis of Variance for Bayesian Inference

This paper develops a multiway analysis of variance for non-Gaussian multivariate distributions and provides a practical simulation algorithm to estimate the corresponding components of variance. It specifically addresses variance in Bayesian predictive distributions, showing that it may be decomposed into the sum of extrinsic variance, arising from posterior uncertainty about parameters, and intrinsic variance, which would exist even if parameters were known. Depending on the application at hand, further decomposition of extrinsic or intrinsic variance (or both) may be useful. The paper shows how to produce simulation-consistent estimates of all of these components, and the method demands little additional effort or computing time beyond that already invested in the posterior simulator. It illustrates the methods using a dynamic stochastic general equilibrium model of the US economy, both before and during the global financial crisis.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

Criteria for estimating the variance function used in the asymptotic quasi-likelihood approach

The estimation of the variance function of a linear regression model used in the asymptotic quasi-likelihood approach is considered. It is shown that the variance function used in the determination of the asymptotic quasi-likelihood estimates encompasses the variance functions commonly found in the literature. Selection criteria of the most appropriate estimate of the variance function for given data are established. These criteria are based on a graphical technique and a chi-squared test.     

Variance Estimation and Asymptotic Confidence Bands for the Mean Estimator of Sampled Functional Data with High Entropy Unequal Probability Sampling Designs

For fixed size sampling designs with high entropy, it is well known that the variance of the Horvitz–Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional, and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz–Thompson estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the Hájek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of 1 week.     

Characterizing Discrete Life Distributions by Properties of Reversed Variance Residual life

The present article discusses the characterization of non negative integer-valued random variable using reversed variance residual life. A special attention is given to the characterizations by relationship between conditional variance and the reversed failure rate. A lower bound to the conditional variance is also established. Our bound is compared to the Cramer-Rao and Chapman-Robbins lower bounds so that construction of minimum variance unbiased estimators of relevant parametric functions in truncated distributions can be possible.