sae under non-normal random effects

Small area models as hierarchical models, random effects models and mixed linear modelsMixed linear models in SAE

Empirical Best Linear Unbiased PredictionApproximation of MSE of EBLUP

MSE of EBLUP under non-normalityMSE estimation without normality

Simulation study

SAE under Non-normal Random Effects

Gauri Sankar Datta

Department of StatisticsUniversity of Georgia

Athens, GA 30602, [email protected]

Joint work with Drs. K. Irimata, J. Maples and E. SludUS Census Bureau

In Likelihood-free Methods of InferenceFebruary 18-19, 2019

Padua, Italy

Gauri Sankar Datta SAE under Non-normal Random Effects




Simulation study

Outline

1 Small area models as hierarchical models, random effectsmodels and mixed linear models

2 Mixed linear models in SAE

3 Empirical Best Linear Unbiased Prediction

4 Approximation of MSE of EBLUP

5 MSE of EBLUP under non-normality

6 MSE estimation without normality

7 Simulation study





Simulation study

Small Area Models

1 Explicit linking models based on random effects that accountfor between area variation beyond that is explained byauxiliary variables are known as “small area models”.

2 Resulting indirect estimators are “model-based estimators”.

3 Model-based indirect estimators are now the norm in SAE

4 Small area models are classified into two broad types:

Area level models relate aggregate direct SA estimators toarea-specific covariates. Such models are used if unit (orelement) level data are not available due to privacy or otherreasons. Has advantage to use design weights.Unit level models relate the unit values of a response variableto unit-specific covariates and/or area-specific covariates.





Simulation study

Frequentist approach to small area estimation

Frequentist approach: empirical best linear unbiasedprediction (EBLUP) or empirical Bayes (EB)

In frequentist approach, the model parameters (such as β, σ2v ,

defined later) are estimated from the marginal distribution ofdata

While frequentist approach is popular, clever approximationsare usually needed for accurate estimates of mean squarederror by accounting for estimation of variance parameters





Simulation study

Bayesian approach to small area estimation

Bayesian approach: Hierarchical Bayes (HB)

In HB approach, a prior distribution is assigned to the modelparameters

While the HB approach is computationally challenging,sensible inference is obtained from the posterior distribution

Since HB approach automatically accounts for varianceparameter estimation, it has an inherent conceptual advantageover the EB/EBLUP approach





Simulation study

Longitudinal Model: A Mixed Linear Model

1 Fay-Herriot model (for area-level data) and nested errorregression model (for unit-level data) are special cases ofmixed linear model

Yi = Xiβ + Zivi + ei , i = 1, . . . ,m,

Xi and Zi are known matrices;

2 v = (vT1 , . . . , vTm )T ∼ MVN(0,G (ψ)), independent of

e = (eT1 , . . . , eTm )T ∼ MVN(0,R(ψ)) .

3 R(ψ) = Block Diag(R1(ψ), · · · ,Rm(ψ)).

4 Here ψ denotes the vector of variance components parameters





Simulation study

Matrix Representation of a Mixed Linear Model

1 Y = (Y T1 , · · · ,Y T

m )T , X = (XT1 , · · · ,XT

m )T (stack the Xi ’s)

2 Z = Block Diag(Z1, · · · ,Zm)⇒ Y = Xβ + Zv + e

3 Y ∼ N(Xβ,Σ(ψ)), Σ(ψ) = R(ψ) + ZG (ψ)ZT

4 R(ψ),G (ψ) are known matrices of ψ

5 If G (ψ) is block diagonal, then Σ(ψ) is block diagonal. Truefor the Fay-Herriot and the Nested Error Regression models.

6 Model with block diagonal variance matrix is termedlongitudinal model by Rao and Molina (2015)

7 Goal: Predict θi = hTi β + λTi v for suitable known hi and λi





Simulation study

Fay-Herriot Model: A Popular Model for Area-Level Data

1 Frequently only area-level summary statistics are available toestimate θi , ith area population mean, i = 1, · · · ,m

2 Area-level summary θi (or Yi ) directly estimates θi3 Fay and Herriot (1979) proposed their model for Yi

4 Sampling model: Yi = θi + ei , eiind∼ N(0,Di ), i = 1, · · · ,m

5 Known sampling variances Di : D = Diag(D1, · · · ,Dm)

6 Fay-Herriot model connects θi to covariate xi by a

7 Linking model : θi = xTi β + vi , viind∼ N(0, σ2

v ), i = 1, · · · ,m8 A matched model

9 Z = Im, G = σ2v Im, Σ(ψ) = D + σ2

v Im.





