Transforming response values in small area prediction

Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

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Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda

Transforming response values in small area prediction Shonosuke Sugasawa a,∗ , Tatsuya Kubokawa b a

The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa-shi, Tokyo 190-8562, Japan

b

Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

article

abstract

info

Article history: Received 15 September 2015 Received in revised form 28 March 2017 Accepted 29 March 2017 Available online xxxx Keywords: Dual power transformation Empirical Bayes estimation Fay–Herriot model Mean squared error Positive-valued data Small area estimation

In real applications of small area estimation, one often encounters data with positive response values. The use of a parametric transformation for positive response values in the Fay–Herriot model is proposed for such a case. An asymptotically unbiased small area predictor is derived and a second-order unbiased estimator of the mean squared error is established using the parametric bootstrap. Through simulation studies, a finite sample performance of the proposed predictor and the MSE estimator is investigated. The methodology is also successfully applied to Japanese survey data. © 2017 Elsevier B.V. All rights reserved.

1. Introduction

1

Sample surveys are indispensable to estimate various characteristics of a population of interest. However, reliability of estimates from sample surveys depends on sample sizes, and direct estimates from small sample surveys have large variability, which is known as a small area estimation problem. In small area estimation methodology, a model-based approach has become very popular to produce indirect and improved estimates by ‘borrowing strength’ from related areas. Importance and usefulness of the model-based small area estimation approach has been emphasized in the literature. For a recent comprehensive review of small area estimation, see Pfeffermann (2014) and Rao and Molina (2015). To describe the detailed setting, we define yi as the direct survey estimator of the area mean θi , noting yi is often unstable because of small area sample sizes. For producing a reliable estimate of θi , the most famous and basic small area model is the Fay–Herriot (FH) (Fay and Herriot, 1979) described as yi = xti β + vi + εi ,

i = 1 , . . . , m,

(1.1)

where vi ∼ N (0, A) and εi ∼ N (0, Di ) for known Di ’s, and the quantity of interest is θi = xti β + vi . It is well known that the best predictor  θi that minimizes the mean squared error is expressed as

 θi = γi yi + (1 − γ ) β, t i xi

which is a weighted combination of the direct estimator yi and the synthetic part β with weight γi = A/(A + Di ). The weight is a decreasing function of Di so that the weight on synthetic part xti β is large when yi is not reliable, that is, the

Corresponding author. E-mail addresses: [email protected] (S. Sugasawa), [email protected] (T. Kubokawa).

http://dx.doi.org/10.1016/j.csda.2017.03.017 0167-9473/© 2017 Elsevier B.V. All rights reserved.

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xti

∗

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S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

θi is obtained by sampling variance Di is large. Since it depends on unknown parameters β and A, the practical form of  plugging estimators  β and  A into  θi , namely  Ayi + Di xti  β  θi =  γi yi + (1 −  γi )xti  β= ,  A + Di which is called the empirical best linear predictor (EBLUP). Until now, the EBLUP and the related topics have been extensively studied in the framework of the Fay–Herriot model. Chatterjee et al. (2008) and Diao et al. (2014) proposed the empirical Bayes confidence intervals of θi with second-order refinement. Li and Lahiri (2010) and Yoshimori and Lahiri (2014) were concerned with the problem of estimating the variance parameter A avoiding 0 estimate. Moreover, Ghosh et al. (2008) and Sinha and Rao (2009) suggested some robust estimating methods for the Fay–Herriot model. The Fay–Herriot model and EBLUP are simple and useful methods, but the setting of the Fay–Herriot model is sometimes inadequate for analysis of real data. Therefore, several extensions of the Fay–Herriot model have been proposed. Opsomer et al. (2008) suggested a nonparametric small area model using penalized spline regression. In relation to the assumption of known Di ’s, Gonzàlez-Manteiga et al. (2010) proposed a nonparametric procedure for estimating Di , and You and Chapman (2006) and Maiti et al. (2014) considered shrinkage estimation of Di by assuming a hierarchical structure for Di . Ybarra and Lohr (2008) and Arima et al. (2015) were concerned with the problem of measurement error in covariate xi . Datta et al. (2011) and Molina et al. (2015) suggested procedures of preliminary testing for existence of the random effect vi . Datta and Mandal (2015) proposed a mixture of two models; one includes a random effect and the other does not include a random effect. Although all these papers treat important problems, the response values of the data are assumed to be normally distributed. However, we often encounter positive-valued data (e.g. income, expense), which have skewed distributions and nonlinear relationships with covariates. For such a data set, the traditional Fay–Herriot model with a linear structure between response values and covariates and a normally distributed error term is not appropriate. A typical alternative approach is using the log-transformed response values as discussed in Slud and Maiti (2006), but the log-transformation is not always appropriate and it may produce inefficient and biased prediction when the log-transformation is misspecified. Thus, a natural way to solve this problem is using a parametric family of transformations which enables us to select a reasonable transformation based on data. A famous family is the Box–Cox transformation (Box and Cox, 1964) defined as hλ (x) = BC

