doubleBootMis: Double bootstrap estimators of prediction accuracy under the...

In qape: Quantile of Absolute Prediction Errors

Double bootstrap estimators of prediction accuracy under the misspecified model

Description

The function computes values of double bootstrap estimators of the MSE and the QAPE prediction accuracy measures of two predictors under the model assumed for one of them.

Usage

doubleBootMis(predictorLMM, predictorLMMmis, B1, B2, p, q)

Arguments

predictorLMM	plugInLMM object, the first predictor used to define the bootstrap model.
predictorLMMmis	plugInLMM object, the second predictor.
B1	the number of first-level bootstrap iterations.
B2	the number of second-level bootstrap iterations.
p	orders of quantiles in the QAPE.
q	estimator bounds assumed for estMSE_db_1_EF and estMSE_db_telesc_EF (which are corrected versions of estMSE_db_1 and estMSE_db_telesc, respectively).

Details

Double-bootstrap method considered by Hall and Maiti (2006) and Erciulescu and Fuller (2013) is used. We use model specification used in predictorLMM. Vectors of random effects and random components are generated from the multivariate normal distribution and REML estimates of model parameters are used. Random effects are generated for all population elements even for subsets with zero sample sizes (for which random effects are not estimated). Double-bootstrap MSE estimator presented in Hall and Maiti (2006) and Erciulescu and Fuller (2013) are taken into account. The QAPE is a quantile of absolute prediction error which means that at least p100% of realizations of absolute prediction errors are smaller or equal to QAPE. The prediction accuracy of two predictors predictorLMM and predictorLMMmis is estimated under the model specified in predictorLMM.

Value

estMSE_param_LMMmis	value/s of the parametric bootstrap MSE estimator of predictorLMMmis. More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_B2_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis computed as the difference of doubled value of estMSE_param and the second-level MSE estimator based on B2 iterations. More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_B2_WDZ_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis computed as the mean of squared first-level bootstraped errors, each corrected by the mean of squared second-level bootstraped errors based on B2 iterations (where correction is made only if their difference is non-negative). More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_B2_HM_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis proposed by Hall and Maiti (2006) equation (2.17). More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_1_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis computed as the difference of doubled value of estMSE_param and the second-level MSE estimator based on B2=1 iteration. More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_1_WDZ_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis computed as the mean of squared first-level bootstraped errors, each corrected by the squared second-level bootstraped error based on 1 iteration (where correction is made only if their difference is non-negative). More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_1_EF_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis proposed by Erciulescu and Fuller (2014) given by equation (13) with correction (17), where the bound for the correction is declared as q. More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_telesc_LMMmis	value/s of the telescoping double bootstrap MSE estimator of predictorLMMmis proposed by Erciulescu and Fuller (2014) given by equation (15). More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_telesc_WDZ_LMMmis	value/s of the double bootstrap MSE estimator of predictorLMMmis computed as the mean of the sums of the following elements: squared first-level bootstraped error, squared first-level bootstrap error for the next iteration and the opposite of second-level bootstraped error based on 1 iteration (but negative sums are replaced by squared first-level bootstraped error). More than one value is computed if in thetaFun more than one population characteristic is defined.
estMSE_db_telesc_EF_LMMmis	value/s of the telescoping double bootstrap MSE estimator of predictorLMMmis proposed by Erciulescu and Fuller (2014) given by equation (15) with correction (17), where the bound for the correction is declared as q. More than one value is computed if in thetaFun more than one population characteristic is defined.
estQAPE_param_LMMmis	value/s of parametric bootstrap estimator of QAPE (Quantile of Absolute Prediction Error) of predictorLMMmis given by a quantile of absolute parametric bootstrap errors. Number of rows is equal the number of orders of quantiles to be considered (declared in p), number of columns is equal the number of predicted characteristics (declared in in thetaFun).
estQAPE_db_B2_LMMmis	value/s of double-bootstrap estimator of QAPE (Quantile of Absolute Prediction Error) of predictorLMMmis given by a quantile of square roots of squared first-level bootstraped errors, each corrected by the mean of squared second-level bootstraped errors based on B2 iterations (where correction is made only if their difference is non-negative). Number of rows is equal the number of orders of quantiles to be considered (declared in p), number of columns is equal the number of predicted characteristics (declared in in thetaFun).
estQAPE_db_1_LMMmis	value/s of double-bootstrap estimator of QAPE (Quantile of Absolute Prediction Error) of predictorLMMmis given by a quantile of square roots of squared first-level bootstraped errors, each corrected by the squared second-level bootstraped error based on 1 iteration (where correction is made only if their difference is non-negative). Number of rows is equal the number of orders of quantiles to be considered (declared in p), number of columns is equal the number of predicted characteristics (declared in in thetaFun).
estQAPE_db_telesc_LMMmis	value/s of double-bootstrap estimator of QAPE (Quantile of Absolute Prediction Error) of predictorLMMmis given by a quantile of square roots of the sums of the following elements: squared first-level bootstraped error, squared first-level bootstrap error for the next iteration and the opposite of second-level bootstraped error based on 1 iteration (but negative sums are replaced by squared first-level bootstraped error). Number of rows is equal to the number of orders of quantiles to be considered (declared in p), number of columns is equal to the number of predicted characteristics (declared in in thetaFun).
error1_LMMmis	the matrix of first-level bootstrap errors of predictorLMMmis. Number of rows is equal to the number of predicted characteristics (declared in in thetaFun), number of columns is equal to B1.
error2_LMMmis	the list of matrices of second-level bootstrap errors of predictorLMMmis. The length of list is equal to the number of predicted characteristics (declared in in thetaFun), the number of rows of each matrix is equal to B1, the number of columns is equal to B2.
corSquaredError1_db_B2_LMMmis	the matrix of corrected squared first-level bootstrap errors of predictorLMMmis defined as doubled squared first-level bootstrap errors minus the mean of squared second-level bootstrap errors (computed for the approriate first-level bootstrap iterations). Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values can be negative.
corSquaredError1_db_1_LMMmis	the matrix of corrected squared first-level bootstrap errors of predictorLMMmis defined as doubled squared first-level bootstrap errors minus the squared second-level bootstrap error (computed once for each first-level bootstrap iteration). Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values can be negative.
corSquaredError1_db_telesc_LMMmis	the matrix of corrected squared first-level bootstrap errors of predictorLMMmis defined by elements from which the average given by equation (15) in Erciulescu and Fuller (2014) is counted. Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values can be negative.
corSquaredError1_db_B2_WDZ_LMMmis	the matrix of squared first-level bootstraped errors of predictorLMMmis, each corrected by the mean of squared second-level bootstraped errors based on B2 iterations (where correction is made only if their difference is non-negative). Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values are non-negative.
corSquaredError1_db_1_WDZ_LMMmis	the matrix of squared first-level bootstraped errors of predictorLMMmis, each corrected by the squared second-level bootstraped error based on 1 iteration (where correction is made only if their difference is non-negative). Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values are non-negative.
corSquaredError1_db_telesc_WDZ_LMMmis	the matrix of corrected squared first-level bootstrap errors of predictorLMMmis defined by sums of the following elements: squared first-level bootstraped error, squared first-level bootstrap error for the next iteration and the opposite of second-level bootstraped error based on 1 iteration (but negative sums are replaced by squared first-level bootstraped error).Number of rows is equal to B1, the number of columns is equal to the number of predicted characteristics (declared in in thetaFun). Values are non-negative.
positiveDefiniteEstGlev1	logical indicating if the estimated covariance matrix of random effects, used to generate bootstrap realizations of the dependent variable at the first level of the double bootstrap, is positive definite.
positiveDefiniteEstGlev2	number of cases ouf of B1 with positive definite estimated covariance matrix of random effects used to generate bootstrap realizations of the dependent variable at the second level of the double bootstrap

