Data generated based on Multivariate Fay Herriot Model (Model 1)

Description

This data is generated based on multivariate Fay-Herriot model (model 1) by these following steps:

1. Generate sampling error e, random effect u, and auxiliary variables X1 X2.

   • For sampling error e, we set e_{d} ~ N_{3}(0, V_{ed}) , where V_{ed} = (\sigma_{dij})_{i,j=1,2,3}, with \sigma_{11} ~ InvGamma(11, 1), \sigma_{22} ~ InvGamma(11, 2), \sigma_{33} ~ InvGamma(11, 3), and \rho_{e} = 0.5.

   • For random effect u, we set u ~ N_{3}(0, V_{u}) , where \sigma_{u11} = 0.2 , \sigma_{u22} = 0.4 , and \sigma_{u33} = 1.2.

   • For auxiliary variables X1 and X2, we set X1 ~ N(5, 0.1) and X2 ~ N(10, 0.2).

2. Calculate direct estimation Y1 Y2 and Y3 , where Y_{i} = X * \beta + u_{i} + e_{i}. We take \beta_{1} = 5 and \beta_{2} = 10.

Auxiliary variables X1 X2, direct estimation Y1 Y2 Y3, and sampling variance-covariance v1 v2 v3 v12 v13 v23 are combined into a dataframe called datasae1.

Usage

datasae1

Format

A data frame with 50 rows and 11 variables:

X1

Auxiliary variable of X1

X2

Auxiliary variable of X2

Y1

Direct Estimation of Y1

Y2

Direct Estimation of Y2

Y3

Direct Estimation of Y3

v1

Sampling Variance of Y1

v12

Sampling Covariance of Y1 and Y2

v13

Sampling Covariance of Y1 and Y3

v2

Sampling Variance of Y2

v23

Sampling Covariance of Y2 and Y3

v3

Sampling Variance of Y3

Reference

Benavent, Roberto & Morales, Domingo. (2015). Multivariate Fay-Herriot models for small area estimation. Computational Statistics & Data Analysis. 100. 372-390. DOI: 10.1016/j.csda.2015.07.013.

Data generated based on Autoregressive Multivariate Fay Herriot Model (Model 2)

Description

This data is generated based on autoregressive multivariate Fay-Herriot model (model 2) by following these steps:

1. Generate sampling error e, random effect u, and auxiliary variables X1 X2.

   • For sampling error e, we set e ~ N_{3}(0, V_{e}) , where V_{e} = (\sigma_{ij})_{i,j=1,2,3}, with \sigma_{11} = 0.1 , \sigma_{22} = 0.2 , \sigma_{33} = 0.3 , and \rho_{e} = 0.5.

   • For random effect u, we set u ~ N_{3}(0, V_{u}) , where \sigma_{u} = 0.4, and \rho_{u} = 0.8.

   • For auxiliary variables X1 and X2, we set X1 ~ N(5, 0.1) and X2 ~ N(10, 0.2).

2. Calculate direct estimation Y1 Y2 and Y3 , where Y_{i} = X * \beta + u_{i} + e_{i}. We take \beta_{1} = 5 and \beta_{2} = 10.

Auxiliary variables X1 X2, direct estimation Y1 Y2 Y3, and sampling variance-covariance v1 v2 v3 v12 v13 v23 are combined into a dataframe called datasae2.

Usage

datasae2

Format

A data frame with 50 rows and 11 variables:

X1

Auxiliary variable of X1

X2

Auxiliary variable of X2

Y1

Direct Estimation of Y1

Y2

Direct Estimation of Y2

Y3

Direct Estimation of Y3

v1

Sampling Variance of Y1

v12

Sampling Covariance of Y1 and Y2

v13

Sampling Covariance of Y1 and Y3

v2

Sampling Variance of Y2

v23

Sampling Covariance of Y2 and Y3

v3

Sampling Variance of Y3

Reference

Benavent, Roberto & Morales, Domingo. (2015). Multivariate Fay-Herriot models for small area estimation. Computational Statistics & Data Analysis. 100. 372-390. DOI: 10.1016/j.csda.2015.07.013.

