bench {hbsae} | R Documentation |
Benchmark small area estimates to conform to given totals at aggregate levels.
bench(x, R, rhs, mseMethod = "no", Omega, Lambda)
x |
sae object to be benchmarked. As an alternative, a list can be supplied with at least components |
R |
restriction matrix, M x r matrix where r is the number of restrictions and M the number of areas; default is a single constraint on the population total.
Note that |
rhs |
r-vector of benchmark totals corresponding to the restrictions represented by (the columns of) |
mseMethod |
if |
Omega |
M x M matrix |
Lambda |
r x r matrix |
This function adjusts the small area estimates EST(x)
, denoted by x_0
, to
x_1 = x_0 + \Omega R_N (R_N' \Omega R_N + \Lambda)^{-1} (t - R_N' x_0)\,,
where
\Omega
is a symmetric M x M matrix. By default, \Omega
is taken to be the covariance matrix V_0
of the input sae-object x
.
R_N = {\rm diag}(N_1,\dots, N_M)\,R
where R
is the matrix passed to bench
and N_i
denotes the population size
of the i
th area, is a M x r matrix describing the aggregate level relative to the area level.
Note that the matrix R
acts on the vector of area totals whereas R_N
acts on the area means to
produce the aggregate totals.
The default for R
is a column vector of 1s representing an additivity constraint to the overall population total.
t
is an r-vector of aggregate-level totals, specified as rhs
, that the small area estimates should add up to.
\Lambda
is a symmetric r x r matrix controlling the penalty associated with deviations from the constraints
R_N' x_1 = t
.
The default is \Lambda=0
, implying that the constraints must hold exactly.
The adjusted or benchmarked small area estimates minimize the expectation of the loss function
L(x_1, \theta) = (x_1 - \theta)' \Omega^{-1} (x_1 - \theta) +
(R_N' x_1 - t)' \Lambda^{-1} (R_N' x_1 - t)
with respect to the posterior for the unknown small area means \theta
.
Optionally, MSE(x)
is updated as well. If mseMethod="exact"
the covariance matrix is adjusted from
V_0
to
V_1 = V_0 - V_0 R_N (R_N' \Omega R_N + \Lambda)^{-1} R_N' V_0\,,
and if mseMethod
is "model"
the adjusted covariance matrix is
V_1 = V_0 + (x_1 - x_0) (x_1 - x_0)'\,.
The latter method treats the benchmark adjustments as incurring a bias relative to the best predictor under the model.
An object of class sae
with adjusted estimates.
G.S. Datta, M. Ghosh, R. Steorts and J. Maples (2011). Bayesian benchmarking with applications to small area estimation. TEST 20(3), 574-588.
Y. You, J.N.K. Rao and P. Dick (2004). Benchmarking Hierarchical Bayes Small Area Estimators in the Canadian Census Undercoverage Estimation. Statistics in Transition 6(5), 631-640.
d <- generateFakeData()
# compute small area estimates
sae <- fSAE(y0 ~ x + area2, data=d$sam, area="area", popdata=d$Xpop)
# calibrate to overall population total
sae.c <- bench(sae, rhs=sum(d$mY0*sae$Narea))
plot(sae, sae.c)