Publications

    Survey Methodology

    Browse by

    4. Variance Estimation

    Jianqiang C. Wang, Jean D. Opsomer and Haonan Wang

    Previous | Next

    While the model-based approach makes it possible to obtain asymptotic distributions and hence perform inference that is asymptotically correct, we are most interested here in the design-based applications of bagging. In the design-based context, the construction of the bagging estimator can be naturally combined with the variance estimation of the original statistic, by taking advantage of the replication samples released by the statistical agencies. In this article, we take stratified simple random sampling as a specific example, with a bootstrap sampling design of stratified SRSWOR.

    We begin by applying a version of the Rao and Wu (1988) bootstrap procedure to estimate the variance of the survey estimators prior to bagging. Let N h ,   n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaaSbaaS qaaiaadIgaaeqaaOGaaGilaiaabccacaWGUbWaaSbaaSqaaiaadIga aeqaaaaa@3ACF@  and k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadIgaaeqaaaaa@377D@  denote the population size, sample size and sub-sample size in the h -th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObGaaeylai aabshacaqGObaaaa@38F3@  stratum, h = 1 , 2 , ⋯ , H . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObGaeyypa0 JaaGymaiaacYcacaaIYaGaaiilaiabl+UimjaaiYcacaWGibGaaiOl aaaa@3E61@  Here, B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbaaaa@363B@  bootstrap samples are drawn by stratified simple random sample without replacement of size k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiaadIgaaeqaaaaa@377D@  for computing the bootstrap variance of the original statistic and the bagging estimator. For each bootstrap sample, we assign a weight of

    N h N ( 1 − k h 1 / 2 ( n h − 1 ) − 1 / 2 ( 1 − n h N h ) 1 / 2 ) 1 n h + N h N k h 1 / 2 ( n h − 1 ) − 1 / 2 ( 1 − n h N h ) 1 / 2 1 k h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaad6 eadaWgaaWcbaGaamiAaaqabaaakeaacaWGobaaamaabmaabaGaaGym aiabgkHiTiaadUgadaqhaaWcbaGaamiAaaqaaiaaigdacaGGVaGaaG OmaaaakmaabmaabaGaamOBamaaBaaaleaacaWGObaabeaakiabgkHi TiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaca GGVaGaaGOmaaaakmaabmaabaGaaGymaiabgkHiTmaalaaabaGaamOB amaaBaaaleaacaWGObaabeaaaOqaaiaad6eadaWgaaWcbaGaamiAaa qabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaa ikdaaaaakiaawIcacaGLPaaadaWcaaqaaiaaigdaaeaacaWGUbWaaS baaSqaaiaadIgaaeqaaaaakiabgUcaRmaalaaabaGaamOtamaaBaaa leaacaWGObaabeaaaOqaaiaad6eaaaGaam4AamaaDaaaleaacaWGOb aabaGaaGymaiaac+cacaaIYaaaaOWaaeWaaeaacaWGUbWaaSbaaSqa aiaadIgaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaale qabaGaeyOeI0IaaGymaiaac+cacaaIYaaaaOWaaeWaaeaacaaIXaGa eyOeI0YaaSaaaeaacaWGUbWaaSbaaSqaaiaadIgaaeqaaaGcbaGaam OtamaaBaaaleaacaWGObaabeaaaaaakiaawIcacaGLPaaadaahaaWc beqaaiaaigdacaGGVaGaaGOmaaaakmaalaaabaGaaGymaaqaaiaadU gadaWgaaWcbaGaamiAaaqabaaaaaaa@71EF@

    to each sampled element in the h -th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGObGaaeylai aabshacaqGObaaaa@38F3@  stratum, and

    N h N ( 1 − k h 1 / 2 ( n h − 1 ) − 1 / 2 ( 1 − n h N h ) 1 / 2 ) 1 n h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaad6 eadaWgaaWcbaGaamiAaaqabaaakeaacaWGobaaamaabmaabaGaaGym aiabgkHiTiaadUgadaqhaaWcbaGaamiAaaqaamaalyaabaGaaGymaa qaaiaaikdaaaaaaOWaaeWaaeaacaWGUbWaaSbaaSqaaiaadIgaaeqa aOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0 IaaGymaiaac+cacaaIYaaaaOWaaeWaaeaacaaIXaGaeyOeI0YaaSaa aeaacaWGUbWaaSbaaSqaaiaadIgaaeqaaaGcbaGaamOtamaaBaaale aacaWGObaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaigda caGGVaGaaGOmaaaaaOGaayjkaiaawMcaamaalaaabaGaaGymaaqaai aad6gadaWgaaWcbaGaamiAaaqabaaaaaaa@5438@

    to the nonsampled elements. We then use the ordinary variance of the replicated sample estimators as variance estimator. The aforementioned weighting scheme is algebraically identical to equation 4.1 of Rao and Wu (1988), in which the finite population correction is incorporated into replication weights. The resampling variance estimator derived from the weighting method reduces to ordinary variance estimator under stratified SRSWOR and guarantees design unbiasedness. In order to combine bagging with bootstrap variance estimator, we use the same bootstrap samples to construct the bagging estimators for the population quantities of interest.

    Under the design-based framework, no analytic variance estimator is available for the bagged estimator in general. For now, we would suggest the following two variance estimation approaches in practice:

    (Var. 1)
    Use the estimated variance of the original estimator even though the bagged estimator may have a smaller variance. This method provides confidence intervals of the same width but outperforms the original confidence interval in having larger coverage rate.
    (Var. 2)
    Multiply the estimated variance of the original estimator by an adjustment factor accounting for the likely improvement in efficiency. One possible choice for such a factor is the efficiency gain assuming the sample is an i i d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbGaamyAai aadsgaaaa@3839@  sample from an infinite superpopulation. The factor can be determined by using the results of Theorems 1 and 2, or by a nonparametric bootstrap experiment. One such possible bootstrap procedure is double bootstrap, which is implemented by drawing ordinary bootstrap resamples to estimate the variance of the original estimator, and another level of SRSWOR resamples to determine the variance of the bagging estimator. One can estimate the ratio of the variance of bagging estimator to original estimator using these nested bootstrap samples, and multiply the design variance of the original estimator by this ratio.

    We will explore both approaches in the simulations in Section 5, but this is clearly an area in which further research is warranted.

    Previous | Next

    Date modified: