Gerente

The multivariate Fay-Herriot Model (MFH) is an extension of the univariate Fay-

Herriot (UFH) model defined in (2.18) and (2.19). Let us consider an (r×1) vector of survey estimators θˆd such that

ˆ

θd=θd+ed, d = 1,2, . . . , D (2.26)

where θd = (θd1, θd2, . . . , θdr)0 is a (r ×1) vector with θd = g( ¯Ydt). Let θˆd =

(ˆθd1,θˆd2, . . . ,θˆdr)0 be a (r×1) vector of survey estimators of θd. The independent

sampling errors, ed = (ed1, ed2, . . . , edr)0, are normally distributed with E(ed) = 0

and Var(ed) = Ψd. Here 0is a (r×1) vector andΨd is a (r×r) matrix. ¯Ydr is the

dth _{small area mean forr}th _{characteristic. The linking model is :}

θd=Zd0β+vd, d= 1,2, . . . , D (2.27)

where β is a (p×1) vector of regression coefficients, vd = (vd1, vd2, . . . , vdr) are

identically, independently and multivariate normally distributed area specific ran-

dom effects with E(vd) = 0,Var(vd) =Σv, and Zd is a (p×r) matrix of auxiliary

variables.

Combining the sampling (2.26) and the linking model (2.27), the MFH model

becomes

ˆ

θd=Z_d0β+vd+ed, d= 1,2, . . . , D (2.28)

The BLUP estimator of θˆd is given as

ˆ θ∗_d=Σv(Ψd+Σv) −1 _ˆ θd+Ψd(Ψd+Σv) −1 Z_d0βˆ (2.29)

(Rao and Molina, 2015, p.236) where βˆis given by ˆ β = " _D X d=1 Z_d0(Ψd+Σv)−1Z_d0 #−1" _D X d=1 Z_d0(Ψd+Σv)−1θˆd # (2.30)

The MSE of the BLUP estimator θˆ_d∗ is obtained as

MSEθˆ_d∗= Ψ−_d1 + Σv−1 −1 + Ψ−_d1+ Σv−1 −1 Σv−1Zd Z_d0(Ψd+Σv)−1Zd −1 Z_d0Σv−1(Ψd+Σv)−1 (2.31)

(Rao and Molina, 2015, p.236). The multivariate model (2.28) is a natural extension

of the UFH model (2.20). Since the multivariate area level Fay-Herriot model in-

corporates the correlations between the components of the direct estimators, it may

provide more efficient estimators of small area statistics compared to those based

on separate univariate models for each variable (Rao, 2003b, pp.236-237).

Gonz´alez-Manteiga et al. (2008) developed a multivariate extension of the Fay-

Herriot model in order to estimate the MSE of multidimensional response variables

in small areas. The method of moments is used to estimate the unknown parameters

(σ_v2) in their study. They estimated MSE using the analytical MSE approximation

(2.25) of Prasad and Rao (1990). Several parametric bootstrap MSE estimators are

also studied by Gonz´alez-Manteiga et al.(2008).

Gonz´alez-Manteiga et al. (2008) conducted a simulation study to examine the

efficiency of the bivariate Fay-Herriot model (BFH) model over the UFH model

as follows: Consider the response vectors θd = (θd1, θd2)

0

, d = 1,2, . . . , D with

the mean vectors θˆd =

ˆ θd1,θˆd2 0 . The covariates (Zd1, Zd2) 0

are generated from

a bivariate normal distribution with mean µZ1 = µZ2 = 10, variances σ2Z1 = 1

and σ2

Z2 = 2 with covariance σZ12 = 1/ √

vd1 = vd2 are generated from a normal distribution with mean 0 and variance 1.

Vectors of sampling errorsed= (ed1, ed2)

0

are generated from the multivariate normal

distribution with mean vector 0and variance-covariance matrix Ψd = (Ψdjk), j, k =

1,2 where Ψdjk = rjk

√

wd and wd is heteroscedasticity weights. For generating ed,

Gonz´alez-Manteiga et al. (2008) used r11 = 1, r22 = 2 and r12 = ρe

p

(r11r22) and

the heteroscedasticity weights wd =

p

Z2

d1+Zd22. The regression coefficients are

β1 = β2 = 1. The two response variables are generated using the model (2.28).

