Jackknife Procedure Research Articles

Improving the jackknife with special reference to correlation estimation

SUMMARY Second-order asymptotic properties of the jackknife procedure are discussed, and the jackknifed estimator is shown to be a vulnerable estimator whose variation can be severely underestimated by the jackknife standard error. Simple robust alternatives to the average pseudovalue are discussed. Particular emphasis is placed on estimation of a correlation coefficient. Numerical examples are given. The jackknife is a method for distribution-free bias reduction and standard error estimation. For a wide class of problems it is known that the jackknife produces consistent results. An excellent review of applications and asymptotic theory is given by Miller (1974). Recently there have been several investigations of small-sample properties of the jackknife procedure (Hin-kley, 1977a, b), which show that some adjustments are necessary in order to obtain accurate confidence intervals using the jackknife. In an unpublished paper, B. Efron ha? shown that the jackknife gives a rough linear approximation to another subsampling method for getting confidence intervals. In the present paper we examine two further aspects of the jackknife, namely the use ol second-order asymptotics in assessing finite-sample properties, and the use of jackknife pseudovalues in obtaining estimates less sensitive to extreme data points. The discussion is illustrated throughout with results for the correlation estimate. A brief summary of the results is as follows: jackknifed estimators can have very large haphazard bias compared to the original estimators; the jackknife estimate of standard error can severely underestimate the standard error of the jackknifed estimator; and use of the jackknife pseudovalues in residual and trimmed-mean analyses can give considerably improved estimators. Section 2 summarizes the standard jackknife method and illustrates it on an artificial data set, where certain difficulties are apparent. Second-order properties of the jackknife are derived in ? 3, and numerical results are given for the correlation example. The same example is used in ? 4, where robust analysis via pseudovalues is discussed. Section 5 gives brief conclusions. Throughout the paper we assume that the basic estimate is obtained from independent, identically distributed random variables. Moreover we assume that the estimate is a regular differentiable functional of the empirical distribution function, with at least two derivatives.

Two-step weighted least squares factor analysis of dichotomized variables

A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.

Robust Interval Estimation of the Innovation Variance of an Arma Model

For the autoregressive-moving average time series model, the normal theory procedure for setting confidence intervals for the error variance is not robust against nonnormality. This paper proposes three asymptotically robust techniques: they are a "standard-error" procedure, an analog of Box's simple data splitting technique, and the jackknife procedure. The large sample distribution of each of these techniques is derived.

The robustness of homogeneity of variance tests for asymmetric distributions: A Monte Carlo study

Robustness properties of three equality of variances tests, the jackknife procedure, the k-sample Box-Andersen test, and Levene’s z test, were investigated and compared in a Monte Carlo study employing small samples from underlying gamma distributions and gamma mixtures. Overall, the jackknife test yielded rejection proportions usually closer to nominal levels under departures from normality, but the Box-Andersen test performed relatively well when departures from normality were not extreme. Four recommendations regarding testing variances were offered.

Some asymptotic properties of jackknife statistics

SUMMARY The jackknife and the original statistic from which it is derived often have the same asymptotic distribution. In fact, their difference often has a variance of smaller order than the original statistic. We give sufficient and almost necessary conditions for this. We prove a similar result also for regular functions of the statistic. Applications are given to sample moments and linear combinations of order statistics. We discuss possible gains and losses in precision when using the jackknife procedure. In particular we give a lower bound on the variance of the jackknife.

Robust Tests for the Equality of Variances

Abstract Alternative formulations of Levene's test statistic for equality of variances are found to be robust under nonnormality. These statistics use more robust estimators of central location in place of the mean. They are compared with the unmodified Levene's statistic, a jackknife procedure, and a χ2 test suggested by Layard which are all found to be less robust under nonnormality.

A Monte-Carlo Study of Asymptotically Robust Tests for Correlation Coefficients

Monte-Carlo simulation is used to compare the small-sample performance of the usual normal theory procedures for inference about correlation coefficients with that of two asymptotically robust procedures, one of which is based on a grouping of the observations and the other on the jackknife technique. The sampled distributions comprise the normal and five nonnormal distributions. The small-sample results support the conclusion based on asymptotic theory that the normal test is not robust. The jackknife procedure works well for most of the sampled distributions.

Robust Large-Sample Tests for Homogeneity of Variances

Abstract Two asymptotically robust tests for equality of variances in the k-sample case are discussed: a simple X 2 test and a test based on the jack-knife procedure. Some Monte Carlo experiments suggest that these tests are reasonably robust for moderately small samples, are more powerful than Box's grouping test and perform similarly to Bartlett's test in the normal case.

Robust Large-Sample Tests for Homogeneity of Variances

Abstract Two asymptotically robust tests for equality of variances in the k-sample case are discussed: a simple X 2 test and a test based on the jack-knife procedure. Some Monte Carlo experiments suggest that these tests are reasonably robust for moderately small samples, are more powerful than Box's grouping test and perform similarly to Bartlett's test in the normal case.

A Monte-Carlo study of asymptotically robust tests for correlation coefficients

SUMMARY Monte-Carlo simulation is used to compare the small-sample performance of the usual normal theory procedures for inference about correlation coefficients with that of two asymptotically robust procedures, one of which is based on a grouping of the observations and the other on the jackknife technique. The sampled distributions comprise the normal and five nonnormal distributions. The small-sample results support the conclusion based on asymptotic theory that the normal test is not robust. The jackknife procedure works well for most of the sampled distributions. Tests of hypotheses and confidence intervals concerning correlation coefficients in bivariate normal populations are commonly based on Fisher's z transformation of the sample correlation coefficient, tanh-1 r. This statistic is taken to be approximately normally distributed with mean tanh-lp and variance 1/(n- 3), the accuracy of the approximation being adequate for moderately small samples and improving as n increases. For a bivariate population, not necessarily normal but having finite fourth moments, it can be shown that Vn(tanh-1 r - tanh-1 p) converges in distribution to N{O,0o2(p)}, where 1 o2(p) + 1+4(1 2)2{P (A40 + AO4) -4P(A31 + A13) + 2(2+p2)A } (1.1) The A's are the standardized cumulants of order four of the bivariate distribution; for example, A2 = K /(0-2 o-), where o2 and o are the variances of the marginal distributions. Details are given in an unpublished Stanford University report by M. W. J. Layard. Note that A40 and A04 are the kurtoses of the marginal distributions. If the distribution is bivariate normal, all the A's vanish, and o-2(p) = 1. If the components are independent, we have p = 0 and A22= 0, and 0-2(0) = 1. Hence the normal theory test for independence based on