Data generated based on Heteroscedastic Autoregressive Multivariate Fay Herriot Model (Model 3)

Description

This data is generated based on heteroscedastic autoregressive multivariate Fay-Herriot model (model 3) by following these steps:

1. Generate sampling error e, random effect u, and auxiliary variables X1 X2.

   • For sampling error e, we set e ~ N_{3}(0, V_{e}) , where V_{e} = (\sigma_{ij})_{i,j=1,2,3}, with \sigma_{11} = 0.1 , \sigma_{22} = 0.2 , \sigma_{33} = 0.3 , and \rho_{e} = 0.5.

   • For random effect u, we set u ~ N_{3}(0, V_{u}) , where \sigma_{u11} = 0.2 , \sigma_{u22} = 0.4 , \sigma_{u33} = 1.2, and \rho_{u} = 0.8.

   • For auxiliary variables X1 and X2, we set X1 ~ N(5, 0.1) and X2 ~ N(10, 0.2).

2. Calculate direct estimation Y1 Y2 and Y3 , where Y_{i} = X * \beta + u_{i} + e_{i}. We take \beta_{1} = 5 and \beta_{2} = 10.

Auxiliary variables X1 X2, direct estimation Y1 Y2 Y3, and sampling variance-covariance v1 v2 v3 v12 v13 v23 are combined into a dataframe called datasae3.

Usage

datasae3

Format

A data frame with 50 rows and 11 variables:

X1

Auxiliary variable of X1

X2

Auxiliary variable of X2

Y1

Direct Estimation of Y1

Y2

Direct Estimation of Y2

Y3

Direct Estimation of Y3

v1

Sampling Variance of Y1

v12

Sampling Covariance of Y1 and Y2

v13

Sampling Covariance of Y1 and Y3

v2

Sampling Variance of Y2

v23

Sampling Covariance of Y2 and Y3

v3

Sampling Variance of Y3

Reference

Benavent, Roberto & Morales, Domingo. (2015). Multivariate Fay-Herriot models for small area estimation. Computational Statistics & Data Analysis. 100. 372-390. DOI: 10.1016/j.csda.2015.07.013.

Transform Dataframe to Matrix R

Description

This function transforms dataframe contains sampling variance to block diagonal matrix R

Usage

df2matR(var.df, r)

Arguments

Value

Block diagonal matrix R

Examples

NULL

EBLUPs based on a Multivariate Fay Herriot (Model 1)

Description

This function gives the EBLUP and MSE based on a multivariate Fay-Herriot model (model 1)

Usage

eblupMFH1(
  formula,
  vardir,
  samevar = FALSE,
  MAXITER = 100,
  PRECISION = 1e-04,
  data
)

Arguments

Value

The function returns a list with the following objects:

eblup

a dataframe with the values of the EBLUP estimators

MSE

a dataframe with the estimated mean squared errors of the EBLUPs for the small domains

randomEffect

a dataframe with the values of the random effect estimators

Rmatrix

a block diagonal matrix composed of sampling errors

fit

a list containing the following objects:

• method : type of fitting method, named "REML"

• convergence : a logical value of convergence of Fisher Scoring algorithm

• iterations : number of iterations performed by Fisher-Scoring algorithm

• estcoef : a dataframe with the estimated model coefficient in the first column, their standard error in the second column, the t statistics in the third column, and the p-values of the significance of each coefficient in the last column

• refvar : a dataframe with the estimated random effect variance

• informationFisher : a matrix of information Fisher of Fisher-Scoring algorithm

Examples

## Load dataset
data(datasae1)

# Compute EBLUP and MSE of Y1 Y2 and Y3  based on Model 1
# using auxiliary variables X1 and X2 for each dependent variable

## Using parameter 'data'
Fo <- list(f1=Y1~X1+X2,
           f2=Y2~X1+X2,
           f3=Y3~X1+X2)
vardir <- c("v1", "v2", "v3", "v12", "v13", "v23")
m1 <- eblupMFH1(Fo, vardir, data=datasae1)

## Without parameter 'data'
Fo <- list(f1=datasae1$Y1~datasae1$X1+datasae1$X2,
           f2=datasae1$Y2~datasae1$X1+datasae1$X2,
           f3=datasae1$Y3~datasae1$X1+datasae1$X2)
vardir <- datasae1[,c("v1", "v2", "v3", "v12", "v13", "v23")]
m1 <- eblupMFH1(Fo, vardir)

m1$eblup   # see the EBLUP estimators
m1$MSE   # see MSE of EBLUP estimators

EBLUPs based on a Autoregressive Multivariate Fay Herriot (Model 2)

Description

This function gives the EBLUP and MSE based on a autoregressive multivariate Fay-Herriot model (model 2).