Simulation study

Best Linear Prediction (BLP) for Mixed Linear Models

For the mixed linear model

Y = Xβ + Zv + e,

we first obtained the best linear predictor (BLP) forθi = hTi β + λTi v . To simplify the problem, first assume that β, ψare known. We predict v by its conditional expectation.

1 v |Y = y ∼ N(G (ψ)Σ−1(ψ)(y−Xβ),G (ψ){I−Σ−1(ψ)G (ψ)))

2 BLP of θi : θi ,BP(β, ψ,Y ) = hTi β + sTi (Y − Xβ), withsTi = λTi G (ψ)Σ−1(ψ).

3 Next we estimate unknown β. Assume ψ is known.

4 Marginally, Y ∼ N(Xβ,Σ(ψ))





Simulation study

Best Linear Unbiased Prediction (BLUP) for θi

1 Estimate β by generalized least squares (GLS) estimator:

2 β(ψ) = (XTΣ−1(ψ)X )−1XTΣ−1(ψ)Y

3 Using β(ψ) in θi ,BP(β, ψ,Y ), leads to BLUP of θi4 θi (ψ) = hTi β(ψ) + sTi (ψ)(Y − X β(ψ))

5 Finally, unknown ψ is estimated from the data by ψ

6 ANOVA, ML, REML and MOM for estimating ψ

7 EBLUP of θi = θi ,EBL = θi (ψ)

8 These estimates of ψ are even functions of OLS residuals





Simulation study

Details for the Fay-Herriot Model

1 Sampling model: Yi = θi + ei , eiind∼ N(0,Di ), i = 1, · · · ,m

2 Sampling variances Di ’s are assumed known

3 Linking model : θi = xTi β + vi , viind∼ N(0, σ2

v ), i = 1, · · · ,m4 vi |Y = y ∼ N( σ2

vσ2v+Di

(yi − xTi β), σ2v{1− (σ2

v + Di )−1σ2

v})

5 vi |Y = y ∼ N( σ2v

σ2v+Di

(yi − xTi β), σ2vDi

σ2v+Di

= g1i (ψ))





Simulation study

BLUP for the Fay-Herriot model

The BP of θi is

θi ,BP(β, ψ,Y ) = xTi β +σ2v

σ2v + Di

(Yi − xTi β)

= Yi − Bi (Yi − xTi β) (1)

BLUP : θi (ψ,Y ) = θi (σ2v ,Y ) = Yi − Bi (Yi − xTi β)

1 BLUP of θi is a WTD Avg. of Yi and synthetic regressionpredictor xTi β, where β = (XTΣ−1(σ2

v )X )−1XTΣ−1(σ2v )Y

2 Bi = Di/(σ2v + Di ) is the shrinkage coefficient, shrinks more

to synthetic regression for large Di

3 Under uniform prior for β, BLUP is the Bayes predictor of θi





Simulation study

MSE of BLUP

θi (σ2v )− θi = {θi ,BP(β, ψ,Y )− θi}+ {θi (σ2

v )− θi ,BP(β, ψ,Y )}= t1i + t2i

1 BLUP needs only first two moments

2 t1i and t2i are uncorrelated

3 t2i = {θi (σ2v )− θi ,BP(β, ψ,Y )} = Bix

Ti {β − β}

4 E{t2i}2 = B2i x

Ti (XTΣ−1Xi )

−1xi = g2i (σ2v )

5 MSE(θi (σ2v )) = g1i (σ

2v ) + g2i (σ

2v ) [⊥ decomposition]

6 g1i (σ2v ) measures uncertainty due to predicting vi

7 g2i (σ2v ) measures uncertainty in estimating β

8 In general, g1i (σ2v ) = O(1), g2i (σ

2v ) = O(m−1)





Simulation study

MSE of EBLUP and Approximation1 Finally, EBLUP of θi is θi ,EBL = Yi − Bi (σ

2v )(Yi − xTi β)

2 θi ,EBL − θi =

{θi ,BP(β, ψ,Y )− θi}+ {θi (σ2v )− θi ,BP}+ {θi ,EBL − θi (σ2

v )}.3 θi ,EBL − θi = t1i + t2i + t3i

4 If σ2v is an even function of OLS residuals, red, blue and green

terms are uncorrelated under normality.

5 MSE(θi ,EBL) = g1i (σ2v ) + g2i (σ

2v ) + E{θi ,EBL − θi (σ2

v )}2

6 Usually, no simple expression of the green term exists

7 So MSE of EBLUP θi usually has no exact expression

8 PR (1990), LR (1995), DL (2000), DRS (2005) and othersapproximated E{θi ,EBL − θi (σ2

v )}2 accurately up to o(m−1).





Simulation study

MSE Approximation and Estimation

1 By Taylor expansion,{θi ,EBL − θi (σ2

v )} ≈ B ′i (σ2v )(σ2

v − σ2v )(Yi − xTi β), and

2 E{θi ,EBL − θi (σ2v )}2 ≈ {B ′i (σ2

v )}2var(σ2v )(Di + σ2

v ) = g3i (σ2v )

3 Note that var(σ2v ) is of the order O(m−1). Thus, the green

term in MSE expression, i.e., g3i (σ2v ) is also of order O(m−1).

4 For any estimator of ψ with bias of the order o(m−1), Prasadand Rao (1990) provided analytical approximation to theestimator of MSE (denoted mse)

5 For other estimators of ψ, the mse estimates are given byDatta and Lahiri (2000), Datta, Rao and Smith (2005)





Simulation study

Second Order Approximations of MSE1 With σ2

v is an estimator of the variance component

MSE [θi ,EBL] = g1i (σ2v ) + g2i (σ

2v ) + g3i (σ

2v ) + o(m−1),

g3i (σ2v ) =

D2i

(σ2v + Di )4

E (Yi − xTi β)2h(σ2v ) =

D2i

(σ2v + Di )3

h(σ2v ),

2 where h(σ2v ) is the asymptotic variance of σ2

v .

3 An estimator mse, is second order unbiased if

E (mse) = MSE + o(1/m).

4 mse is called an accurate estimator of MSE(θi ,EBL)

5 Accurate estimation of MSE(θi ,EBL) is a major issue in SAE





Simulation study

Second Order Unbiased Estimation of MSE

If σ2v is an unbiased estimator of σ2

v , such as the Prasad-Raoestimator, then

E [g1i (σ2v )] = g1i (σ

2v )−g3i (σ

2v ) + o(1/m).

Also, if σ2v is asymptotically unbiased,

E [g2i (σ2v )] = g2i (σ

2v ) + o(1/m), E [g3i (σ

2v )] = g3i (σ

2v ) + o(1/m).

With an unbiased estimator σ2v , a second order unbiased estimator

of MSE is

mse(σ2v ) = g1i (σ

2v ) + g2i (σ

2v ) + 2g3i (σ

2v ).





Simulation study

Various methods of variance estimation

1 ANOVA or Henderson’s method (Prasad and Rao, 1990):

2 σ2vPR = (m − p)−1[

∑mi=1 u

2i −

∑mi=1 Di (1− hii )],

σ2vPR = max{σ2

vPR , 0}3 ui OLS residual and hii = xTi (XTX )−1xi .

4 σ2vPR is an unbiased estimator (Normality not needed). Then

msePR(σ2vPR) = g1i (σ

2vPR) + g2i (σ

2vPR) + 2g3i (σ

2vPR)

5 Note: Instead of truncating σ2vPR at a lower bound zero, we

suggested using a larger bound

LF =

√2DH√m

(due to Fuller).





Simulation study

Other estimators of σ2v : Fay-Herriot method

1 PR (1990) estimator of σ2v is subject to large variability

2 Note: E [Y T{Σ−1 − Σ−1X (XTΣ−1X )−1XTΣ−1}Y ] = m − p

3 FH (1979) suggested the estimating equation

4 Q(σ2v )

def=

∑mi=1

(Yi−xTi β)2

σ2v+Di

= m − p [Estimating equation]

5 Q(σ2v ) is a decreasing function of σ2

v with Q(σ2v ) ↓ 0

6 FH estimator of σ2v is

σ2v ,FH = I (Q(0) > m − p)Q−1(m − p) + I (Q(0) ≤ m − p)LF .

7 The estimator is biased.





Simulation study

ML and REML methods

1 DL (2000) discussed ML and REML in SAE

2 Likelihood function: L(β, σ2v ) = |Σ|−

12 exp[−1

2

∑mi=1

(Yi−xTi β)2

σ2v+Di

]

3 Profile likelihood function: LP(σ2v ) = L(β, σ2

v )

4 MLE of σ2v : σ2

v ,ML maximizes LP(σ2v )

5 Likelihood equation for σ2v :

m∑i=1

[(Yi − xTi β)2

(σ2v + Di )2

− 1

σ2v + Di

] = 0

6 DL(2000) showed that σ2v ,ML is negatively biased

7 DL(2000) showed that bias of σ2v ,RE is o(m−1);





Simulation study

Fay-Herriot model under non-normality

Yi = θi + ei , θi = xTi β + vi , i = 1, · · · ,mE (ei ) = 0, V (ei ) = Di , E (vi ) = 0, V (vi ) = σ2

v

Error terms (sampling and model) ei ’s and vi ’s are allassumed uncorrelated

Under moments assumptions, EBLUPs of θi can be derived

The MSE depends on the fourth moments of ei ’s, usually notavailable from the area summary statistics

Assume sampling errors approximately normal, may be okay

Lahiri & Rao (1995) derived MSE with non-normal vi ’s

[Bayesian version of the problem was considered by Datta &Lahiri (1995) with normal mixture]





Simulation study

EBLUP and MSE

Lahiri & Rao estimated σ2v by PR estimator and β by GLS

Prediction error: θi ,EBL − θi =

{θi ,BP(β, ψ,Y )− θi}+ {θi (σ2v )− θi ,BP}+ {θi ,EBL − θi (σ2

v )}.θi ,EBL − θi = t1i + t2i + t3i

t1i & t2i are uncorrelated; corr(t2i , t3i ) = O(m−1)

MSE (θi ,EBL) = g1i + g2i + E [t23i ] + 2E [t1i t3i ] + o(m−1)

The last two terms,O(m−1), depend on σ2v and µ4,v = E (v4

i )





Simulation study

Estimation of MSE

LR (1995) showed that the analytic estimator

msePR(σ2v ,PR) = g1i (σ

2v ,PR) + g2i (σ

2v ,PR) + 2g3i (σ

2v ,PR)

is an accurate estimator of MSE even under non-normality

This implies robustness of msePR under non-normality

Hall and Maiti (2006) considered non-normal SAE prediction

They considered for unit-level data the NER model undermoments assumption for the error terms eij ’s and vi ’s

The MSE approximation depends on E (e4ij) = µ4,e and µ4,v

They proposed bootstrap estimation of the MSE





Simulation study

Estimation equations for non-normal FH model

Estimating equation for β: GLS

m∑i=1

yi − xTi β

Di + σ2v

xi = 0

Estimating equation for σ2v :

m∑i=1

{(yi − xTi β)2

(Di + σ2v )a− (Di + σ2

v )1−a} = 0

(i) a = 0 corresponds to usual MOM (PR version)(ii) a = 1 corresponds to FH version(iii) a = 2 corresponds to ML (for normality) version





Simulation study

MSE approximation for EBLUP

Let φ = (σ2v , µ4,v )

MSE (θi ,EBL) = g1i + g2i + g3i + g4i + o(m−1)

E (t21i ) = g1i (σ

2v )

E (t22i ) = g2i (σ

2v ) = O(m−1)

E [t23i ] = g3i (φ) + o(m−1)

2E [t1i t3i ] = g4i (φ) + o(m−1)

g3i = O(m−1), g4i = O(m−1)

MSE (θi ,EBL) = g1i (σ2v ) + gi (φ) + o(m−1)

gi (φ) = O(m−1)





Simulation study

Bootstrap bias correction

Pfeffermann and Corea (2012) suggested parametricbootstrap to estimate MSE

Hall and Maiti (2006) suggested nonparametric bootstrap

From the property of σ2v we know

E [g1i (σ2v )] = g1i (σ

2v ) + bi (φ) + o(m−1)

We estimate bi (φ) by bootstrapping

MSE approximation depends on 2nd and 4th moments of vi

Estimate µ4,v from the residuals





Simulation study

Bootstrap estimation of MSE

1 Bootstrap from a population matching estimated 2nd and 4thmoments of vi

2 As in Hall and Maiti, we generate v∗i from a 3−pointdistribution

3 Generate e∗i from N(0,Di )

4 Based on the estimator φ and FH model, we generate F sets

of {θ(f )i : i = 1, · · · ,m}, f = 1, · · · ,F and datasets

{Y (f ) = (Y(f )1 , · · · ,Y (f )

m )T : f = 1, · · · ,F}





Simulation study

Bootstrap estimation of MSE (continued)

1 Corresponding to the f th dataset we obtain φ(f ) and EBLUP

θ(f )i ,EBL of θ

(f )i . Based on the F generated sets of data and

small area means, we define an empirical MSE of θi ,EBL by

m(F )i =

1

F

F∑f =1

(θ(f )i ,EBL − θ

(f )i )2. (2)





Simulation study

Bootstrap estimation of MSE (continued)

1 Eφ[m(F )i ] = g1i (φ) + bi (φ) + gi (φ) + o(m−1)

2 Eφ[g1i (φ(f ))] = g1i (φ) + 2bi (φ) + o(m−1)

3 m(F )i − { 1

F

∑Ff =1 g1i (φ

(f ))− g1i (φ)} is a second-order

unbiased estimator of the MSE(θi ,EBL(Yi ; φ)).

4 This estimator uses the known form of g1i (φ)





Simulation study

Simulation results

We compare our bootstrap method of estimating MSE withLahiri-Rao method

Following Lahiri-Rao we use three distributions (normal,exponential, double exponential), two sample sizes (m=20,m=30) and two sampling variance patterns (Pattern (a) withσ2v = .2 and four groups of sampling variances, and Pattern

(b) with σ2v = 1 and five groups of sampling variances)





Simulation study

Empirical MSE x 102: m=20, Pattern (a), σ2v = 0.2

Di 0.5 0.33 0.25 0.2γi 0.29 0.375 0.44 0.50

NormalLR(R) 16.9 14.8 13.1 11.7a=0 16.9 14.8 13.1 11.8a=1 16.8 14.7 13.0 11.7

Double ExponentialLR(R) 16.4 14.4 12.7 11.3a=0 16.3 14.3 12.7 11.4a=1 16.3 14.3 12.6 11.3

Shifted ExponentialLR(R) 16.3 14.0 12.4 11.0a=0 16.1 14.0 12.4 11.1a=1 16.1 13.9 12.3 11.0Gauri Sankar Datta SAE under Non-normal Random Effects




Simulation study

Empirical MSE x 102: m=20, Pattern (b), σ2v = 1

Di 0.5 0.33 0.25 0.2γi 0.29 0.375 0.44 0.50

NormalLR(R) 55.1 44.4 36.4 30.9a=0 55.2 44.8 37.0 31.6a=1 55.2 44.6 36.7 31.2

Double ExponentialLR(R) 53.49 43.0 35.4 30.3a=0 53.3 43.2 35.9 30.9a=1 53.62 43.2 35.6 30.5

Shifted ExponentialLR(R) 52.7 41.9 34.6 29.6a=0 52.4 42.2 35.1 30.2a=1 52.9 42.1 34.8 29.7Gauri Sankar Datta SAE under Non-normal Random Effects




Simulation study

Empirical MSE x 102: m=30, Pattern (a), σ2v = 0.2

Di 0.5 0.33 0.25 0.2 0.14Normal

LR 16.3 14.4 12.7 11.6 10.5LR(R) 15.9 13.9 12.5 11.2 10.3a=0 16.0 14.0 12.6 11.3 10.5a=1 15.9 13.9 12.5 11.2 10.3

Double ExponentialLR 15.8 14.0 12.5 11.2 10.4

LR(R) 15.7 13.7 12.2 11.0 10.0a=0 15.7 13.7 12.4 11.2 10.2a=1 15.7 13.7 12.2 11.0 10.0

Shifted ExponentialLR 15.8 13.7 12.2 11.1 10.0

LR(R) 15.7 13.5 12.0 10.8 9.8a=0 15.5 13.5 12.1 11.0 10.0a=1 15.6 13.5 12.0 10.8 9.8





Simulation study

Empirical MSE x 102: m=30, Pattern (b), σ2v = 1

Di 1 0.67 0.5 0.4 0.33γi 0.5 0.6 0.67 0.71 0.75

NormalLR 53.6 42.8 35.4 30.3 26.3

LR(R) 53.0 42.3 35.3 30.0 26.0a=0 53.1 42.5 35.5 30.2 26.3a=1 53.0 42.4 35.4 30.0 26.1


LR(R) 52.2 41.6 34.7 29.7 25.6a=0 52.0 41.7 35.0 30.0 26.9a=1 52.3 41.7 34.8 29.8 25.7


LR(R) 51.7 41.0 34.1 29.4 25.3a=0 51.3 41.0 34.4 29.8 25.7a=1 51.8 41.1 34.2 29.4 25.3





Simulation study

Relative Bias: m=20, Pattern (a), σ2v = 0.2

Di 0.5 0.33 0.25 0.2γi 0.29 0.375 0.44 0.50

NormalLR 5.0 9.0 12.0 17.0

LR(R) 9.6 12.8 17.1 20.5B(a=0) -9.7 -11.4 -11.3 -11.5

Double ExponentialLR 6.0 11.0 14.0 21.0

LR(R) 12.4 15.8 20.4 24.6B(a=0) -7.9 -10.7 -10.9 -11.1B(a=1) -8.3 -10.8 -11.2 -11.4

Shifted ExponentialLR 7.0 13.0 16.0 23.0

LR(R) 13.5 18.4 23.3 27.8B(a=0) -7.8 -9.6 -9.9 -10.2B(a=1) -8.1 -9.7 -10.3 -10.7





Simulation study

Relative Bias: m=20, Pattern (b), σ2v = 1

Di 1 0.67 0.5 0.4γi 0.5 0.6 0.67 0.71

NormalLR -0.2 0.9 0.4 2.3

LR(R) -1.1 -0.9 0.6 1.9B(a=0) -2.9 -3.9 -3.1 -2.5B(a=1) -2.7 -3.6 -2.8 -2.2

Double ExponentialLR -0 1.6 1.2 3.0

LR(R) -0.6 0.3 2.3 3.4B(a=0) -3.5 -4.7 -4.2 -4.4B(a=1) -3.8 -4.6 -4.1 -4.0

Shifted ExponentialLR 0.2 2.0 2 4.5

LR(R) -0.8 1.6 3.7 5.2B(a=0) -4.9 -5.5 -5.4 -5.3B(a=1) -5.1 -5.3 -5.1 -5.1





Simulation study

Relative Bias: m=30, Pattern (a), σ2v = 0.2

Di 0.5 0.33 0.25 0.2 0.14γi 0.29 0.375 0.44 0.50 0.54

NormalLR 1.6 2.7 3.5 5.0 5.0

LR(R) 2.7 4.5 5.3 7.4 8.3B(a=0) -6.5 -7.4 -8.6 -8.3 -8.9B(a=1) -5.6 -6.7 -7.9 -7.8 -8.3


LR(R) 3.5 5.3 7.0 8.8 11.1B(a=0) -6.2 -7.6 -8.7 -9.2 -8.8B(a=1) -5.6 -7.1 -8.0 -8.5 -8.2


LR(R) 3.5 6.5 8.8 10.0 13.1B(a=0) -6.9 -7.7 -8.4 -10.0 -9.0B(a=1) -6.5 -7.5 -8.2 -9.6 -8.8





Simulation study

Relative Bias: m=30, Pattern (b), σ2v = 1

Di 1 0.67 0.5 0.4 0.33γi 0.5 0.6 0.67 0.71 0.75

NormalLR 0.1 0.6 0.2 0.8 0

LR(R) -0.8 -0.2 -0.6 0.2 0B(a=0) 0.1 0.2 -0.3 0.6 0.2B(a=1) 0.0 0.4 -0.01 0.5 0.2

Double ExponentialLR 0.9 0 0.4 0.6 0.1

LR(R) -1.3 -0.4 -0.4 0 0.9B(a=0) -0.6 -0.6 -1.2 -1.0 -0.4B(a=1) -0.6 -0.4 -1.0 -1.0 -0.3

Shifted ExponentialLR -0.4 0.4 0.8 0.4 1.7

LR(R) 1.8 0 0.3 0.2 1.5B(a=0) -1.6 -1.1 -1.6 -2.4 -1.5B(a=1) -1.5 -0.9 -1.4 -2.1 -1.4


sae under non-normal random effects

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