Author(s)

Alicja Wolny-Dominiak, Tomasz Zadlo

References

1. Erciulescu, A. L. and Fuller, W. A. (2013) Parametric Bootstrap Procedures for Small Area Prediction Variance. JSM 2014 - Survey Research Methods Section, 3307-3318.

2. Hall, P. and Maiti, T. (2006) On Parametric Bootstrap Methods for Small Area Prediction. Journal of the Royal Statistical Society. Series B, 68(2), 221-238.

Examples

library(lme4)
library(Matrix)
library(mvtnorm)


data(invData) 
# data from one period are considered: 
invData2018 <- invData[invData$year == 2018,] 
attach(invData2018)

N <- nrow(invData2018) # population size

con <- rep(1,N) 
con[c(379,380)] <- 0 # last two population elements are not observed 

YS <- log(investments[con == 1]) # log-transformed values
backTrans <- function(x) exp(x) # back-transformation of the variable of interest
fixed.part <- 'log(newly_registered)'
random.part.mis <- '(1|NUTS4type)'
random.part <- '(1|NUTS2)'

reg <- invData2018[, -which(names(invData2018) == 'investments')]
weights <- rep(1,N) # homoscedastic random components

### Characteristics to be predicted:
# values of the variable for last two population elements  
thetaFun <- function(x) {x[c(379,380)]}
set.seed(123456)

predictorLMM <- plugInLMM(YS, fixed.part, random.part, reg, con, weights, backTrans, thetaFun)
predictorLMM$thetaP

predictorLMMmis <- plugInLMM(YS, fixed.part, random.part.mis, reg,con,weights,backTrans,thetaFun)
predictorLMMmis$thetaP

### Estimation of prediction accuracy
# in the first column
# for the predictor of the value of the variable for population element no. 379,
# in the second column
# for the predictor of the value of the variable for population element no. 380:
doubleBootMis(predictorLMM, predictorLMMmis, 3, 3, c(0.5,0.9), 0.77) 
#q=0.77 assumed as in Erciulescu and FUller (2014) eq. (17)

detach(invData2018)

qape documentation built on Aug. 21, 2023, 5:07 p.m.