Usage

eblupMFH2(formula, vardir, MAXITER = 100, PRECISION = 1e-04, data)

Arguments

Value

The function returns a list with the following objects:

eblup

a dataframe with the values of the EBLUP estimators

MSE

a dataframe with the estimated mean squared errors of the EBLUPs for the small domains

randomEffect

a dataframe with the values of the random effect estimators

Rmatrix

a block diagonal matrix composed of sampling errors

fit

a list containing the following objects:

• method : type of fitting method, named "REML"

• convergence : a logical value of convergence of Fisher Scoring algorithm

• iterations : number of iterations performed by Fisher-Scoring algorithm

• estcoef : a dataframe with the estimated model coefficient in the first column, their standard error in the second column, the t statistics in the third column, and the p-values of the significance of each coefficient in the last column

• refvar : a dataframe with the estimated random effect variance

• rho : a dataframe with the estimated rho of random effect variance and their rho parameter test based on Model 2

• informationFisher : a matrix of information Fisher of Fisher-Scoring algorithm

Examples

## Load dataset
data(datasae2)

# Compute EBLUP and MSE of Y1 Y2 and Y3  based on Model 2
# using auxiliary variables X1 and X2 for each dependent variable

## Without parameter 'data'
Fo <- list(f1=Y1~X1+X2,
           f2=Y2~X1+X2,
           f3=Y3~X1+X2)
vardir <- c("v1", "v2", "v3", "v12", "v13", "v23")
m2 <- eblupMFH2(Fo, vardir, data=datasae2)

## Without parameter 'data'
Fo <- list(f1=datasae2$Y1~datasae2$X1+datasae2$X2,
           f2=datasae2$Y2~datasae2$X1+datasae2$X2,
           f3=datasae2$Y3~datasae2$X1+datasae2$X2)
vardir <- datasae2[,c("v1", "v2", "v3", "v12", "v13", "v23")]
m2 <- eblupMFH2(Fo, vardir)

m2$eblup   # see the EBLUP estimators
m2$MSE   # see MSE of EBLUP estimators

EBLUPs based on a Heteroscedastic Autoregressive Multivariate Fay Herriot (Model 3)

Description

This function gives the EBLUP and MSE based on a heteroscedastic autoregressive multivariate Fay-Herriot model (model 3).

Usage

eblupMFH3(formula, vardir, MAXITER = 100, PRECISION = 1e-04, data)

Arguments

Value

The function returns a list with the following objects:

eblup

a dataframe with the values of the EBLUP estimators

MSE

a dataframe with the estimated mean squared errors of the EBLUPs for the small domains

randomEffect

a dataframe with the values of the random effect estimators

Rmatrix

a block diagonal matrix composed of sampling errors

fit

a list containing the following objects:

• method : type of fitting method, named "REML"

• convergence : a logical value of convergence of Fisher Scoring algorithm

• iterations : number of iterations performed by Fisher-Scoring algorithm

• estcoef : a dataframe with the estimated model coefficient in the first column, their standard error in the second column, the t statistics in the third column, and the p-values of the significance of each coefficient in the last column

• refvar : a dataframe with the estimated random effect variance

• refvarTest : homogeneity of random effect variance test based on Model 3

• rho : a dataframe with the estimated rho of random effect variance and their rho parameter test based on Model 2

• informationFisher : a matrix of information Fisher of Fisher-Scoring algorithm

Examples

## Load dataset
data(datasae3)

# Compute EBLUP and MSE of Y1 Y2 and Y3  based on Model 3
# using auxiliary variables X1 and X2 for each dependent variable

## Using parameter 'data'
Fo <- list(f1=Y1~X1+X2,
           f2=Y2~X1+X2,
           f3=Y3~X1+X2)
vardir <- c("v1", "v2", "v3", "v12", "v13", "v23")
m3 <- eblupMFH3(Fo, vardir, data=datasae3)

