Benchmarked Empirical Bayes Estimators for Multiplicative

Benchmarked Empirical Bayes Estimators forMultiplicative Area Level Models

Malay Ghosh,Tatsuya Kubokawa and Yuki KawakuboUniversity of Florida and University of Tokyo

September, 2014

Malay Ghosh Small Area Estimation

Outline

• INTRODUCTION

• MULTIPLICATIVE MODELS

• LOSS FUNCTIONS AND EB ESTIMATORS

• RISK EVALUATION AND ESTIMATION OF RISK

• SIMULATION STUDY

• RISK EVALUATION OF BENCHMARKED ESTIMATORS

• DATA ANALYSIS

• SUMMARY AND CONCLUSION


INTRODUCTION

• There is a worldwide growing demand for reliable small area(small domain) estimates.

• The need is felt both in the public and private sectors.

• There are multiple uses, for instance, allocation of governmentfunds, regional planning strategies, marketing decisions atlocal level, and the list can go on and on.

• Some important current day examples include estimation ofK-12 children under poverty at the county and lower levels ofgeography, per capita income for small places, unemploymentrates for local areas, proportion of people without healthinsurance for small domains etc.


• The ‘direct’ survey estimates, based only on domain-specificsample data may be adequate for large domains or areas, butare usually accompanied with large standard errors andcoefficients of variation for small domains, due to thesmallness of small sizes.

• The simple reason is that the original survey was designed toachieve a targeted level of accuracy for larger areas ordomains, and one does not have the resources to carry outnew surveys for these small areas or domains to achieve adesired level of accuracy.

• This makes it a necessity to borrow strength, or connectdifferent small areas through some models.

• These models provide the necessary link by bringing inrelevant auxiliary information, often collected from multipleadministrative sources.


• Bayesian and the related shrinkage methods have beenextensively used and actively studied in small-area estimation.

• However, one potential difficulty of the shrinkage estimators isthat the overall estimator for a larger geographical area, whichis often a weighted sum of the shrinkage estimators ofindividual small areas, is not necessarily equal to thecorresponding overall direct estimator.

• One way to resolve this issue is the benchmarking approach,which modifies the shrinkage estimators so that one gets thesame weighted aggregate direct estimator for largergeographical areas.

• Benchmarked estimators in the small area and relatedcontexts have been addressed in a series of articles byPfeffermann and Barnard (1991), Isaky, Tsay and Fuller(2000), You and Rao (2002; 2003), Wang, Fuller and Qu(2008), Datta, Ghosh, Storts and Maples (2011), Bell, Dattaand Ghosh (2013), Steorts and Ghosh (2013), Ghosh andSteorts (2013) among others.

• A good review of the topic is in Pfeffermann (2014).Malay Ghosh Small Area Estimation

• The current literature in small area estimation is dominatedby linear models along with normality of errors as well asnormality of random effects.

• There are a few exceptions, especially for the analysis ofbinary or count small area data, where generalized linearmixed models are mostly used.

• These assertions apply both to the frequentist and Bayesiananalysis.

• However, in many surveys, the response consists of positiveoutcomes, such as income, revenue, harvest yield, productionand many other similar quantites of interest.

• Their disributions are quite often positively skewed, and needsuitable transformations for normality to hold.

• Log transformation is one of many such transformations.

• I will be considering throughout the log-normal model in thistalk.


MULTIPLICATIVE MODELS

• Positive responses yi , i = 1, . . . ,m.

• Multiplicative model: yi = θiηi , i = 1, . . . ,m.

• The object of estimation are the θi , i = 1, . . . ,m.

• Log transformation: zi = log(yi ) = φi + ei ,where φi = logθi and ei = logηi .

• Hierarchical Model:zi |φi , β

ind∼ N(φi , di ); φi |βind∼ N(xT

i β, τ2); β ∼ uniform(Rp).

• Random Effect Formulation: zi = xTi β + ui + ei , ui , ei

mutually independent, uiiid∼ N(0, τ2), ei

ind∼ N(0, di ).

• For the moment, τ2 will be assumed known. This will berelaxed later.

• di will be assumed known throught to avoid non-identifiability.


• Notations: z = (z1, . . . , zm), θ = (θ1, . . . , θm)T ,φ = (φ1, . . . , φm)T , XT = (x1, . . . , xm), γi = di/(di + τ2),i = 1, . . . ,m.

• Σ = diag(d1 + τ2, . . . , dm + τ2).

• Assume rank(X) = p, so that XT X is nonsingular.

