Jackknife Variance Estimator Research Articles

Jackknife variance estimators in linear models

An alternative to the weighted jackknife variance estimator of Wu (1986) is introduced. Motivated by the comments of Efron (1986), Ghosh (1986) and Tibshirani (1986), the performance of Wu's estimator is compared with the proposed estimator in terms of the average loss in some linear models. Some asymptotic properties of the proposed estimator are discussed

Jackknife variance estimation with survey data under hot deck imputation

SUMMARY Hot deck imputation is commonly employed for item nonresponse in sample surveys. It is also a common practice to treat the imputed values as if they are true values, and then compute the variance estimates using standard formulae. This procedure, however, could lead to serious underestimation of the true variance, when the proportion of missing values for an item is appreciable. We propose a jackknife variance estimator for stratified multistage surveys which is obtained by first adjusting the imputed values for each pseudo-replicate and then applying the standard jackknife formula. The proposed jackknife variance estimator is shown to be consistent as the sample size increases, assuming equal response probabilities within imputation classes and using a particular hot deck imputation.

Jackknife variance estimators of the location estimator in the one-way random-effects model

In a one-way random-effects model, we frequently estimate the variance components by the analysis-of-variance method and then, assuming the estimated values are true values of the variance components, we estimate the population mean. The conventional variance estimator for the estimate of the mean has a bias. This bias can become severe in contaminated data. We can reduce the bias by using the delta method. However, it still suffers from a large bias. We develop a jackknife variance estimator which is robust with respect to data contamination.

Some Asymptotic Results for Jackknifing the Sample Quantile

This note studies the asymptotic behavior of the delete-$d$ jackknife in the irregular case of a sample $p$-quantile, calculated from a sample of $n$ iid r.v.'s. Two results are obtained: (a) an almost sure rate of convergence of the delete-$d$ jackknife histogram to the normal distribution; (b) almost sure convergence of the delete-$d$ jackknife variance estimate to the asymptotic variance of the sample $p$-quantile.

Consistency of jackknife variance estimators jun

A class of delete-d jackknife estimators of the asymptotic variances of point estimators are shown to be consistent when d, the number of observations removed from the original sample, is a fraction of n and the point estimators are generated from a sta¬tistical functional which possesses a weak differentiability property. The computation of the delete-∧ jackknife estimators is almost as easy as the traditional delete-1 jackknife estimator. The results are applied to problems in robust M-estimation

On using the jackknife to estimate quantile variance

AbstractWe 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.

Using the jackknife to estimate the variance of regression estimators from repeated measures studies

We discuss methods for analyzing data from repeated measures studies. The marginal distribution of the response at each time is assumed to be from the exponential family. The maximum likelihood regression estimators, assuming the repeated measurements on the same individual are independent, have been shown to be consistent by Liang and Zeger (1986). Liang and Zeger (1986) propose a ‘robust’ estimator of the asymptotic variance of the estimated regression parameters. We show that a ‘one-step’ jackknife estimator of variance is asymptotically equivalent to the Liang and Zeger variance estimator; we perform simulation runs to compare the small sample performance of the jackknife and Liang and Zeger's estimator. We also show that SAS Proc Jackreg (1986), which does jackknife regression for ordinary least squares, can be used to calculate jackknife estimates of the regression parameters and the estimated parameters' covariance matrix.

A Monte Carlo Study of Some Censored Data Wilcoxon Rank Tests

AbstractThe variance estimators usually applied for the generalized censored data Wilcoxon rank tests by Gehan and Peto & Prentice, are heavily biased in unbalanced problems. This paper reports the results of a Monte Carlo simulation study, where jackknifing is used to construct estimators of variance. Size, power and variance properties are compared for five variance estimators, when using different combinations of group sizes, failure and censoring patterns. The variance estimators are the permutational, the conditional permutational and the jackknife variance estimators for the statistic of Gehan and the asymptotic and the jackknife variance estimators for the statistic of Peto & Prentice. It appears that observed size, power and variance properties may be improved by using the jackknife variance estimator, when comparing to the variance estimators usually applied.

The weighted jackknife for ratio estimation

Using the idea of impirical influence function, Hinkley (1977), the weighted jackknife technique is extended to ratio estimation. A weighted jackknife variance estimator for the ratio estimator is developed. Using the prediction theory approach, the properties of the weighted jackknifed variance estimator are examined. The implications of the failures of regression model on the behaviour of the weighted jackknifed variance estimator, for ratio estimation, are also studied.

A General Theory for Jackknife Variance Estimation

The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$ are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, $d$ has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample $p$-quantiles, the jackknife variance estimators with $d$ satisfying $n^{1/2}/d \rightarrow 0$ and $n - d \rightarrow \infty$ are consistent and asymptotically unbiased.

The Efficiency and Consistency of Approximations to the Jackknife Variance Estimators

Abstract The problem considered is the computation reduction for general delete-d jackknife variance estimators. The delete-d jackknife estimator was proved consistent (Shao and Wu 1986), and in this article its mean squared error is shown to have order o(n –2), where n is the sample size. These properties are not shared by the traditional delete-1 jackknife in some situations. Use of the delete-d jackknife, however, requires ( n d ) recomputations of a point estimate θ, which increases rapidly as n and d increase. Using techniques from survey sampling, a shortcut can be taken with θ evaluated only m times, m ≪ ( n d ). The efficiency and consistency of the resulting jackknife-sampling (hybrid) variance estimators (JSVE's) are studied. If m is chosen so that n/m → 0, the increase in mean squared error by using the JSVE is relatively negligible. For the consistency of JSVE, m → ∞ is sufficient. Hence the JSVE with m < n can also be used to alleviate the computational burden for the delete-1 jackknife in the case where n is large and evaluating θ needs large computations. The performance of JSVE is also studied in a simulation study.

The Efficiency and Consistency of Approximations to the Jackknife Variance Estimators

Abstract The problem considered is the computation reduction for general delete-d jackknife variance estimators. The delete-d jackknife estimator was proved consistent (Shao and Wu 1986), and in this article its mean squared error is shown to have order o(n –2), where n is the sample size. These properties are not shared by the traditional delete-1 jackknife in some situations. Use of the delete-d jackknife, however, requires ( n d ) recomputations of a point estimate θ, which increases rapidly as n and d increase. Using techniques from survey sampling, a shortcut can be taken with θ evaluated only m times, m ≪ ( n d ). The efficiency and consistency of the resulting jackknife-sampling (hybrid) variance estimators (JSVE's) are studied. If m is chosen so that n/m → 0, the increase in mean squared error by using the JSVE is relatively negligible. For the consistency of JSVE, m → ∞ is sufficient. Hence the JSVE with m < n can also be used to alleviate the computational burden for the delete-1 jackknife in th...

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.

Jackknifing R-estimators

SUMMARY Sufficient conditions are given for the consistency of the jackknife variance estimator for R-estimators of location in the one- and two-sample problems. In particular, the jackknife is shown to produce strongly consistent estimates of the variance of the Hodges-Lehmann estimator. An efficient algorithm for computing the variance estimator is presented. For the two sample sizes considered in this paper, calculations for the jackknife estimator of the variance of the Hodges-Lehmann estimator are about 50 times as fast as the bootstrap variance estimator with B = 100. Some Monte Carlo evidence of the small-sample efficiency of this jackknife variance estimator is reported. Extensions of the results to R-estimators in the linear model context are discussed.

On Resampling Methods for Variance and Bias Estimation in Linear Models

Let $g$ be a nonlinear function of the regression parameters $\beta$ in a heteroscedastic linear model and $\hat{\beta}$ be the least squares estimator of $\beta.$ We consider the estimation of the variance and bias of $g(\hat{\beta})$ [as an estimator of $g(\beta)$] by using three resampling methods: the weighted jackknife, the unweighted jackknife and the bootstrap. The asymptotic orders of the mean squared errors and biases of the resampling variance and bias estimators are given in terms of an imbalance measure of the model. Consistency of the resampling estimators is also studied. The results indicate that the weighted jackknife variance and bias estimators are asymptotically unbiased and consistent and their mean squared errors are of order $o(n^{-2})$ if the imbalance measure converges to zero as the sample size $n \rightarrow \infty$. Furthermore, based on large sample properties, the weighted jackknife is better than the unweighted jackknife. The bootstrap method is shown to be asymptotically correct only under a homoscedastic error model. Bias reduction, a closely related problem, is also discussed.