## Without parameter 'data'
Fo <- list(f1=datasae3$Y1~datasae3$X1+datasae3$X2,
           f2=datasae3$Y2~datasae3$X1+datasae3$X2,
           f3=datasae3$Y3~datasae3$X1+datasae3$X2)
vardir <- datasae3[,c("v1", "v2", "v3", "v12", "v13", "v23")]
m3 <- eblupMFH3(Fo, vardir)

m3$eblup   # see the EBLUP estimators
m3$MSE   # see MSE of EBLUP estimators

EBLUPs based on a Univariate Fay Herriot (Model 0)

Description

This function gives the EBLUP and MSE based on a univariate Fay Herriot model (model 0)

Usage

eblupUFH(
  formula,
  vardir,
  samevar = FALSE,
  MAXITER = 100,
  PRECISION = 1e-04,
  data
)

Arguments

Value

The function returns a list with the following objects:

eblup

a dataframe with the values of the EBLUP estimators

MSE

a dataframe with the estimated mean squared errors of the EBLUPs for the small domains

randomEffect

a dataframe with the values of the random effect estimators

Rmatrix

a block diagonal matrix composed of sampling errors

fit

a list containing the following objects:

• method : type of fitting method, named "REML"

• convergence : a logical value of convergence of Fisher Scoring algorithm

• iterations : number of iterations performed by Fisher-Scoring algorithm

• estcoef : a dataframe with the estimated model coefficient in the first column, their standard error in the second column, the t statistics in the third column, and the p-values of the significance of each coefficient in the last column

• refvar : a dataframe with the estimated random effect variance

• informationFisher : a matrix of information Fisher of Fisher-Scoring algorithm

Examples

## Load dataset
data(datasae1)

# Compute EBLUP and MSE of Y1 Y2 and Y3  based on Model 0
# using auxiliary variables X1 and X2 for each dependent variable

## Using parameter 'data'
Fo <- list(f1=Y1~X1+X2,
           f2=Y2~X1+X2,
           f3=Y3~X1+X2)
vardir <- c("v1", "v2", "v3", "v12", "v13", "v23")
un <- eblupUFH(Fo, vardir, data=datasae1)

## Without parameter 'data'
Fo <- list(f1=datasae1$Y1~datasae1$X1+datasae1$X2,
           f2=datasae1$Y2~datasae1$X1+datasae1$X2,
           f3=datasae1$Y3~datasae1$X1+datasae1$X2)
vardir <- datasae1[,c("v1", "v2", "v3", "v12", "v13", "v23")]
un <- eblupUFH(Fo, vardir)

un$eblup   # see the EBLUP estimators
un$MSE   # see MSE of EBLUP estimators

msae : Multivariate Fay Herriot Models for Small Area Estimation

Description

Implements multivariate Fay-Herriot models for small area estimation. It uses empirical best linear unbiased prediction (EBLUP) estimator. Multivariate models consider the correlation of several target variable and borrow strength from auxiliary variables to improve the effectiveness of a domain sample size. Models which accommodated by this package are univariate model with several target variables (model 0), multivariate model (model 1), autoregressive multivariate model (model 2), and heteroscedastic autoregressive multivariate model (model 3). Functions provide EBLUP estimators and mean squared error (MSE) estimator for each model. These models were developed by Roberto Benavent and Domingo Morales (2015) <doi:10.1016/j.csda.2015.07.013>.

Author(s)

Novia Permatasari, Azka Ubaidillah

Maintainer: Novia Permatasari 16.9335@stis.ac.id

Functions

eblupUFH

Gives the EBLUPs and MSE of Univariate SAE (Model 0)

eblupMFH1

Gives the EBLUPs and MSE of Multivariate SAE (Model 1)

eblupMFH2

Gives the EBLUPs and MSE of Autoregressive Multivariate SAE (Model 2)

eblupMFH3

Gives the EBLUPs and MSE of Heteroscedastics Autoregressive Multivariate SAE (Model 3)

Reference

• Benavent, Roberto & Morales, Domingo. (2015). Multivariate Fay-Herriot models for small area estimation. Computational Statistics & Data Analysis. 100. 372-390. DOI: 10.1016/j.csda.2015.07.013.

• Rao, J.N.K & Molina. (2015). Small Area Estimation 2nd Edition. New York: John Wiley and Sons, Inc.

• Ubaidillah, Azka et al. (2019). Multivariate Fay-Herriot models for small area estimation with application to household consumption per capita expenditure in Indonesia. Journal of Applied Statistics. 46:15. 2845-2861. DOI: 10.1080/02664763.2019.1615420.