• Let β̂ = (XT Σ−1X)−1XT Σ−1z

• To estimate β, we consider the following two cases:

• (I) β is a random vector of coefficients having the hierarchicalprior β ∼ uniform(Rp), the uniform distribution over Rp.

• (II) β is an unknown parameter which will be estimated fromthe marginal distribution of z.

• Cases (I) and (II) lead to different posterior distributions of φi .


• (I) The posterior of φi given z isφi |z, τ2 ∼ N (φ̂HB

i (τ2), ki (τ2)),

ki (τ2) = τ2γi + γ2

i xTi (XT Σ−1X)−1xi .

• (II) The posterior of φi given z isφi |z,β, τ2 ∼ N (φ̂B

i (β, τ2), τ2γi (τ2)),

φ̂Bi (β, τ2) = (1− γi )zi + γix

Ti β.


• The posterior means E [φi |z] for Cases I and II are givenrespectively by φ̂HB

i (τ2) and φ̂Bi (β, τ2).

• Substituting the GLS β̂(τ2) into φ̂Bi (β, τ2) yields the EB

estimator φ̂EBi (τ2) = (1− γi )zi + γix

Ti β̂(τ2), which is

identical to the HB estimator φ̂HBi (τ2).

• This is not so for the θi , since the posterior means of the θifor Cases (I) and (II) are given respectively byθ̂HBi (τ2) = exp

{φ̂HB

i (τ2) + ki (τ2)/2

}and

θ̂Bi (β, τ2) = exp

{φ̂B

i (β, τ2) + τ2γi/2}

.

• Substituting β̂(τ2) into θ̂Bi (β, τ2) gives the EB estimator

θ̂EBi (τ2) = exp

{φ̂EB

i (τ2) + τ2γi/2}

.

• This is not identical to the HB estimator θ̂HBi (τ2).

• Under mild conditions, the difference between θ̂HBi (τ2) and

θ̂EBi (τ2) is O(m−1).


• One may originally assume both β and τ2 to be known, andobtain the Bayes estimator θ̂B

i (ω) of the θi and then obtainthe EB estimator of θi by substituting the GLS estimator of βas before. Here ω = (β, τ2).

• We prefer the present approach due to the fact thatθ̂HBi (τ2) = E [θi |z, τ2] = E

[E [θi |z,β, τ2]|z, τ2

]=

E [θ̂Bi (ω)|z, τ2].

• By the Rao-Blackwell theorem, θ̂HBi (τ2) has smaller risk than

that of θ̂Bi (ω) for any convex loss.


LOSS FUNCTIONS AND ESTIMATORS

• Consider the benchmarking problem of estimating the positiveparameters θi ’s by estimators θ̂i ’s under the constraint∑m

i=1 wi θ̂i = M(y) and (θ̂1, . . . , θ̂m)T ∈ Θ.

• Here wi ’s are weights such that wi > 0 and∑m

i=1 wi = 1,while Θ = {(θ1, . . . , θm)T |θ1 > 0, . . . , θm > 0}.

• M(y) can be a required constant or a random variable likeM(y) =

∑mi=1 wiyi .

• A quadratic loss LQ(θ, θ̂) =∑m

i=1 ξi (θ̂i − θi )2 for positiveconstants ξi ’s gives the constrained Bayes estimator

θ̂Ci = E [θi |yi ]−

wiξ−1iPm

j=1 w2j ξ−1j

(∑mj=1 wjE [θj |yj ]−M(y)

).

• But θ̂Ci takes negative values with a positive probability.


• This suggests that we should derive a constrained Bayesestimator under the additional restriction θ̂i > 0, i = 1, . . . ,m.

• However, such a constrained Bayes estimator cannot beexpressed in a closed form under quadratic loss.

• Another drawback of the quadratic loss (θ̂i − θi )2 is that itpenalizes θ̂i much less for θ̂i < θi than for θ̂i > θi , since(θ̂i − θi )2 converges to the finite value θ2

i as θ̂i → 0, but tends

to ∞ as θ̂i →∞.

• An alternative loss is the log-transformed quadratic lossfunction given by LTQ(θ, θ̂) =

∑mi=1 ξi (log θ̂i − log θi )

2, whichis quite natural since the multiplicative model is transformedinto the additive model via the log transformation.

• If the constraint is given by the geometric mean∏mi=1 θ̂

1/mi =

∏mi=1 y

1/mi , then the benchmark problem in the

multiplicative model can be reduced to the problem in theadditive model with the constraintm−1

∑mi=1 log θ̂i = m−1

∑mi=1 log yi .