Discussion of the Papers by Hinkley and Diciccio and Romano

Discussion of the Papers by Hinkley and Diciccio and Romano

A note on the delete- d jackknife variance estimators

In this note, a new delete- d jackknife variance estimator V j(d) ∗ is proposed, which differs from the original delete- d jackknife estimator V j( d) . It is shown that V j(d) ∗ is less biased than V j( d) .

Boostrapping, Jackknife and James-Stein estimators

In this article, the most recent results in resampling methods in regression analysis are reviewed. Different classes of estimators are studied and compared the simple jackknife the weighted jackknife variance estimator for the case of heteroscedastic errors, the variable jackknife, and the general estimator. A brief description of Stein-rule estimators in the case of pre-testing estimation then follows. As an application, it is shown how to improve on pre-testing efficiency, following a Stein-rule estimator, by applying bootstrap methods to the estimated variances.

Heteroscedasticity-Robustness of Jackknife Variance Estimators in Linear Models

The asymptotic unbiasedness and consistency of three types of jackknife variance estimators in the presence of error variance heteroscedasticity in linear models are studied. The results are given in terms of the number of observations deleted and measures of imbalance of the model. The consistency of a class of Wu's weighted jackknife variance estimators for nonlinear parameters is also studied. A necessary and sufficient condition is given for the asymptotic unbiasedness and consistency of the unweighted delete-1 jackknife variance estimator and Hinkley's weighted delete-1 jackknife variance estimator. This condition is more stringent than those required for Wu's weighted jackknife. Comparison of the three delete-1 jackknife variance estimators in terms of their biases also favors the latter method.

Jackknifing and Bootstrapping Goodness-of-Fit Statistics in Sparse Multinomials

Abstract Many goodness-of-fit problems can be put in the following form: n = (n 1, …, nk ) is a multinomial (N, π) vector, and it is desired to test the composite null hypothesis H 0: π = p(θ) against all possible alternatives. The usual tests used are Pearson's (1900) statistic or the likelihood ratio statistic (Neyman and Pearson 1928) . Cressie and Read (1984) pointed out that both of these statistics are in the power family of statistics , with λ = 1 and 0, respectively; they suggested an alternative statistic with . Although all of these statistics are asymptotically χ2 in the usual situation of K fixed and N → ∞, this is not the case if the multinomial is sparse; specifically, Morris (1975) showed that, under certain regularity conditions with K and N → ∞, both X 2 and G 2 are asymptotically normal (with different mean and variance) under a simple null hypothesis. Cressie and Read (1984) extended these results to the general 2NI λ family. Although these results have not been proven for composite nulls, it is certainly reasonable to expect that they continue to hold. Clearly, testing would require an estimate of the variance of the statistic that is valid under composite hypotheses in the sparse situation. In this article the use of nonparametric techniques to estimate these variances is examined. Simulations indicate (and heuristic arguments support) that although the bootstrap (Efron 1979) does not lead to a consistent variance estimate, the parametric bootstrap, the jackknife (Miller 1974) and a “categorical jackknife” (in which cells are deleted rather than observations) each leads to a consistent estimate. Simulations indicate that the jackknife is the nonparametric estimator of choice, and it is superior to the usual asymptotic formula for sparse data. Although these comparisons are based on the unconditional variance of the statistics, it is shown that the unconditional variance and the variance conditional on fitted parameter estimates are asymptotically equal if the underlying probability vector is from the general exponential family. Simulations also indicate that the jackknife estimate of variance is the estimator of choice in general parametric models for multinomial data.