Jackknife Variance Estimator Research Articles

Jackknife methods for left-truncated data

Let U * and V * be two independent positive random variables with continuous distribution functions F and G, respectively. Under random truncation, both U * and V * are observable only when U * ≥ V * . Let F ¯ n be the nonparametric maximum likelihood estimate ( Lynden-Bell, 1971) of F ¯ = 1 − F . In this paper, it is shown that the jackknife variance estimate of F ¯ n consistently estimates the limit variance. In a simulation study, we show that the jackknife works satisfactorily for moderate sample size. As an illustration we also apply our method to a real set of AIDS data.

Imputation of mean of ratios for missing data and its application to PPSWR sampling

In practical survey sampling, nonresponse phenomenon is unavoidable. How to impute missing data is an important problem. There are several imputation methods in the literature. In this paper, the imputation method of the mean of ratios for missing data under uniform response is applied to the estimation of a finite population mean when the PPSWR sampling is used. The imputed estimator is valid under the corresponding response mechanism regardless of the model as well as under the ratio model regardless of the response mechanism. The approximately unbiased jackknife variance estimator is also presented. All of these results are extended to the case of non-uniform response. Simulation studies show the good performance of the proposed estimators.

Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response

In this paper, bias-adjustment in the jackknife estimator of variance accredited to Rao and Sitter (1995) has been considered. Then the bias-adjusted Rao and Sitter (1995) estimator has been calibrated such that its expected value under the imputing superpopulation model remains the same as the expected value of the mean squared error of the ratio estimator in the presence of non-response. A simulation study has been performed to compare the six different estimators of variance: out of them four estimators belong to Rao and Sitter (1995) and the other two proposed estimators are named as bias-adjusted and bias-adjusted-cum-calibrated estimators. The empirical relative bias and empirical relative efficiency of the two proposed estimators with respect to the four existing estimators accredited to Rao and Sitter (1995) have been investigated through simulations. The bias-adjusted-cum-calibrated estimator has been found to be an efficient estimator in the case of heteroscadastic populations. The present paper considers the situation of simple random and without replacement sampling. The possibility of obtaining a negative estimate of variance by the estimator due to Kim et al. (2006) has been pointed out.

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.

IGF-I, IGFBP-3, and IGF-I/IGFBP-3 Ratio: No Association with Incident Colorectal Cancer in the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study

The insulin-like growth factor (IGF) signaling system is thought to affect tissue growth and development ([1][1]), with IGF-I and IGF binding protein-3 (IGFBP-3) having putative procarcinogenic and anticarcinogenic properties, respectively. A recent meta-analysis of four prospective studies found

Jackknife method for estimating the variance of the age composition using two-phase sampling with an application to commercial catches of swordfish ( Xiphias gladius)

An estimate of the age composition of a commercial swordfish catch and its variance was obtained using the replicate full jackknife estimator for two-phase sampling where the sampling units in the first phase are clusters (vessel-trips) and the units of the second phase are individual fishes. The jackknife variance estimator was compared with an analytical estimator of variance commonly used in fisheries studies which ignores the clustering, and assumes simple random sampling of individual fish. The application of these naïve estimators to the Chilean South Pacific swordfish fishery results in an underestimation of the variance. This is a consequence of using an inappropriate primary sampling unit of analysis in the first-phase sample where the vessel-trip unit is replaced by an individual fish unit, thus ignoring the intra-cluster correlation in age samples. The jackknife estimator of variance is an alternative useful for complex two-phase sampling designs.