• However, the constraint considered in this paper is theweighted mean constraint

∑mi=1 wi θ̂i = M(y), and the

constrained Bayes estimator under this constraint cannot bederived explicitly.

• To see this, let φi = log θi and φ̂i = log θ̂i .

• Goal: minimization of∑m

i=1 wiE [(φ̂i − φi )2|z] with respect to

estimator φ̂i subject to the constraint∑mi=1 wi exp(φ̂i ) = M(y),

• Resulting constrained Bayes estimators φ̂Ci ’s are solutions of

the nonlinear equations M(y){φ̂Ci − E [φi |zi ]} =

exp(φ̂Ci )∑m

j=1 wj{φ̂Cj − E [φj |zj ]}, i = 1, . . . ,m.

• Unfortunately, the solution cannot be expressed in a closedform, and more importantly, we do not know about existenceand uniqueness of the solution.

• Moreover, it is not easy to study any properties of theseestimators.


• For estimating positive quantities, there are other optionssuch as the Kullback-Leibler or the entropy loss and otherrelated loss functions.

• James and Stein (1961) used such a loss for estimating thecovariance matrix.

• KL Loss: LKL(θ, θ̂) =∑m

i=1 wi{θ̂i/θi − log(θ̂i/θi )− 1}.• The resulting Bayes estimator of θi is not the posterior mean,

but the harmonic mean 1/E [θ−1i |zi ].

• The constrained Bayes estimator is then (E [θ−1i |yi ] + λwi )

−1

where λ is the solution of the equation∑mj=1 wj/(E [θ−1

j |yj ] + λwj) = M(y).


• An alternative loss, a weighted KL loss, is given byLw

KL(θ, θ̂) =∑m

i=1 wiθi{θ̂i/θi − log(θ̂i/θi )− 1

}=∑m

i=1 wi

{θ̂i − θi − θi log(θ̂i/θi )

}.

• LwKL(θ, θ̂)→∞ as θ̂i → 0 or as θ̂i →∞.

• The resulting Bayes estimator of θi is the posterior meanE [θi |yi ].

• The resulting constrained Bayes estimator is a ratio adjustedestimator of the posterior mean, and is given byθ̂Ci = E [θi |yi ]

M(y)Pmj=1 wjE [θj |yj ]

.

• In the case when E [θ̂i/θi ] is a positive constant, the unbiasedestimator of θi is the best among estimators c θ̂i for anyconstant c .

• With uniform prior for (β, τ2), the constrained HB estimator

of θi is θ̂CHBi (τ2) = θ̂HB

i (τ2) M(y)Pmj=1 wj θ̂

HBj (τ2)

.

• Substitution of an estimator of τ2 now yields the constrainedEB estimator of θi .


• Empirical Bayes estimation of the θi needs estimation of theunknown τ2.

• Possible approaches: (a) the iterative method of momentsestimator by Fay and Herriot (1979) and Morris (1983), (b)the Prasad-Rao (1990) method of moment estimator and (c)the ML and REML estimators proposed by Datta and Lahiri(2000).

• (a) and (c) do not provide closed form estimators, but (b)does.

• The Prasad-Rao estimator of τ2 isτ̂2 = max[0, (m − p)−1

∑mj=1{(zj − xT

j β̃)2 − dj(1− hj)}].• β̃ = (XT X)−1XT z.

• The EB estimator of θi is θ̂EBi = exp[φ̂B(τ̂2) + (1/2)ki (τ̂

2)].


• A general loss function:LmKL(θ, θ̂, t) = θt

t2 {(θ̂/θ)t − log(θ̂/θ)t − 1}, t real.

• limt→0LmKL(θ, θ̂, t) = (logθ̂ − logθ)2/2.

• For the general loss, the Bayes estimator of θi isθ̂Bt = {E (θt |y)}1/t .

• For t = 1 and t = −1, the Bayes estimators correspond to theposterior mean E [θ|y] and the posterior harmonic mean1/E [θ−1|y], respectively.

• For t → 0, the Bayes estimator tends to the geometric meanexp{E [log(θ)|y]}.


RISK EVALUATION AND ESTIMATION OF RISK

• Goal: second order correct (namely, correct up to O(m−1))expression for the risk of θ̂EB

i under the loss

Li = θ̂EBi − θi − θi log(θ̂EB

i /θi ).• Assumptions: (A1) XT X/m converges to a p.d. matrix;

(A2) max1≤i≤mxTi (XT X)−1xi = O(m−1);

(A3) 0 < dL ≤ min1≤i≤mdi ≤ max1≤i≤mdi ≤ dU <∞;(A4) τ̂2(z− Xα) = τ̂2(z) for every z and every α andτ̂2(z) = τ̂2(−z) for every z;(A5) τ̂2 − τ2 = Op(m−1/2).