 −1 λ λ (x − 1) log x

However, it suffers from a truncation problem that the range of the Box–Cox transformation is not the whole real line if λ ̸= 0, which leads to inconsistency of the maximum likelihood estimator of λ. Moreover, the inverse transformation cannot be defined on whole real line, so that we cannot define a back-transformed predictor in the original scale. Alternatively, Yang (2006) suggested a novel family of transformations called the dual power transformation (DPT): hλ (x) =



(2λ)−1 (xλ − x−λ ) log x

41 42 43 44 45 46 47 48

λ>0 λ = 0,

BC which can be seen as the average of two Box–Cox transformations, namely hλ (x) = {hBC λ (x)+ h−λ (x)}/2. The main advantage of the DPT is that its range is the whole real line for all λ ≥ 0 so that DPT does not suffer from the truncation problem. Sugasawa and Kubokawa (2015) proposed the Fay–Herriot model in which response variables are transformed by general parametric transformations. In this paper, we focus on the FH model with DPT transformation described as

hλ (yi ) = xti β + vi + εi ,

i = 1, . . . , m,

(1.2)

where vi ∼ N (0, A) and εi ∼ N (0, Di ) for known Di ’s. Although Sugasawa and Kubokawa (2015) derived EBLUP of θi = xti β + vi and the MSE estimator, the parameter of most 1 −1 interest in the model (1.2) is µi = h− λ (θi ) rather than θi , where hλ (·) is the inverse transformation of DPT:



40

λ ̸= 0 λ = 0.

1 h− λ (x) = λx +

1/λ  λ2 x2 + 1 .

Thus, the method developed in Sugasawa and Kubokawa (2015) is not enough for practical applications. In this paper, we focus on the prediction of µi with its risk evaluation. Specifically, we derive the best predictor of µi as the conditional expectation and the empirical best predictor by plugging the parameter estimates in the best predictor. For risk evaluation, we construct a second order unbiased MSE estimator based on the parametric bootstrap. The paper is organized as follows. In Section 2, we derive the best predictors of µi as well as the maximum likelihood estimation of model parameters. A second-order unbiased estimator of the mean squared error of the small area predictor is derived based on the parametric bootstrap. In Sections 3 and 4, we show some simulation studies and empirical applications, respectively. In Section 5, we give some concluding remarks. The technical details are given in the Appendix.

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

3

2. Transformed Fay–Herriot model

1

2.1. Model setup and best predictor

2

We consider the following parametric transformed Fay–Herriot (PTFH) model for area-level data: hλ (yi ) = xti β + vi + εi ,

3

i = 1, . . . , m

(2.1)

where vi ∼ N (0, A), εi ∼ N (0, Di ) for known Di ’s, β and xi are p-dimensional vectors of regression coefficients and covariates, respectively. The unknown models parameters are denoted by φ = (βt , A, λ)t and we aim to estimate (predict) 1 t µi = h − λ (θi ) with θi = xi β + vi . Note that when λ = 0, the model (2.1) reduces to the log-transformed Fay–Herriot model studied by Slud and Maiti (2006). It is well known that the best predictor of θi under the squared error loss is given by

 θi = γi hλ (yi ) + (1 − γi )xti β,

(2.2) (S )

1  = h− λ (θi ). However,  µi is not suitable for predicting µi , because  µi has a non-ignorable bias for predicting µi , namely E[ µi −µi ] ̸= 0 even when m is large. On the other hand, Slud and Maiti (2006) considered the bias corrected predictor of F (θi ) for a general function F (·), which leads to the following form:

where γi = A/(A + Di ). Hence, one possible way to predict µi is using the simple back-transformed predictor  µi (S )

 µ(i SM ) =

(S )

1 E[ h− λ (θi )] hλ ( θi ) 1  E[ h− λ (θi )]

E[µi ]

=

(S )

E[ µi ]

(S )

 µ(i S ) .

is an unbiased predictor of µi while it does not necessarily minimize − µi ] = 0, that is,  µi It clearly holds that E[ µi 1 the squared error loss. We here use the conditional expectation  µi = E[h− λ (θi )|yi ] with known φ as a predictor of µi , which 2   minimizes the squared error loss. Since θi |yi ∼ N (θi , σi ) with θi given in (2.2) and σi2 = ADi /(A + Di ) under the model (2.1), the conditional expectation  µi can be expressed as

 µi ≡  µi (yi ; φ) =



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10

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15

(SM )

(SM )

4

16 17 18 19

∞

−∞

1  2 h− λ (t )φ(t ; θi , σi )dt ,

(2.3) (SM )

where φ(·; a, b) denotes the density function of N (a, b). It should be noted that  µi =  µi

when λ = 0, namely are not necessarily identical when λ > 0. = exp(x). However,  µi and  µi Since the model parameters φ is unknown in practice, we estimate them by maximizing the marginal likelihood function, and the details are given in the next section. Let  φ be the corresponding estimator of φ. Then, replacing φ with  φ in (2.3) leads to the empirical form of  µi :  ∞ −1  µi ≡  µ(yi ;  φ) = h (t )φ(t ;  θi ,  σi2 )dt , λ (SM )

1 h− λ (x)

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26

−∞

which is known as empirical best predictor (EBP). Note that  µi is no longer the conditional expectation but  µi converges to  µi as m → ∞ under some regularity conditions. Since  µi cannot be obtained in an analytical form, we rely on numerical techniques for computing  µi . A typical method is the Monte Carlo integration by generating a large number of random samples from ( θi ,  σi2 ). However, we here use Gaussian–Hermite quadrature which is known to be more accurate than the

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Monte Carlo integration.

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2.2. Estimation of model parameters

32

Under normality assumption of vi and εi , it follows that hλ (yi ) ∼ N (xti β, A + Di ) and hλ (yi ), i = 1, . . . , m are mutually independent. Then, the maximum likelihood estimator  φ of φ is defined as the maximizer of L(φ), where L(φ) = −

m  i=1

m  hλ (yi ) − xti β A + Di i =1



log(A + Di ) −

2 +2

m 

log yiλ−1 + yi−λ−1 .





(2.4)

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i=1

Note that the third term in (2.4) comes from the Jacobian of the transformation. When λ is given, maximizing (2.4) with respect to β and A coincides to maximizing the log-likelihood function of the classical Fay–Herriot model. Hence, the value of profile likelihood function of λ is easily computed, so that we may estimate λ by grid search over a specified region or golden section method (Brent, 1973). Though the parameter space of λ is [0, ∞), it would be sufficient to consider the space [0, λm ] for moderately large λm .

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S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

For asymptotic properties of the estimator  φ, we assume the following conditions. Assumption 2.1.  1. There exist D and D independent to m such that D ≤ Di ≤ D for i = 1, . . . , m. m 2. maxi=1,...,m xti ( j=1 xj xtj )−1 xi = O(m−1 ). These conditions are usually assumed in the context of small area estimation, see Datta and Lahiri (2000) and Butar and Lahiri (2003). Under these conditions, we have the following lemma.

√

6 7

Lemma 2.1. Under Assumption 2.1, as m → ∞, m( φ − φ) asymptotically follows the multivariate normal distribution N (0, V (φ)) with a covariance matrix V (φ), and it holds E[ φ − φ] = m−1 b(φ) + o(m−1 ) with a smooth function b(φ).

14

The asymptotic normality of  φ immediately follows from Sugasawa and Kubokawa (2015). Moreover, from the proof of Theorem 1 in Lohr and Rao (2009), the bias b(φ) can be expressed by partial derivatives of L(φ) given in (2.4), so that the latter statement in Lemma 2.1 follows. Other estimators of A are the restricted maximum likelihood estimator (Jiang, 1996), the Prasad–Rao estimator (Prasad and Rao, 1990), the Fay–Herriot estimator (Fay and Herriot, 1979) and the adjusted maximum likelihood estimator (Li and Lahiri, 2010). These methods can be easily implemented and their asymptotic properties are discussed in Sugasawa and Kubokawa (2015). However, for simplicity, we do not treat these estimators in this paper.

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2.3. Mean squared error of the empirical best predictor

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In small area estimation, mean squared errors (MSEs) of small area estimators are used for risk evaluation, and their importance has been addressed in many papers including Lahiri and Rao (1995) and Datta et al. (2005). Following this convention, we evaluate the MSE of the empirical best predictor  µi . To begin with, we note that the MSE can be decomposed as MSEi ≡ E[( µi − µi )2 ] = E[( µi − µi )2 ] + E[( µi −  µi ) 2 ]

≡ g1i (φ) + g2i (φ),

16 17

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22

because  µi = E [µi |yi ] is the conditional expectation. In what follows, we use the explicit notation  µi (yi , φ) instead of  µi if necessary. The first term g1i (φ) is expressed as g1i (φ) = E

 2  1 t  µi (xti β + vi + εi , φ) − h− ( x β + v ) , i i λ

which has no analytical expression. The direct Monte Carlo integration by generating random samples of vi and εi requires a large computational burden because we need another Monte Carlo integration for computing  µi for each sample (vi , εi ). However, as shown in the Appendix, it turns out to have the following more simple expression of g1i (φ): 1 t −1 t −1 t 2 g1i (φ) = E {h− λ (xi β + z1 )} − hλ (xi β + c1i z1 + c2i z2 )hλ (xi β + c1i z1 − c2i z2 ) ,





(2.5)

24

√ where z1 , z2 ∼ N (0, A), c1i = (1 + ai )/2, and c2i = (1 − ai )/2 for ai = A/(A + Di ). Hence, g1i (φ) can be easily calculated by generating a large number of random samples of z1 and z2 . On the other hand, the second term g2i (φ) can be evaluated

25

as the following lemma, where the proof is given in the Appendix.

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Lemma 2.2. Under Assumption 2.1, it holds

√

23

27

g2i (φ) =

1 m



tr V (φ)E



∂ µi ∂ µi ∂φ ∂φt



+ o(m−1 ).

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Since the MSE depends on unknown parameter φ, we need to estimate it for practical use. To this end, we obtain a second-order unbiased estimator of the MSE. Here, an estimator  B is called second order unbiased if E[ B] = B + o(m−1 ). −1 −1 From Lemma 2.2, it follows that g2i (φ) = m c1 (φ) + o(m ) with the smooth function c1 (φ), thereby the plug-in estimator g2i ( φ) is second-order unbiased. However, the plug-in estimator g1i ( φ) has a second-order bias since g1i (φ) = O(1), so that we need to correct the bias. Hence, we propose the parametric bootstrap method to correct the bias of g1i ( φ) and computing g2i ( φ). The procedure is given in the following.

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Parametric bootstrap method for the MSE estimation

35

1. Generate bootstrap samples y∗i from the estimated model;

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36

37

∗ t ∗ ∗ h λ (yi ) = xi β + vi + εi ,

i = 1 , . . . , m,

where εi and vi are generated from N (0, Di ) and N (0,  A), respectively ∗

∗

(2.6)

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

5

∗

2. Based on (y∗i , xi ), i = 1, . . . , m, compute the maximum likelihood estimate  φ and the predicted values of  µi =  µi (yi ,  φ)

1

∗

and  µ∗i =  µi (yi ,  φ ). 3. Derive the bootstrap estimates of g1i and g2i via



∗

2

 2  ∗  µ∗i −  µi g2i (φ) = E ∗ 



bc  φ ) , g1i (φ) = 2g1i ( φ) − E ∗ g1i (

1 t where  µ∗i = h− (xi β + vi∗ ) and E ∗ [·] denotes the expectation with respect to the bootstrap samples generated from λ (2.6). The second-order unbiased estimator of the MSE based on the parametric bootstrap is given by

∗  i = g bc ( MSE 1i φ) + g2i (φ).

(2.7)

The resulting MSE estimator (2.7) is second-order unbiased as shown in the following theorem, which is proved in the Appendix.

i be the parametric bootstrap MSE estimator given in (2.7). Then, under Assumption 2.1, we have Theorem 2.1. Let MSE



3 4

5

6 7

8



i = MSEi + o(m−1 ), E MSE

9

where the expectation is taken with respect to yi ’s following the model (2.1).

10

∗

In (2.7), the bias correction of g1i ( φ) is carried out via using the additive form g1ibc ( φ) = 2g1i ( φ) − E ∗ [g1i ( φ )], where E ∗ denotes the expectation with respect to bootstrap samples. Hall and Maiti (2006) suggested other bias-correcting methods ∗ including a multiplicative bias correcting method of the form g1i ( φ)2 /E ∗ [g1i ( φ )]. The multiplicative form for bias correction can avoid negative estimates of the MSE while the additive form for bias correction gives negative estimates of the MSE with a positive probability. Although those bias corrections give second-order unbiased estimates of g1i (φ), in this paper, we use the additive-type bias correction, because it has been frequently used in the literatures (e.g. Butar and Lahiri, 2003).

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3. Simulation studies

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3.1. Evaluation of prediction errors

18

We evaluated prediction errors of the proposed PTFH model and some existing models. As a data generating process, we considered the following PTFH model: hλ (yi ) = β0 + β1 xi + vi + εi ,

i = 1, . . . , 30,

(3.1)

where vi ∼ N (0, A), εi ∼ N (0, Di ) with β0 = 1, β1 = 1 and A = 1.5. For λ, we treated the four cases λ = 0.1, 0.4, 0.7 and 1.0. The covariates xi were initially generated from the uniform distribution on (0, 4) and fixed in simulation runs. Concerning sampling variance Di , we divided 30 areas into 5 groups (from G1 to G5 ), and areas within the same group have the same Di value. The Di -pattern we considered was (0.2, 0.4, 0.6, 0.8, 1.0). The true small area parameters are µi = hλ−1 (β0 + β1 xi + vi ). For comparison, we considered the log-transformed FH (log-FH) model and the traditional Fay–Herriot (FH) model, which are described as

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log-FH: log yi = β0 + β1 xi + vi + εi FH: yi = β0 + β1 xi + vi + εi . It is noted that the data generating process (3.1) get close to log-FH as λ gets smaller. Since we do not known the true Di in practice, we computed the estimates of µi with estimated Di as investigated in Bell (2008). To this end, we generated the auxiliary observation zik from the model: hλ (zik ) = εik ,

i = 1, . . . , 30, k = 1, . . . , 10,

(3.2)

where εij ∼ N (0, Di ). In applying log-FH and FH, we computed the estimates of Di as the sampling variances of {log zi1 , . . . , log zi10 } and {zi1 , . . . , zi10 }, respectively. Then we computed the estimates of µi using EBLUP in FH and the bias-corrected estimator used in Slud and Maiti (2006) in log-FH, where the model parameters β0 , β1 and A are estimated via the maximum likelihood method. For fitting PTFH, we first define Di ≡ Di (λ) =

10 1 

9 k=1

hλ (zik ) − hλ (z )i

2

,

hλ ( z ) i =

10 1 

10 k=1

hλ (zik ),

so that we regard Di as a function of λ and replace Di with Di (λ) in (2.4). Since Di (λ) can be immediately computed under the given λ, we can maximize the profile likelihood function of λ in the similar manner to that presented in Section 2.2. Once the estimate  λ is computed, Di can be calculated as Di ( λ).

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Table 1 Simulated percent coefficient of variation (CV) and absolute relative biases (ARB) of the parametric transformed Fay–Herriot with use of true Di (PTFH-t) and estimated Di (PTFH), the log-transformed Fay–Herriot (log-FH) model, and the Fay–Herriot (FH) model under λ = 0.1, 0.4, 0.7 and 1.0. Method

1 2

3

CV

ARB

0.1

0.4

0.7

1.0

0.1

0.4

0.7

1.0

G1

PTFH-t PTFH log-FH FH

46.49 47.46 47.11 50.24

33.28 32.90 35.34 32.94

23.49 23.12 28.63 22.84

17.95 17.53 23.57 16.87

13.67 13.97 13.16 9.74

6.89 6.89 4.18 3.82

3.69 3.51 3.85 1.78

2.43 2.21 2.39 1.13

G2

PTFH-t PTFH log-FH FH

63.92 66.32 66.79 82.25

47.64 47.84 53.25 52.67

37.09 36.92 46.70 38.61

31.61 31.38 41.76 31.11

20.20 21.28 22.25 19.95

11.85 12.24 15.14 9.16

8.13 8.31 13.96 6.27

6.69 6.74 13.86 5.98

G3

PTFH-t PTFH log-FH FH

75.92 79.86 77.70 116.09

59.42 60.35 60.99 74.04

48.51 48.22 53.25 53.70

40.60 40.35 47.15 40.88

25.67 27.67 26.56 30.55

16.79 17.74 15.97 17.10

12.85 13.15 13.04 11.77

9.59 9.67 11.30 9.25

G4

PTFH-t PTFH log-FH FH

86.92 92.97 86.81 156.12

67.82 69.13 65.36 91.68

53.66 53.38 53.46 61.04

44.73 43.83 46.36 45.61

32.29 35.12 30.88 45.29

20.84 22.13 14.42 25.10

14.11 14.43 8.29 15.15

10.82 10.63 8.94 11.25

G5

PTFH-t PTFH log-FH FH

92.90 101.81 95.91 198.30

72.74 75.26 71.73 112.23

59.86 60.39 61.69 73.57

49.86 49.54 53.58 53.02

33.87 37.62 34.91 57.58

22.99 24.73 20.57 31.04

17.30 18.07 15.30 19.05

13.04 13.24 12.87 13.93

Based on R = 10,000 simulation runs, we computed the coefficient of variation (CV) and the absolute relative bias (ARB), defined as

  R  (r ) 2 1   µi − µ(i r )  CVi = R r =1 µ(i r )2

R 1   µ(i r ) − µ(i r )  ARBi =  , R r =1 µ(i r )



and



(r )

18

µ(r ) is the estimated value from PTFH, log-FH or FH, in the rth iteration. Table 1 shows the where µi is the true value and  percent CV and ARB averaged within the same groups for each case of λ. For comparison, we also show the results for PTFH with true Di values, denoted by PTFH-t in Table 1. From Table 1, we can observe that difference CV or ARB between PTFH-t and PTFH tends to be large when λ is small and Di is large while two methods perform similarly when λ is large or Di is small. Moreover, it is revealed that the larger Di would lead to larger CV and ARB values in all the methods. Concerning comparison among PTFH, log-FH and FH, it can be seen that PTFH performs better than FH except for λ = 0.9, and PTFH performs better than log-FH except for λ = 0.1. Moreover, the differences between PTFH and log-FH in λ = 0.1 get larger as Di gets larger. Regarding PTFH-t, it performs best in most cases. However, it is observed that log-FH and FH produce more accurate estimates than PTFH-t in some cases. We next investigated the prediction errors when the true distribution of vi is not normal. Here, we considered a t-distribution with 5 degrees of freedom for vi , where the variance is scaled to A, and the other settings for the data generation are the same as (3.1). Under the scenario, we again computed the values of CV and ARB of the four methods based on 10,000 simulation runs, and the results are reported in Table 2. It is observed that the simulated RMSE values in Table 2 are larger than those in Table 1 due to misspecification of the distribution of vi . However, relationships of CV and ARB among three methods are similar to Table 1.

19

3.2. Finite sample performance of the MSE estimator

4 5 6 7 8 9 10 11 12 13 14 15 16 17

20 21 22 23 24 25 26 27

28

We next investigated a finite sample performance of the MSE estimator (2.7). Following Datta et al. (2005), we considered the following data generating process without covariates: hλ (yi ) = µ + vi + εi ,

i = 1, . . . , 30,

with µ = 0, vi ∼ N (0, A) with A = 1 and εi ∼ N (0, Di ). As a value of λ, we considered the three cases λ = 0.2, 0.6, 1.0. For setting of Di , we divided Di ’s into five groups G1 , . . . , G5 , where Di ’s were the same values over the same group, and the following three patterns of Di ’s were considered:

(a) 0.3, 0.4, 0.5, 0.6, 0.7,

(b) 0.2, 0.4, 0.5, 0.6, 2.0,

(c) 0.1, 0.4, 0.5, 0.6, 4.0.

Based on R1 = 5000 simulation runs, we calculated the simulated values of the MSE as MSEi =

R1 1 

R1 r =1

( µ(i r ) − µ(i r ) )2 ,

(r ) 1 µ(i r ) = h− λ (β0 + vi )

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

7

Table 2 Simulated percentage coefficient of variation (CV) and percentage absolute relative biases (ARB) of the parametric transformed Fay–Herriot with use of true Di (PTFH-t) and estimated Di (PTFH), the log-transformed Fay–Herriot (log-FH) model, and the Fay–Herriot (FH) model under λ = 0.1, 0.4, 0.7 and 1.0, when the distribution of vi is a t-distribution with 5 degrees of freedom. Method

