bootParMis {qape}

Parametric bootstrap estimators of prediction accuracy under the misspecified model

Description

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

Usage

bootParMis(predictorLMM, predictorLMMmis, B, p)

Arguments

predictorLMM

plugInLMM object, the first predictor used to define the bootstrap model.

predictorLMMmis

plugInLMM object, the second predictor.

B

number of iterations in the bootstrap procedure.

p

orders of quantiles in the QAPE.

Details

We use bootstrap model presented by Chatterjee, Lahiri and Li (2008) p. 1229 but assumed for all population elements. We use model specification used in predictorLMM. Vectors of random effects and random components are generated from the multivariate normal distribution where 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). We use the MSE estimator defined as the mean of squared bootstrap errors considered by Rao and Molina (2015) p. 141 and given by equation (6.2.22). 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. It is estimated as a quantile of absolute bootstrap errors as proposed by Zadlo (2017) in Section 2. The prediction accuracy of two predictors predictorLMM and predictorLMMmis is estimated under the model specified in predictorLMM.

Value

estQAPElmm

estimated value/s of QAPE of predictorLMM - 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 thetaFun).

estRMSElmm

estimated value/s of RMSE of predictorLMM (more than one value is computed if in thetaFun more than one population characteristic is defined).

estQAPElmmMis

estimated value/s of QAPE of predictorLMMmis - 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 thetaFun).

estRMSElmmMis

estimated value/s of RMSE of predictorLMMmis (more than one value is computed if in thetaFun more than one population characteristic is defined).

predictorLMMSim

bootstrapped values of predictorLMM.

predictorLMMmisSim

bootstrapped values of predictorLMMmis.

thetaSim

bootstrapped values of the predicted population or subpopulation characteristic/s.

Ysim

simulated values of the (possibly tranformed) variable of interest.

errorLMM

differences between bootstrapped values of predictorLMM and bootstrapped values of the predicted characteristic/s.

errorLMMmis

differences between bootstrapped values of predictorLMMmis and bootstrapped values of the predicted characteristic/s.

positiveDefiniteEstG

logical indicating if the estimated covariance matrix of random effects, used to generate bootstrap realizations of the dependent variable, is positive definite.

Author(s)

Alicja Wolny-Dominiak, Tomasz Zadlo

References

1. Butar, B. F., Lahiri, P. (2003) On measures of uncertainty of empirical Bayes small-area estimators, Journal of Statistical Planning and Inference, Vol. 112, pp. 63-76.

2. Chatterjee, S., Lahiri, P. Li, H. (2008) Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models, Annals of Statistics, Vol. 36 (3), pp. 1221?1245.

3. Rao, J.N.K. and Molina, I. (2015) Small Area Estimation. Second edition, John Wiley & Sons, New Jersey.

4. Zadlo T. (2017), On asymmetry of prediction errors in small area estimation, Statistics in Transition, 18 (3), 413-432.

Examples

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


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 <- '(1|NUTS2)'
random.part.mis <- '(1|NUTS4type)'

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)]}


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

set.seed(123456)
### Estimation of prediction accuracy under the model used to define predictorLMM
est_accuracy <- bootParMis(predictorLMM, predictorLMMmis, 10, c(0.75,0.9))

# Estimation of prediction RMSE of predictorLMM 
est_accuracy$estRMSElmm

# Estimation of prediction RMSE of predictorLMMmis
est_accuracy$estRMSElmmMis

# Estimation of prediction QAPE of predictorLMM 
est_accuracy$estQAPElmm

# Estimation of prediction QAPE of predictorLMMmis
est_accuracy$estQAPElmmMis

detach(invData2018)