• Notation: si = exp[xTi β + (1/2)τ2].

• Theorem. Under assumptions (A1)-(A3), and the loss Li , therisk of the estimator θ̂EB

i of θi is

Rβ,τ2(θ̂HBi ) = Eβ,τ2Li =

1

2[di (1− γi ) + (γ2

i /2)hi ]si

+γ2

i

2{(2− γi )

2 +4γi

di}siV (τ̂2) + O(m−3/2).


• Outline of proof:Since E (θi ) = E [E (θi |τ2, z)] = E (θ̂B

i ),

E [θ̂EBi − θi − θi log(θ̂EB

i /θi )] =

E [θ̂EBi − θ̂B

i + θi log(θi )− θ̂Bi log(θ̂B

i )− θ̂Bi log(θ̂EB

i /θ̂Bi )].

• We need the fact that if W ∼ N(µ, σ2), thenE [W exp(W )] = (µ+ σ2)exp(µ+ 1

2σ2).

• Then

E (θi log(θi )] = E [φiexp(φi )] = E [E{φexp(φi )|z}]= E [{φ̂B

i + ki (τ2)}exp(φ̂B

i + (1/2)ki (τ2))]

• E [θ̂Bi log(θ̂B

i )] = E [(φ̂Bi + (1/2)ki (τ

2))exp(φ̂Bi + (1/2)ki (τ

2))].

• E (θi logθi − θ̂Bi logθ̂B

i ) =

(1/2)ki (τ2)exp{(1/2)ki (τ

2)}E [exp(φ̂Bi )].


• E [θ̂Bi log(θ̂EB

i /θ̂Bi )] =

E [θ̂Bi {φ̂B

i (τ̂2)− φ̂Bi (τ2)}+ (1/2){ki (τ̂

2)− ki (τ2)].

• E [θ̂EBi − θ̂B

i − θ̂Bi log(θ̂EB

i /θ̂Bi )] = 1

2E [θ̂Bi (τ2)(τ̂2 −

τ2)2{(φ̂Bi )′(τ2) + 1

2k ′i (τ̂2)− ki (τ

2)}2] + O(m−3/2).

• E (τ̂2 − τ2)2] = 2m−2∑m

j=1(dj + τ2)2 + O(m−2).


• The first term in the right hand side of the expression forRβ,τ2(θ̂EB

i ) is O(1).

• The remaining terms are of O(m−1).

• For the terms which are O(m−1), one needs to plug inestimators for β and τ2.

• The bias of these plugged in estimators is O(m−3/2).

• Thus the problem reduces to find an estimator ofui (τ

2) = (1/2)di (1− γi )si whose bias is of the orderO(m−3/2).

• First step is the Taylor expansionui (τ̂

2) =ui (τ

2) + (τ̂2 − τ2)u′i (τ2) + 1

2(τ̂2 − τ2)2u′′i (τ2) + O(m−3/2).

• This gives

E [ui (τ̂2)] = ui (τ

2) + (1/2)γi si{τ2hi + (τ2 + 2γi )Bias(τ̂2)

+ (−2γ2

i

di+ γi +

τ2

4)V (τ̂2)}+ O(m−3/2).


SIMULATION STUDY

• Framework of Datta, Rao and Smiith (Biometrika, 2005).

• There are five groups G1, . . . ,G5 and three small areas in eachgroup.

• The sampling variances di are the same for area within thesame group.

• Transformed Fay-Herriot Model with m = 15, τ2 = 1 and twodi patterns.

• (a): di = .7, .6, .5, .4, .3; (b) di = 2.0, 0.6, 0.5, 0.4, 0.2.

• Intercept model: the model only has a constant term.

• Simulated MSE of yi and θ̂EBi are based on 100,000

simulations.

• The relative bias (Rbias) and the relative mean squared error(RMSE) of the R̂i , the estimates of the MSE are alsocalculated.