Adaptive functional mixed NHPP models for the analysis of recurrent event panel data

An adaptive semi-parametric model for analyzing longitudinal panel count data is presented. Panel data refers here to data collected as the number of events occurring between specific followup times over a period or disjoint periods of observation of a subject. The counts are assumed to arise from a mixed nonhomogeneous Poisson process where frailties account for heterogeneity common to this type of data. The generating intensity of the counting process is assumed to be a smooth function modeled with penalized splines. A main feature is that the penalization used to control the amount of smoothing, usually assumed to be time homogeneous, is allowed to be time dependent so that the spline can more easily adapt to sharp changes in curvature regimes. Splines are also used to model time dependent covariate effects. Penalized quasi-likelihood (PQL; [Breslow, N.E., Clayton, D.G., 1993. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9–25]) is used to derive estimating equations for this adaptive spline model so that only low moment assumptions are required for inference. Both jackknife and bootstrap variance estimators are developed. The finite sample properties of the proposed estimating functions are investigated empirically by simulation. Comparisons with a model assuming a time homogeneous penalty are made. The methods are used in an analysis of data from an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth. Recommendations are provided on when the simpler model with a time homogeneous penalty may provide a fair approximation to data and where such an approach will be lacking, calling for the more complicated adaptive methods.

The asymptotic distribution of the delete- [formula omitted] Jackknife variance estimator for smooth functionals

The delete- d Jackknife estimate of variance is commonly used in survey sampling. We derive the asymptotic distribution of this estimator for a class of second-order differentiable statistical functionals, including the useful case of estimating the variance of an estimated coefficient in logistic regression. The asymptotic distribution of the variance estimator depends on the quadratic term of the functional through its Hajék projection. We relate these results to existing results in which the variance estimator is seen to have the asymptotic distribution of a “linearized” version. A practical consequence is that d = 1 is the best choice for variance estimates of many common statistics.

A Jackknife Variance Estimator for Unistage Stratified Samples with Unequal Probabilities

Existing jackknife variance estimators used with sample surveys can seriously overestimate the true variance under unistage stratified sampling without replacement with unequal probabilities. A novel jackknife variance estimator is proposed which is as numerically simple as existing jackknife variance estimators. Under certain regularity conditions, the proposed variance estimator is consistent under stratified sampling without replacement with unequal probabilities. The high entropy regularity condition necessary for consistency is shown to hold for the Rao–Sampford design. An empirical study of three unequal probability sampling designs supports our findings.

Jackknife variance estimation under two-phase sampling

Two new Jackknife methods, as the counterparts of two existing Bootstrap methods of variance estimation under two-phase sampling, have been proposed. A simulation study has been conducted under both design-based and Conditional inference frameworks by generating two-phase samples from an infinite population for comparison of the proposed methods with five existing Jackknife and Bootstrap methods. The first method, the two-phase post-stratified Jackknife, reduces to an existing Jackknife variance estimation method considered under sampling from infinite population set up. The performance of the second method, the two-phase proportionate Jackknife, was better than two existing Jackknife methods while performing at par with another Jackknife method as well as with the two Bootstrap methods considered.

Algorithms for Constructing Combined Strata Variance Estimators

A jackknife or balanced repeated replication variance estimator in a large survey typically requires a large number of replicates and replicate weights. Reducing the number of replicates may have important advantages for computations and for limiting the risk of data disclosure from public use data files. This article proposes algorithms adapted from scheduling theory to combine variance strata and, thus, reduce the number of replicates. The algorithms are simple and efficient and can be adapted to easily account for vector characteristics and analytic domains. An important concern with combining strata is that the resulting variance estimators may be inconsistent. We establish conditions for the consistency of the combined variance estimator and show that the proposed algorithms ensure they are met. We also derive bounds on the degrees of freedom that the algorithms will assure. The algorithms are applied both to a real sample survey and to samples from simulated populations, and the algorithms perform very well, attaining variance estimators with precision levels close to the upper bounds.

Imputation for missing data and variance estimation when auxiliary information is incomplete

In this paper, a new imputation is developed for the treatment of missing data, which removes the bias of usual ratio imputation. The Jackknife variance estimators are provided and they are proved to be approximately design-unbiased under uniform response. Our method is compared with the existing method of Sitter and Rao (9) through simulation studies. The simulation studies show that the new method for missing data does reduce the bias without increasing the variance significantly.