Both the BFH and UFH models are fitted and the empirical MSE (2.25) of the

corresponding EBLUP (ˆθEd1 and ˆθEd2) are computed. For comparison, the median

percentage reduction from the UFH to the BFH is obtained for each of the 1000

datasets.

The correlation between sampling errors (ρe) was varied from -0.5 to 0.5. Their

study showed that the reduction in MSE from the UFH to the BFH model is greater

for the second variable since the auxiliary variables and the sampling errors are

generated with larger variance. The reduction in MSE is as large as 60% when

ρe = −0.5 for the first variable and up to about 30% for ρe = −0.5 for the sec-

ond variable. In this thesis, a simulation experiment has been conducted following

Gonz´alez-Manteiga et al. (2008) in Chapter 3. Based on the simulation study in

Chapter 3 (Section 3.5), it appears that Gonz´alez-Manteiga et al. (2008) may per-

haps have actually used heteroscedasticity weights wd = 1.

Benavent and Morales (2016) introduced a general class of multivariate Fay-

Herriot models from Gonz´alez-Manteiga et al. (2008) allowing for different covari-

small area parameters. Several simulation studies are performed by Benavent and

Morales (2016) in order to assess the behavior of multivariate EBLUP and to com-

pare the multivariate Fay-Herriot model with univariate Fay-Herriot by calculating

MSE estimators. Their developed methods are applied to data from the 2005 and

2006 Spanish living condition surveys and estimated the poverty proportions and

gaps at province level. The empirical research of Benavent and Morales (2016) based

on Monte Carlo simulations showed that the multivariate EBLUPs have lower MSE

than the corresponding univariate EBLUPs.

Porter et al. (2015) improved multivariate estimates by considering both multi-

variate outcomes and latent spatial dependence. They consider three models, one in

which the outcome-by-space dependence structure is separable, one that accounts

for the cross dependence between spatial location and outcome variables through

the use of a generalized multivariate conditional autoregressive (GMCAR) structure

and the another one is the state-of-the-art multivariate model with unstructured

dependence between outcomes and no spatial dependence. They demonstrated that

GMCAR-FH model shows excellent results. A simulation study of Porter et al.

(2015) based on the Missouri county lattice structure finds gains in MSE of around

10% when cross-correlations are weaker, and around 30-70% when they are stronger,

compared to a non-spatial multivariate model (2.28).

The multivariate extension of the univariate Fay-Herriot model (2.20) for small

area estimation has been suggested by Arima et al. (2017) for two reasons: one is

to produce improved small area predictions by capturing dependence between the

different population characteristics, for example, an estimate of year-to-year change

and an associated measure of uncertainty. In small area estimation, it is also im-

portant to account random error of the covariates in the modelling (Arima et al.,

2017). Ybarra and Lohr (2008) observed that when covariates for a small area are

measured with sufficient error, then model-based small area estimators yield worse

predictions (higher mean-squared error) than the direct estimators.

Arima et al.(2015) developed a Bayesian approach of the univariate Fay-Herriot

model (2.20) when some of the covariates are measured with error and their approach

is extended to multivariate generalization by Arimaet al.(2017). Arimaet al.(2017)

provided a Bayesian analysis of a multivariate Fay-Herriot model (2.28) with func-

tional measurement errors accounting for random errors in some of the covariates.

The functional measurement error model of Arima et al. (2017) assumes that the

underlying values of the covariates (Zd in (2.28)) are fixed but unknown quanti-

ties. They applied their approach to data on 2010 and 2011 US country poverty

rates of school-aged children, for predicting 2011 poverty rates and the 2010-2011

changes. The measurement error model showed results in great improvements in pre-

diction of country poverty rates compared to the direct estimates. The authors also

showed that for estimating yearly changes of poverty rates, the bivariate model that

accounts the dependence of the model errors of each year leads to lower standard

errors compared to univariate model that assumes independence of the measurement

2.6

Small Area Estimation: Univariate Unit