Table 1. Simulated MSE’s, MSE Estimates, Relative Biases andRelative MSE’s for Pattern (i) using Fay-Herriot Estimates of τ2.

i di yi θ̂EBi R̂i Rbias RMSE

1 0.7 1.87 1.11 1.08 -.03 .302 0.6 1.56 1.00 0.97 -.02 .323 0.5 1.26 0.88 0.85 -.13 .284 0.4 0.99 0.74 0.73 -.00 .285 0.3 0.73 0.59 0.58 -.00 .23


Table 2. Simulated MSE’s, MSE Estimates, Relative Biases andRelative MSE’s for Pattern (ii) using Fay-Herriot Estimates of τ2.

i di yi θ̂EBi R̂i Rbias RMSE

1 2.0 7.81 2.04 1.83 -1.04 .472 0.6 1.56 0.97 0.98 -0.26 .303 0.5 1.27 0.85 0.87 -0.18 .274 0.4 1.00 0.72 0.74 -0.18 .255 0.2 0.47 0.40 0.42 +0.10 .19


RISK EVALUATION OF BENCHMARKED ESTIMATORS

• Recall θ̂CEBi (τ̂2) = θ̂EB

i (τ̂2) M(y)Pmj=1 wj θ̂

HBj (τ̂2)

.

• Also, recall the loss LmKL(θi , θ̂i ) = θi (θ̂i/θi − log(θ̂i/θi )− 1),the i-th component of the weighted KL loss.

• Risk(θ̂CEBi (τ̂2)) = Risk(θ̂EB

i (τ̂2))− K1i + K2i ;

• K1i = E [θ̂Bi log M(y)Pm

j=1 wj θ̂EBJ

];

• K2i = E [θ̂EBi {

M(y)Pmj=1 wj θ̂

EBJ

− 1}].• Assume the conditions:

(A1) XT X/m converges to a positive definite matrix;(A2) max1≤i≤m xT

i (XT X)−1xi = O(m−1);(A3) 0 < dL ≤ di ≤ dU <∞ for all i = 1, . . . ,m, where dL

and dU do not depend on m.(A4) τ̂2(z− Xα) = τ̂2(z) and τ̂2(−z) = τ̂2(z) ∀ α and z.(A5) τ̂2 − τ2 = Op(m−1/2).(A6) wi > 0 for all i and

∑mj=1 w2

j = O(m−1).


• Let M(y) =∑m

j=1 wjyj =∑m

j=1 wjexp(zj).

• Theorem. Assume (A1)-(A6). Then∑mj=1 wjyj =

∑mj=1 wjsjexp(dj/2) + Op(m−1/2);∑m

j=1 wj θ̂EBj =

∑mj=1 wjsj + Op(m−1/2), where

sj = exp(xTj β + τ2/2).

• This impliesPm

j=1 wjyjPmj=1 wj θ̂

EBj

=Pm

j=1 wj sjexp(dj/2)Pmj=1 wj sj

+ Op(m−1/2).

• E (θ̂Bj ) = sj and limm→∞θ̂

EBj = sj .

• limm→∞(−K1i + K2i ) = si (C − log(C )− 1);

C =Pm

j=1 wj sjexp(dj/2)Pmj=1 wj sj

.


• Theorem. Assume (A1)-(A6). Thenlimm→∞[R(θ̂CEB

i )− R(θ̂EBi )] = si (C − log(C )− 1).

• Now to estimate −K1i + K2i , we write−K1i + K2i = E (K̂i ) + K3i .

• K̂i = θ̂EBi [

Pmj=1 wjyjPm

j=1 wj θ̂EBj

− logPm


EBj

− 1];

K3i = E [(θ̂EBi − θ̂B

i )logPm


EBj

].

• K̂i is an unbiased estimator of Ki .

• Need a second order unbiased estimator of K3i .

• We use parametric bootstrap.


• Generate z∗i = φ∗i + ε∗i , φ∗i = xTi β̂(τ̂2) + u∗i ;

• u∗i ∼ N(0, τ̂2), ε∗i ∼ N(0, di ).

• Let y∗i = exp(z∗i ), θ∗i = exp(φ∗i ), η∗i = exp(ε∗i ) so thaty∗i = θ∗i η

∗i .

• τ̂2∗ same as τ̂2, but based on y∗i .

• K ∗3i = E∗[{θ̂EBi (τ̂2∗)− θ̂B

i (β̂, τ̂2)}logPm

j=1 wjy∗jPm

j=1 wj θ̂EBj (τ̂2∗)

|y].

• Result: E (K ∗3i ) = K3i + O(m−3/2).

• Theorem. Assume (A1)-(A6). Then the second orderunbiased estimator of the risk of θCEB

I is

R̂∗(θ̂CEBi ) = R̂∗(θ̂EB

i ) + K̂i + K ∗3i .

