datasaem {sae.prop}

Data generated based on Multivariate Fay Herriot Model with Additive Logistic Transformation

Description

This data is generated based on multivariate Fay-Herriot model and then transformed by using inverse Additive Logistic Transformation (alr). The steps are as follows:

1. Set these following variables:

   • q = 4

   • r_{1} = r_{2} = r_{3} = 2, r = 6

   • \beta_{1} = (\beta_{11}, \beta_{12})' = (1, 1)', \beta_{2} = (\beta_{21}, \beta_{22})' = (1, 1)', \beta_{3} = (\beta_{31}, \beta_{32})' = (1, 1)'

   • \mu_{x1} = \mu_{x2} = \mu_{x3} and \sigma_{x11} = 1, \sigma_{x22} = 3/2, \sigma_{x33} = 2

   • for k = 1, 2, \dots, q -1 and d = 1, \dots, D, generate X_{d} = diag(x_{d1}, x_{d2}, x_{d3})_{(q-1) \times r}, where:

     • x_{d1} = (x_{d11}, x_{d11})

     • x_{d1} = (x_{d21}, x_{d22})

     • x_{d1} = (x_{d31}, x_{d31})

     • x_{d11} = x_{d21} = x_{d31} = 1

     • U_{dk} \sim U(0, 1)

     • x_{d12} = \mu_{x1} + \sigma_{x11}^{1/2}U_{d1}

     • x_{d22} = \mu_{x2} + \sigma_{x22}^{1/2}U_{d2}

     • x_{d32} = \mu_{x3} + \sigma_{x33}^{1/2}U_{d3}

2. For random effects u, u_{d} \sim N_{q-1}(0, V_{ud}), where \theta_{1} = 1, \theta_{2} = 3/2, \theta_{3} = 2, \theta_{4} = -1/2, \theta_{5} = -1/2, \theta_{6} = 0

3. For sampling errors e, e_{d} \sim N_{q-1}(0, V_{ed}), where c = -1/4

4. The generated data is transformed using inverse alr transformation, so the data will be within the range of proportion.

Auxiliary variables X_{1}, X_{2}, X_{3}, direct estimation Y_{1}, Y_{2}, Y_{3}, and sampling variance-covariance v_{1}, v_{2}, v_{3}, v_{12}, v_{13}, v_{23} are combined into a data frame called datasaem. For more details about the structure of covariance matrix, it is available in supplementary materials of Reference.

Usage

datasaem

Format

A data frame with 30 rows and 12 columns:

Y1

Direct Estimation of Y1

Y2

Direct Estimation of Y2

Y3

Direct Estimation of Y3

X1

Auxiliary variable of X1

X2

Auxiliary variable of X2

X3

Auxiliary variable of X3

v1

Sampling Variance of Y1

v2

Sampling Variance of Y2

v3

Sampling Variance of Y3

v12

Sampling Covariance of Y1 and Y2

v13

Sampling Covariance of Y1 and Y3

v23

Sampling Covariance of Y2 and Y3

Reference

Esteban, M. D., Lombardía, M. J., López-Vizcaíno, E., Morales, D., & Pérez, A. (2020). Small area estimation of proportions under area-level compositional mixed models. Test, 29(3), 793–818. https://doi.org/10.1007/s11749-019-00688-w.