Adjusted Jackknife for Imputation under Unequal Probability Sampling Without Replacement

SummaryImputation is commonly used to compensate for item non-response in sample surveys. If we treat the imputed values as if they are true values, and then compute the variance estimates by using standard methods, such as the jackknife, we can seriously underestimate the true variances. We propose a modified jackknife variance estimator which is defined for any without-replacement unequal probability sampling design in the presence of imputation and non-negligible sampling fraction. Mean, ratio and random-imputation methods will be considered. The practical advantage of the method proposed is its breadth of applicability.

Outlier Robustness of the Estimators of Variance of the Ratio Estimator

This paper investigates the sensitivities of the variance estimators for the ratio estimator. The model-based estimators V D and V L are found to be more sensitive to outliers than the rest of the estimators while the jack-knife variance estimator is the best on this criterion. Journal of Agriculture, Science and Technology Vol.4(1) 2002: 66-74

Jackknife variance estimator with reimputation for randomly imputed survey data

Reimputing after each deletion of a unit in the jackknife is known to lead to inconsistent variance estimators in the presence of random imputation in complex surveys. We use rescaling to adjust the reimputation and show that the resulting variance estimator performs somewhat better than the adjusted jackknife variance estimator.

Analysis of Performance Measures in Experimental Designs Using Jackknife

Experimental designs with performance measures as responses are common in industrial applications. The existing analysis methods often regard performance measures as sole response variables without replicates. Consequently, no degrees of freedom are left for error variance estimation in these methods. In reality, performance measures are obtained from replicated primary-response variables. Precious information is hence lost. In this paper, we suggest a jackknife-based approach on the replicated primary responses to provide an estimate of error variance of performance measures. The resulting tests for factor effects become easy to construct and more reliable. We compare the proposed method with some existing methods using two real examples and investigate the consistency of the jackknife variance estimate based on simulation studies.

A Jackknife Variance Estimator for Unequal Probability Sampling

SummaryThe jackknife method is often used for variance estimation in sample surveys but has only been developed for a limited class of sampling designs. We propose a jackknife variance estimator which is defined for any without-replacement unequal probability sampling design. We demonstrate design consistency of this estimator for a broad class of point estimators. A Monte Carlo study shows how the proposed estimator may improve on existing estimators.

AN EDGEWORTH EXPANSION OF A CONVEX COMBINATION OF U-STATISTICS BASED ON STUDENTIZATION

As an estimator of an estimable parameter, Toda and Yamato (2001) introduce Y-statistic which is a convex combination of U-statistics including V-statistic and LB-statistic. We give the Edgeworth expansions of studentized Y-statistic about the estimable parameter using a jackknife variance estimator, with remainder o(n−1).

Estimating equations for association structures

This paper investigates generalized estimating equations for association parameters, which are frequently of interest in family studies, with emphasis on covariance estimation. Separate link functions are used to connect the mean, the scale, and the correlation to linear predictors involving possibly different sets of covariates, and separate estimating equations are proposed for the three sets of parameters. Simulations show that the robust 'sandwich' variance estimator and the jackknife variance estimator for the correlation parameters are generally close to the empirical variance for the sample size of 50 clusters. The results contradict Ziegler et al. and Kastner and Ziegler, where the 'sandwich' estimator obtained from the software MAREG was shown to be unsuitable for practical usage. The problem appears to arise because the MAREG variance estimator does not account for variability in estimation of the scale parameters, but may be valid with fixed scale. We also find that the formula for the approximate jackknife variance estimator in Ziegler et al. is deficient, resulting in systematic deviations from the fully iterated jackknife variance estimator. A general jackknife formula is provided and performs well in numerical studies. Data from a study on the genetics of alcoholism is used to illustrate the importance of reliable variance estimation in biomedical applications.

Asymptotic results in jackknifing nonsmooth functions of the sample mean vector

The asymptotic behavior of jackknife estimators and jackknife variance estimators is investigated for nonsmooth functions of the sample mean vector. An application of jackknifing a suitable estimator of the intrinsic diversity profile is also presented.