bench | 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 Ω in objective function, see details. By default this is the covariance matrix of the small area estimates. |
Lambda |
r x r matrix Λ in objective function, see details. By specifying |
This function adjusts the small area estimates EST(x)
, denoted by x0, to
x1 = x0 + Ω S (S' Ω S + Λ)^{-1} (t - S' x0)
where
Ω
is a symmetric M x M matrix. By default, Ω is taken to be the covariance matrix V0
of the input sae-object x
.
S = diag(N1,N2,…)R
where R is the matrix passed to bench
and Ni denotes the population size
of the ith 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 S 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.
Λ is a symmetric r x r matrix controlling the penalty associated with deviations from the constraints S' x1 = t. The default is Λ=0, implying that the constraints must hold exactly.
The adjusted or benchmarked small area estimates minimize the expectation of the loss function
L(x1, θ) = (x1 - θ)' Ω^{-1} (x1 - θ) + (S' x1 - t)' Λ^{-1} (S' x1 - t)
with respect to the posterior for the unknown small area means θ.
Optionally, MSE(x)
is updated as well. If mseMethod="exact"
the covariance matrix is adjusted from
V0 to
V1 = V0 - V0 S (S' Ω S + Λ)^{-1} S' V0
and if mseMethod
is "model"
the adjusted covariance matrix is
V1 = V0 + (x1 - x0) (x1 - x0)'
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.
sae-class
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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.