• R̂∗(θ̂CEBi ) = R̂∗(θ̂EB

i ) + O(m−3/2)-


DATA ANALYSIS

• Data: urvey of Family Income and Expenditure (SFIE) inJapan.

• Investigate the performances of the second-order unbiasedestimators of the risks of the EB and CEB estimators throughanalysis of the data.

• In this study, we use the data of the disbursement item‘Education’ in the survey in November, 2011.

• The average disbursement (scaled by 1,000 Yen) at eachcapital city of 47 prefectures in Japan is denoted by yi fori = 1, . . . , 47.

• Each variance di is calculated based on data of thedisbursement item ‘Education’ at the same city everyNovember in the past ten years.


• Although the average disbursements in SFIE are reportedevery month, the sample sizes are around 100 for mostprefectures, and data of the item ‘Education’ have highvariability.

• On the other hand, we have data in the National Survey ofFamily Income and Expenditure (NSFIE) for 47 prefectures.

• In this study, we use the log-transformed data of the item‘Education’ of NSFIE in 2009, which is denoted by xi fori = 1, . . . , 47.


• For i = 1, . . . , 47, the observation yi follows the multiplicativemodelyi = θiηi , i = 1, . . . , 47.

• zi = log yi , φi = log θi and εi = log ηi follow the model givenearlier.

• For τ2, we used the Fay-Herriot estimator, which yieldsτ̂2FH = 0.0520214 in this example.

• The value of the ratio∑m

j=1 wjyj/∑m

j=1 wj θ̂EBj (τ̂2FH) is

1.0876.

• CEB estimates are obtained by multiplying the empirical EBestimates by 1.0876.

• The sample size in each area (prefecture) is denoted by ni .

• The second-order unbiased estimators of the risk of θ̂EBi , the

analytical estimator and the parametric bootstrap alternativeare denoted by REB and R∗EB , respectively.

• The second-order unbiased estimator of the risk of θ̂CEBi based

on the parametric bootstrap is denoted by R∗CEB .


• Among 47 prefectures in Japan, we select the sevenprefectures in the Kanto region around Tokyo.

• The first table gives the values of ni , di , yi , EB, CEB.

• The second table gives the values of ni , di ,REB , R∗EB andR∗CEB multiplied by 100.

• The risk estimates REB and R∗EB are close each other, whichsuggest that the parametric bootstrap estimates R∗EB are notbad.

• It also shows that the estimation errors of the EB estimatorfor Chiba and Kanagawa are large, while those for Tokyo issmall since ni is large and di is small for Tokyo.

• The estimation errors R∗CEB of the CEB estimator are slightlylarger than those of R∗EB , and the estimator R∗CEB works well.


Table 3. Direct, EB and CEB Estimates

Perfecture ni di yi θ̂EBi θ̂CEB

i

Ibaraki 95 9.56 8.10 8.88 9.66Tochigi 95 26.96 10.03 9.47 10.30Gunma 94 8.41 5.21 7.70 8.38Saitama 95 3.93 12.33 12.72 13.83Chiba 94 37.83 30.71 13.05 14.20Tokyo 386 2.30 15.45 14.44 15.70Kanagawa 142 15.99 23.25 14.02 15.25


Table 4. Risk Estimates

Perfecture ni di θ̂EBi θ̂CEB

i

Ibaraki 95 9.56 .191 .224Tochigi 95 26.96 .234 .273Gunma 94 8.41 .194 .223Saitama 95 3.93 .197 .250Chiba 94 37.83 .315 .370Tokyo 386 2.30 .127 .182Kanagawa 142 15.99 .293 .350


SUMMARY AND CONCLUSION

• We considered the problem of estimating positive quantitieswith the transformed Fay-Herriot multiplicative model.

• EB estimators using weighted Kullback-Leibler loss functionare found.

• Second-order approximation of the risk of the EB estimatorsunder the weighted Kullback-Leibler loss is calculated.

• Second-order unbiased estimator of the risk via twoapproaches, namely, the analytical method based on the Taylorseries expansion and the parametric bootstrap are found.


• Benchmarked estimators are obtained and their and secondorder estimates of their risks are calculated via bootstrap.

• The performance is compared both through simualation and anumerical example.

• As future projects, one can extend the results to the unit-levelmultiplicative models and to the benchmark issues.

• It is also of interest to consider the problem of constructingconfidence intervals of the θi with second-order accuracy.


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Benchmarked Empirical Bayes Estimators for Multiplicative