PREDICTION OF A SMALL AREA MEAN FOR A FINITE POPULATION WHEN THE VARIANCE COMPONENTS ARE RANDOM

FOR A FINITE POPULATION WHEN

THE VARIANCE COMPONENTS ARE RANDOM

MARIUS S¸TEFAN

In this paper we propose a new model with random variance components for estimating a small area mean. Under the proposed model and in the finite popu- lation case we derive the empirical best linear unbiased predictor for the small area mean, an approximation to terms of ordero(1/m) and an estimator whose bias is of ordero(1/m) for its mean squared error, wheremis the number of small areas in the population.

AMS 2000 Subject Classification: 62D05.

Key words: small areas, direct and indirect estimation, finite population, empirical best linear unbiased predictor.

1. INTRODUCTION

National surveys are generally designed for estimating parameters at na- tional level. The inference is based on the sampling distribution and the size of the sample insures an adequate level of precision for the resulting estimators.

There is a growing demand for estimations at sub national level, region or county for instance. In the sample survey theory these sub populations are called domains or areas. If we base the inference on the sampling distribu- tion which is the distribution coming from the sampling design the resulting estimator for some area characteristics called direct estimator has the same formula as the estimator for the same parameter at the national level but it uses only the observations falling in the area. Because the national survey was designed for national estimation not for the area under study, it happens more often than not that these observations are not enough to insure an adequate level of precision for the direct estimator of the area parameter. This is why in this case the area is called small area and a new theory is needed to estimate small area parameters.

The new theory is called small area estimation theory. A detailed ac- count is given in [4]. In small area estimation the inference is model based.

This means that first one has to find a model for the population values of the

MATH. REPORTS12(62),3 (2010), 301–313

variable under study, then check the model fit and finally base the inference on the model to get the estimator as well as a measure for its precision. The model based estimator for the small area characteristics called indirect esti- mator uses the entire national sample not only the small area sample because the population model acts like a link between different areas of the popula- tion. For this reason it is said that the indirect estimator borrows strength from related small areas and its precision is generally better than that of the direct estimator.

The parameter of interest is the small area mean. We consider only the finite population case (the infinite population case will be presented else- where). In Section 2 of our paper we propose a new model which can be used for estimating the mean of a small area and give the formula for the best linear unbiased predictor of the mean under the proposed model. In Section 3, first we derive an approximation to terms of order o(1/m) for the precision of the predictor measured by its mean squared error. Then we will obtain an estima- tor of bias of order o(1/m) for the precision of the predictor. In Section 4 we present a Monte Carlo simulation showing the accuracy of both the approxi- mation and the estimator of the precision. Finally, in Section 5 we draw some conclusions and give a few directions for future research.

2. A NEW MODEL

In our paper we will be interested in estimating the meanµiof small area ifrom a population composed ofmsmall areas. We suppose that the sampling design is non informative which means that the model for the population values holds true for the sample values. A wide range of models have been proposed and used in the literature for estimating µ_i. In [6] the authors proposed a two- fold nested error regression model with constant variances useful in situations when the individuals are grouped in primary units and each small area is composed of several primary units. Cleary their model for the population and the sample has the form

y_ijk=x^t_ijkβ+v_i+u_ij +e_ijk, (1)

k= 1, . . . , Nij(nij), j= 1, . . . , M_i⁰(m⁰_i), i= 1, . . . , m,

where m is the number of small areas in the population; M_i⁰ and m⁰_i are the population respectively the sample number of primary units in small area i;

N_ij andn_ij are the population number of individuals and the sample number of individuals in primary unitjfrom small areai;x^t_ijkis the vector of auxiliary variable for individual k in primary unit j from small areai; β is the vector of fixed effects, v_i is the random small area effect following v_i ∼ N(0, σ²_v);

uij is the random primary unit effect following uij ∼ N(0, σ_u²); eijk is the

model error following eijk ∼ N(0, σ_e²); the effects vi, uij and eijk are inde- pendent. The variance componentsσ² = (σ²_v, σ_u², σ²_e)^t are fixed and unknown.

The sample is selected as follows: from the small area i a sample of m⁰_i pri- mary units is selected and from each selected primary units a sample of nij

individuals is selected.

Model (1) supposes that the variancesσ²_u andσ_e²are the same regardless of the small area or the primary unit. In practice this hypothesis may be restrictive especially if we deal with non homogenous small areas. For this reason we propose a less restrictive model with σ²_u and σ²_e depending on the small area. Clearly, our model is given by

y_ijk =µ+vi+uij +e_ijk, (2)

k= 1, . . . , N(n), j= 1, . . . , M⁰(m⁰), i= 1, . . . , m,

where v_i ∼N(0, σ_v²), u_ij|σ_i² ∼ N(0, σ²_i), e_ijk|τ_i² ∼N(0, τ_i²), σ²_i ∼law(β₁, α₁) and τ_i² ∼law(β₂, α₂) (law(β, α) designates an arbitrary distribution of mean β and variance α). The effectsvi, uij and e_ijk are conditionally independent The other notations in (2) has the same meaning as in (1).

Remark 1. In (2) we don’t have auxiliary variables and the model sup- poses that we have both a balanced population and sample: N and nare the same regardless of iand j; M⁰ and m⁰ are the same regardless ofi; we make these restrictive hypotheses for theoretical reasons.

Remark 2. (2) supposes that τ_i² depend only on i and not onj; a more general model should incorporate τ_ij² instead of τ_i² but the theoretical results would be more difficult to obtain under this less restrictive model.

The idea of passing from a constant to a random vector of variance components is not new. It can be found in [3] where it was done for a one-fold nested error regression model, that is a model appropriate for a population without primary units. Initially, an one-fold nested error regression model with constant variances was proposed in [1]. In [3] the authors transformed the model in [1] into a model with random variances.

We divide the individuals in two parts: the observed and the unobserved ones. Then the mean of small area ican be written as

(3) µ_i = 1

M⁰N m⁰

X

j=1 n

X

k=1

y_ijk+

m⁰

X

j=1 N

X

k=n+1

y_ijk+

M⁰

X

j=m⁰+1 N

X

k=1

y_ijk

,

where the first term is the known sum of the observed individuals, the second term is the unknown sum of the unobserved individuals from the selected primary units and the third term is the sum of all the individuals in the primary units that were not selected. As a consequence, in order to predictµi

we will have to predict the second and the third terms. We will do it by taking the best (in the sense of the mean squared error) linear unbiased predictor which will be of the form

µe^F_i = 1 M⁰N

m⁰

X

j=1 n

X

k=1

y_ijk+l^ty

,

whereyis the vector of all the sample observations anlwill have to be chosen so thatE(µe^F_i ) =E(µ_i) andM SE(eµ^F_i ) takes its minimum (F stands for finite population case). It can be verified (see [5, Chapter 4] for technical details) that under (2), µe^F_i is given by

µe^F_i = 1 M⁰N

m⁰n¯yi··+1

δ[(M⁰N−m⁰n)σ²_v+ (N−n)β₁]¯yi··+ (4)

+ 1

m⁰nδ[nN(M⁰−m⁰)β₁+ (M⁰N−m⁰n)β₂]¯y

, where β = β1 + ^β_n², δ = σ_v²+ _m^β10 + _m^β20n, ¯yi·· = _m¹0n

Pm⁰ j=1

Pn

k=1y_ijk and ¯y =

1 mm⁰n

Pm i=1

Pm⁰ j=1

Pn

k=1y_ijk. We can also compute exactly the mean squared error of (4) (see [5, Chapter 4] for technical details) which will be given by

M SE(µe^F_i ) = 1 (M⁰N)²

σ_v²(M⁰N −m⁰n)²+β2(M⁰N −m⁰n)+

(5) +β₁

m⁰(N −n)²+ (M⁰−m⁰)N²

−1 δ

σ_v²(M⁰N −m⁰n) +β₁(N−n)2

+ +δ

m h

(M⁰N−m⁰n)−1

δ(σ_v²(M⁰N−m⁰n) +β1(N −n)) i2

.

The best linear unbiased predictor µe^F_i cannot be used to predict µi be- cause as can be seen from (4) it depends on β, β₁, β₂, δ and σ²_v which are unknown. In practice we look for (if possible) unbiased estimators of β, β1, β₂, δ and σ²_v, plug them into (4) and get what is called the empirical best linear unbiased predictor of µ_i which we will denote byµb^F_i . Clearly, it can be proved (see [5, Chapter 4] for details) that the estimators β,b βb₁, βb₂,bδ and σb²_v given below are unbiased

βb= 1 m(m⁰−1)

m

X

i=1 m⁰

X

j=1

(¯yij·−y¯i·)², δb= 1 m−1

m

X

i=1

(¯yi··−y)¯ ², (6)

βb₂ = 1 mm⁰(n−1)

m

X

i=1 m⁰

X

j=1 n

X

k=1

(y_ijk−y¯ij·)², βb₁ =βb−βb2

n, bσ²_v =δb− βb m⁰,

where ¯yij·= _n¹Pn

k=1yijk. Thenµb^F_i will be given by

eµ^F_i = 1 M⁰N

m⁰n¯yi··+1 bδ

(M⁰N−m⁰n)bσ²_v+ (N−n)βb1

y¯i··+ (7)

+ 1

m⁰nbδ

nN(M⁰−m⁰)βb₁+ (M⁰N−m⁰n)βb₂

¯ y

(7) is the predictor that can be used in practice to predict µi.

3. APPROXIMATION AND ESTIMATION OF M SE(µb^Fi )

We now have to measure the precision of µb^F_i , that is to compute its mean squared errorM SE(µb^F_i ). Contrary toM SE(µe^F_i ),M SE(µb^F_i ) cannot be computed exactly and it would be na¨ıve to consider that they are equal. In fact we will considerM SE(µe^F_i ) as the na¨ıve mean squared error of bµ^F_i , that is M SE_N(µb^F_i ) =M SE(µe^F_i ). µb^F_i contains in its formula the estimatorsβ,b βb1, βb₂, bδ and bσ_v² compared to eµ^F_i which contains the fixed unknown constants β, β₁, β₂, δ and σ_v², instead. For this reason we expect µb^F_i to have a larger variability than µe^F_i , that is M SE(µb^F_i ) > M SE(µe^F_i ). In Section 4 numerical results will show that M SEN(µb^F_i ) can represent a serious under estimation for M SE(bµ^F_i ).

The following theorem gives an approximation to terms of ordero(1/m) forM SE(µb^F_i ) (O(1) represents a quantity that remains bounded whenm→ ∞ and o(1/m) represents a quantity which multiplied by m tends to zero when m→ ∞).

Theorem 1. Let the model (2) and the following regularity conditions:

a) M⁰ =O(1), m⁰ =O(1), N =O(1) and n=O(1); b) σ²_i and τ_i² have finite twelfth order moments. Let’s define

T₁= β₂

m⁰n− β₂² m(m⁰n)²δ

m−3− 2 m⁰(n−1)

+ 6α₂β₂ m(m⁰n)³δ²−

− 3αβ₂²

mm⁰⁴n²δ³ − α2

m(m⁰n)²δ

1− 2

m⁰(n−1)

, T2 = β

m⁰ − β²

m⁰²δ − 3α

mm⁰²δ + β²

mm⁰²δ + 2(α+β²)

mm⁰(m⁰−1)δ + 6αβ

mm⁰³δ² − 3β²α mm⁰⁴δ³, T₃= β2

m⁰n− β2β m⁰²nδ

1− 3

m

− α2

mm⁰²n²δ + 3β2α

mm⁰³nδ² + 3βα2

mm⁰³n²δ² − 3β2βα mm⁰⁴nδ,

where α=α1+^α_n2². Then M SE(µb^F_i ) =m⁰²(N−n)²

(M⁰N)² T1+ (M⁰−m⁰)²

M⁰² T2+ 2m⁰(N−n)(M⁰−m⁰) M⁰²N T3+ (8)

+M⁰−m⁰

M⁰² β+ 1 M⁰²

M⁰N −m⁰n

N² −M⁰−m⁰ n

β₂+o

1 m

. Proof. See [5, Chapter 4, pag. 156–181] for technical details.

(8) shows that the approximation correct to terms of order o(1/m) will be given by

M SE_A(µb^F_i ) =m⁰²(N−n)²

(M⁰N)² T₁+(M⁰−m⁰)²

M⁰² T₂+2m⁰(N−n)(M⁰−m⁰) M⁰²N T₃+ (9)

+M⁰−m⁰

M⁰² β+ 1 M⁰²

M⁰N −m⁰n

N² −M⁰−m⁰ n

β2.

If the numberm of small areas in the population is large enough (gene- rally, m ≥ 30 is sufficient) than M SEA(µb^F_i ) is a good approximation of M SE(µb^F_i ), the relative error being negligible.

M SE_A(µb^F_i ) is unknown because it depends on unknown constants. As a consequence it cannot be used for estimating M SE(µb^F_i ). The na¨ıve approach would be to replace in the formula of M SE_N(µb^F_i ) the unknown quantities by their estimators resulting in the na¨ıve estimator mseN(µb^F_i ). By simulation (see Section 4) we checked that mseN(µb^F_i ) can have large negative bias. The following theorem gives an estimator for M SE(µb^F_i ) of bias of order o(1/m)

Theorem2. Let the model (2)and the same regularity conditions as in the theorem above. Let’s define

bγ_i²= 1 m⁰−1

m⁰

X

j=1

(¯yij·−y¯i··)², αb= m⁰−1 m(m⁰+ 1)

m

X

i=1

(bγ_i²)²−βb²,

bτ_i² = 1 m⁰(n−1)

m⁰

X

j=1 n

X

k=1

(y_ijk−y¯ij·)², αb₂ = m⁰(n−1) m⁰(n−1) + 2

1 m

m

X

i=1

(τb_i²)²−βb₂²,

Tb1= βb2

m⁰n− m−1 m(m⁰n)²

βb₂²

δb + 4 m(m⁰n)²

"

βb₂²

bδ + βb2αb2

m⁰nδb²+ αb2

m⁰(n−1)bδ+ βb₂² m⁰(n−1)bδ

# ,

Tb₂= βb

m⁰ −m−1 mm⁰²

βb² bδ + 4

mm⁰

"

βb²

(m⁰−1)bδ + βbαb

m⁰²δb² + αb m⁰(m⁰−1)bδ

# ,

and

Tb3 = βb2

m⁰n− m−1 mm⁰²n

βb2βb

bδ + 2 mm⁰²n

2βb2βb

bδ + αb2βb

m⁰nbδ² + αbβb m⁰bδ²

! . Let’s consider mse(µb^F_i ) be given by

mse(µb^F_i ) = (N−n)²m⁰²

(M⁰N)² Tb1+(M⁰−m⁰)² M⁰² Tb2+ (10)

+2(N−n)(M⁰−m⁰)m⁰

M⁰²N Tb₃+M⁰−m⁰

M⁰² βb+ 1 M⁰²

M⁰N−m⁰n

N² − M⁰−m⁰ n

βb₂. Then the bias of mse(µb^F_i ) as an estimator ofM SE(µb^F_i ) is of order o(1/m).

Proof. See [5, Chapter 4, pag. 156–181] for technical details.

As above, whenmis large enough (m≥30), mse(bµ^F_i ) can be used as an estimator of negligible bias.

4. MONTE CARLO STUDY

In this section we will present the results of Monte Carlo simulations showing howM SEA(µb^F_i ) andmse(µb^F_i ) perform as approximation and estima- tion of M SE(µb^F_i ) respectively. The results are shown for the first small area of the population. The computer program written in S-Plus is in the Annexe.

Without loss of generality we fixed µ = 0, m = 30, M⁰ = 8, m⁰ = 2, N = 8, n= 2 and β₂ = 300. Then we took several values forσ²_v and β₁ equal to 15, 30, 60, 150, and 300 such that the ratios β1/β2 and σ²_v/β2 take the values 0.05, 0.1, 0.2, 0.5, and 1. Then we computed for the first small area of the population (i = 1) the values of M SEN(µb^F_i ) and M SEA(µb^F_i ) given by (5) and (9). The Monte Carlo simulated parameters are computed from G= 10000 samplesy_ijk,i= 1, . . . , m;j= 1, . . . , m⁰ and k= 1, . . . , n selected through simple random sampling of primary units and then individuals from G= 10000 populations, each population being generated as follows:

– we generatedσ₁²,σ²₂, . . . , σ₃₀² fromχ(β1) (⇒α1= 2β1) andτ₁²,τ₂², . . . , τ₃₀² from χ(β₂) (⇒α₂= 2β₂);

– we generatedeijkfromN(0, τ_i²),uij fromN(0, σ_i²) andvifromN(0, σ²_v), fori= 1, . . . ,30,j= 1, . . . ,8,k= 1, . . . ,8 and computed the population values by y_ijk =v_i+u_ij +e_ijk;

– having generated the whole population we could compute the true mean µ_1g of the first small area;

– from each population we selected the sample observations through sim- ple random sampling of primary units and then from each selected primary units we sampled the individuals again by simple random sampling;

– using the sample valuesyijkwe obtainedµb^F_1g,mse(µb^F₁)gandmseN(µb^F₁)g

according to their respective formulas.

Then we computed the Monte Carlo mean squared error ofµb^F₁ given by (11) M SE_{M C}(µb^F₁) = 1

10000

10000

X

g=1

(µb^F_1g−µ_1g)².

Using (11) we computed the relative errors ofM SE_A(µb^F₁) andM SE_N(bµ^F₁) by RE= 100M SEA(µb^F₁)−M SEM C(µb^F₁)

M SEM C(µb^F₁) , RE_N = 100M SE_N(µb^F₁)−M SE_{M C}(µb^F₁)

M SE_{M C}(µb^F₁) . (12)

The Monte Carlo value ofE(mse(µb^F₁)) was simulated by (13) E_{M C}(mse(bµ^F₁)) = 1

10000

10000

X

g=1

mse(µb^F₁)_g.

The same formula will be used forE(mseN(µb^F₁)). Using (13) the relative bias of mse(µb^F₁) will be given by

(14) RB= 100EM C(mse(µb^F₁))−M SEM C(µb^F₁) M SEM C(µb^F₁) .

A similar formula will be used for mseN(µb^F₁). The two tables below present the numerical results of our simulations.

Table 1. Relative errors (%) ofM SEN(µb^F₁)/M SEA(µb^F₁) m= 30,M⁰= 8,N= 8,m⁰= 2,n= 2

β₁/β₂

0.05 0.1 0.2 0.5 1

0.05 −29.31/−2.99 −26.84/−0.60 −25.77/0.47 −27.21/−0.89 −27.23/−0.83 σ²_v/β₂ 0.1 −21.97/−3.85 −18.90/−0.47 −19.18/0.03 −20.77/0.12 −23.04/−0.56 0.2 −13.19/−2.12 −11.06/0.49 −13.09/−0.57 −14.74/0.05 −19.54/−2.17 0.5 −4.91/0.18 −4.47/0.97 −8.22/−2.07 −7.93/−0.04 −10.78/−0.51 1 −4.36/−1.65 −2.32/0.57 −3.20/0.09 −4.96/−0.52 −6.49/−0.37

Table 2. Relative bias (%) ofmseN(bµ^F₁)/mse(bµ^F₁) m= 30,M⁰= 8,N= 8,m⁰= 2,n= 2

β₁/β₂

0.05 0.1 0.2 0.5 1

0.05 −55.93/−1.82 −53.37/1.65 −53.67/2.51 −54.91/0.60 −54.62/1.15 σ_v²/β2 0.1 −39.42/−2.61 −38.15/0.83 −39.91/1.20 −42.52/2.09 −45.69/1.22 0.2 −24.15/−1.16 −23.12/1.87 −26.14/0.64 −30.13/1.33 −37.33/−0.90 0.5 −10.00/1.01 −10.20/1.65 −14.47/−1.43 −16.61/0.57 −21.25/0.69

1 −7.17/−1.38 −5.55/0.82 −6.65/0.49 −9.33/0.23 −13.26/−0.12

From Tables 1 and 2 we can see that the underestimations ofM SEN(µb^F₁) and mseN(µb^F₁) can be important. REN and RBN decrease in absolute value when β₁/β₂ is fixed and σ_v²/β₂ increases. On the other hand, they increase when σ_v²/β2 is fixed andβ1/β2 increases. As far asREand RBare concerned they are small for all the values ofβ₁/β₂ and σ_v²/β₂ considered in our simula- tions which confirms the accuracy of theoretical results in Theorems 1 and 2.

5. CONCLUSION

In this paper we propose a two-fold nested error regression model with random variances. Under this model and in the finite population case we derive the empirical best linear unbiased predictor together with an approximation and an estimator for its mean squared error.

Presently, we study the infinite population case. In this case the total number of primary units M⁰ is large enough so that µi ≈ µ+vi. In this case predicting µ_i is equivalent to predicting a linear combination of some parameters of model (2). The empirical best linear unbiased predictor will have to be derived together with an approximation and an estimator of bias of order o(1/m) for its mean squared error. The results will be the subject of another paper.

Often in practice statisticians deal with time series data. For instance we have data which could be modelled using an one-fold nested error regression model but if moreover the observations are done in time then a supplementary indextfor time is needed and an one-fold model will transform into a two-fold model. Thus the theory in this paper can be used to handle such data. Of course the independence hypotheses on the effectsv_i,u_ij ande_ijk will have to be dropped because in time the observations on the same units are correlated.

In dealing with such a model one can use a result presented in [2]. Clearly, they show that given the model z = Wβ+ψ where ψ is a vector of mean 0 and of variance-covariance matrix V, it is possible to find a transformation matrix T such that the errors ψ^∗ = Tψ of the transformed model Tz = TWβ+Tψ be non-correlated and with constant variances. We can use the results presented in this paper under the model Tz=TWβ+Tψ and then return to the initial model via the transformation T.

APPENDIX

The S-Plus functiongenereazabazadatePFcomputes the relative errors and the relative bias in the case of a finite population:

genereazabazadatePF<-(miu,sigmav,beta1,beta2,m,Mprim,N, mprim,n,G){

alpha1<-2*beta1; alpha2<-2*beta2; alpha<-alpha1+alpha2/n^2 beta<-beta1+beta2/n;delta<-sigmav+beta/mprim

T1<-beta2/(mprim*n)-beta2^2/((mprim*n)^2*delta)+

3*beta2^2/(m*delta*(mprim*n)^2)+2*beta2^2/(m*mprim^3*n^2*

(n-1)*delta)+6*beta2*alpha2/(m*(mprim*n)^3*delta^2)- 3*alpha*beta2^2/(m*mprim^4*n^2*delta^3)-alpha2/

(m*(mprim*n)^2*delta)+2*alpha2/(m*mprim^3*(n-1)*delta*n^2) T2<-beta/mprim-beta^2/(mprim^2*delta)-3*alpha/(m*mprim^2*delta) +beta^2/(m*mprim^2*delta)+2*(alpha+beta^2)/(m*mprim*(mprim-1)*

delta)+6*alpha*beta/(m*mprim^3*delta^2)-3*alpha*beta^2/

(m*mprim^4*delta^3)

T3<-beta2/(mprim*n)-(m-3)*beta2*beta/(m*mprim^2*n*delta)-

alpha2/(m*mprim^2*n^2*delta)+3*beta2*alpha/(m*mprim^3*n*delta^2) +3*beta*alpha2/(m*mprim^3*n^2*delta^2)-3*beta2*beta*alpha/

(m*mprim^4*n*delta^3)

EQMmiuchap1PF<-(mprim*(N-n))^2*T1/(Mprim*N)^2+

(N*(Mprim-mprim))^2*T2/(Mprim*N)^2+2*mprim*N*(N-n)*

(Mprim-mprim)*T3/(Mprim*N)^2+N^2*(Mprim-mprim)*beta1/

(Mprim*N)^2+(Mprim*N-mprim*n)*beta2/(Mprim*N)^2

EQMmiuchap1PFN<-sigmav*(Mprim*N-mprim*n)^2/(Mprim*N)^2+

beta2*(Mprim*N-mprim*n)/(Mprim*N)^2+beta1*(mprim*(N-n)^2+

(Mprim-mprim)*N^2)/(Mprim*N)^2-(sigmav*(Mprim*N-mprim*n)+

beta1*(N-n))^2*(m-1)/(delta*(Mprim*N)^2*m)+

delta*(Mprim*N-mprim*n)^2/(m*(Mprim*N)^2)-

2*(Mprim*N-mprim*n)*(sigmav*(Mprim*N-mprim*n)+beta1*(N-n))/

(m*(Mprim*N)^2)

taui<-rep(0,m);ybari<-rep(0,m);taui<-rep(0,m);

sigmai<-rep(0,m)

listayPF<-array(,dim=c(N,Mprim,m));

listay<-array(,dim=c(n,mprim,m)) listaePF<-array(,dim=c(N,Mprim,m))

u<-matrix(,m,Mprim);ybarij<-matrix(,m,mprim) EQMMmiuchap1PF<-0;EMeqmmiuchap1PF<-0;

EMeqmmiuchap1PFN<-0

for(g in 1:G) {

sigma<-rchisq(m,beta1);tau<-rchisq(m,beta2) v<-rnorm(m,mean=0,sd=sqrt(sigmav))

v1<-sort(sample(1:N,n));v2<-sort(sample(1:Mprim,mprim)) betachap2<-0

for(i in 1:m) {

u[i,]<-rnorm(Mprim,mean=0,sd=sqrt(sigma[i])) for(j in 1:Mprim)

{

listaePF[,j,i]<-rnorm(N,mean=0,sd=sqrt(tau[i])) listayPF[,j,i]<-miu+v[i]+u[i,j]+listaePF[,j,i]

{

listay[,,i]<-listayPF[v1,v2,i]

betachap2<-betachap2+sum(colVars(listay[,,i],SumSquares=TRUE)) taui[i]<-sum(colVars(listay[,,i],SumSquares=TRUE))/(mprim*(n-1)) ybarij[i,]<-colMeans(listay[,,i])

ybari[i]<-mean(listay[,,i]) }

betachap2<-betachap2/(m*mprim*(n-1))

sigmai<-rowVars(ybarij,SumSquares=TRUE)/(mprim-1)

betachap<-sum(rowVars(ybarij,SumSquares=TRUE))/(m*(mprim-1)) deltachap<-sommecarres(ybari)/(m-1);

sigmavchap<-deltachap-betachap/mprim

alphachap<-(mprim-1)*mean(sigmai^2)/(mprim+1)-betachap^2 betachap1<-betachap-betachap2/n

alphachap2<-mprim*(n-1)*mean(taui^2)/(mprim*(n-1)+2)-betachap2^2 miu1<-mean(listayPF[,,1])

miuchap1PF<-ybari[1]-(ybari[1]-mean(ybari))*betachap*

(Mprim-mprim)/(mprim*Mprim*deltachap)-(ybari[1]-mean(ybari))*

(N-n)*betachap2/(Mprim*N*n*deltachap)

EQMMmiuchap1PF<-EQMMmiuchap1PF+(miu1-miuchap1PF)^2 T1chap<-betachap2/(mprim*n)-(m-5)*betachap2^2/

(m*mprim^2*n^2*deltachap)+4*alphachap2*betachap2/

(m*mprim^3*n^3*deltachap^2)+4*alphachap2/

(m*mprim^3*n^2*(n-1)*deltachap)+4*betachap2^2/

(m*mprim^3*n^2*(n-1)*deltachap)

T2chap<-betachap/mprim-(m-1)*betachap^2/

(m*mprim^2*deltachap)+4*betachap^2/(m*mprim*(mprim-1)

*deltachap)+4*betachap*alphachap/(m*mprim^3*deltachap^2)+

4*alphachap/(m*mprim^2*(mprim-1)*deltachap)

T3chap<-betachap2/(mprim*n)-(m-1)*betachap2*betachap/

(m*mprim^2*n*deltachap)+4*betachap2*betachap/(m*mprim^2*n*

deltachap)+2*alphachap2*betachap/(m*mprim^3*n^2*deltachap^2)+

2*alphachap*betachap2/(m*mprim^3*n*deltachap^2) eqmmiuchap1PF<-(mprim*(N-n))^2*T1chap/(Mprim*N)^2+

(Mprim-mprim)^2*T2chap/Mprim^2+2*mprim*(N-n)*(Mprim-mprim)*

T3chap/(Mprim^2*N)+(Mprim-mprim)*betachap/Mprim^2+

(Mprim*N-mprim*n)*betachap2/(Mprim*N)^2-(Mprim-mprim)*

betachap2/(Mprim^2*n)

EMeqmmiuchap1PF<-EMeqmmiuchap1PF+eqmmiuchap1PF

eqmmiuchap1PFN<-sigmavchap*(Mprim*N-mprim*n)^2/(Mprim*N)^2+

betachap2*(Mprim*N-mprim*n)/(Mprim*N)^2+betachap1*(mprim*(N-n)^2+

(Mprim-mprim)*N^2)/(Mprim*N)^2-(sigmavchap*(Mprim*N-mprim*n)+

betachap1*(N-n))^2*(m-1)/(deltachap*(Mprim*N)^2*m)+

deltachap*(Mprim*N-mprim*n)^2/(m*(Mprim*N)^2)-2*(Mprim*N-mprim*n)*

(sigmavchap*(Mprim*N-mprim*n)+betachap1*(N-n))/(m*(Mprim*N)^2) EMeqmmiuchap1PFN<-EMeqmmiuchap1PFN+eqmmiuchap1PFN

}

EQMMmiuchap1PF<-EQMMmiuchap1PF/G

RE<-100*(EQMmiuchap1PF-EQMMmiuchap1PF)/EQMMmiuchap1PF EMeqmmiuchap1PF<-EMeqmmiuchap1PF/G;

EMeqmmiuchap1PFN<-EMeqmmiuchap1PFN/G

RB<-100*(EMeqmmiuchap1PF-EQMMmiuchap1PF)/EQMMmiuchap1PF REN<-100*(EQMmiuchap1PFN-EQMMmiuchap1PF)/EQMmiuchap1PF RBN<-100*(EMeqmmiuchap1PFN-EQMMmiuchap1PF)/EQMMmiuchap1PF listarezultate<list(relerror=RE,relbias=RB,EQM=EQMmiuchap1PF, EQMMC=EQMMmiuchap1PF,EeqmPF=EMeqmmiuchap1PF,naifrelerror=REN, naifrelbias=RBN,EQMN=EQMmiuchap1PFN,EeqmN=EMeqmmiuchap1PFN) return(listarezultate)

}

Acknowledgement. This paper is supported by the Sectoral Operational Pro- gram for Human Resources Development (SOP HRD), financed from the Euro- pean Social Fund and by the Romanian Government under the contract number POSDRU/89/1.5/S/64109.

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Received 18 December 2008 Polytechnic University of Bucharest mastefan@gmail.com Free University of Brussels

mastefan@ulb.ac.be