CV

ARB 0.4

0.7

1.0

G1

PTFH-t PTFH log-FH FH

47.19 48.42 48.09 51.54

39.58 39.84 41.08 41.09

34.16 34.40 39.98 33.00

28.94 28.34 35.31 27.84

14.10 14.70 14.00 10.26

9.77 10.07 8.14 6.92

7.31 7.38 4.95 5.18

5.43 5.18 5.65 4.28

G2

PTFH-t PTFH log-FH FH

66.06 68.38 67.79 83.45

53.12 53.52 56.17 58.51

41.12 40.86 46.04 41.73

35.83 35.18 43.20 34.32

21.41 22.59 21.84 21.03

14.08 14.78 13.32 12.63

8.99 9.10 7.32 7.67

6.92 6.80 6.73 6.18

G3

PTFH-t PTFH log-FH FH

79.92 83.75 83.53 121.91

62.21 62.26 65.45 73.92

49.91 47.41 55.47 50.11

42.61 41.62 52.08 41.05

25.50 27.39 27.40 31.49

16.81 17.60 18.20 17.11

10.91 11.05 12.93 10.35

8.69 8.66 13.22 8.25

G4

PTFH-t PTFH log-FH FH

89.82 98.67 99.13 155.72

78.86 75.36 81.96 108.51

61.81 59.27 62.46 65.22

53.21 51.18 57.54 52.10

30.06 32.81 31.16 44.14

21.02 22.26 19.98 25.07

15.63 15.83 13.48 15.96

12.03 12.01 11.01 12.80

G5

PTFH-t PTFH log-FH FH

104.87 123.97 120.17 224.09

86.53 86.69 87.54 128.60

77.05 74.22 75.41 87.85

71.23 64.63 67.96 69.25

34.72 38.87 34.96 61.85

26.73 28.84 22.70 40.85

22.00 22.83 15.61 29.11

17.04 17.06 11.36 21.47

(r )

0.1

0.4

0.7

1.0

0.1

(r )

where  µi and vi are the predicted value and the realized value of vi in the rth iteration. Then based on R2 = 2000 simulation runs, we calculated the relative bias (RB) and the coefficient of variation (CV) defined as RBi =

CV2i =

R2  1 

R2 r =1

i MSE

R2  1 

R 2 r =1

(r )

i MSE

(r )

 − MSEi /MSEi , − MSEi

2

/MSE2i .

For calculation of the MSE estimates in each iteration, we used 100 bootstrap replication for the MSE estimator and 10,000 Monte Carlo samples for computing g1i . We also investigated the performance of the MSE estimator when we used the estimated sampling variances instead of known Di . To this end, similarly to the previous section, we generated the auxiliary observation from (3.2), and calculate Di ’s using these data in each simulation run. Based on the same number of simulation runs, we computed the values of RB and CV. Tables 3 and 4 show the maximum, mean and minimum values of RB and CV within the same group. In both tables, the simulated values of RB and CV of the MSE estimator with estimated Di are given in the parenthesis. It is seen that the proposed MSE estimator with known Di provides reasonable estimated values in almost all cases in terms of both RB and CV. On the other hand, the MSE estimator with estimated Di performs worse than the MSE estimator with known Di since the former estimator is affected by the variability of estimating Di . Moreover, it is observed that performances of both MSE estimators get better in the order of Pattern (a), (b) and (c).

10

4. Application to survey data in Japan

11

We consider an application of the proposed method together with some existing methods to the data from the Survey of Family Income and Expenditure (SFIE) in Japan. Especially, we used the data on the spending item ‘Health’ and ‘Education’ in the survey in 2014. For the spending item ‘Health’ and ‘Education’, the annual average spending data at each capital city of 47 prefectures are available. The estimates are both unreliable since the sample sizes are around 50 for most prefectures. As a covariate, we used data from the National Survey of Family Income and Expenditure (NSFIE) for 47 prefectures. Since NSFIE is based on much larger sample than SFIE, the reported values are more reliable, but this survey has been implemented every five years. Although the joint bivariate modeling of the two items ‘Health’ and ‘Education’ would be preferable as proposed in Benavent and Morales (2016), we here consider applying univariate models separately to each item for simplicity. In what follows, yi and xi denote the direct estimate (scaled by 1000) from SFIE and the covariate (reliable estimate) from NSFIE, respectively, on the item ‘Health’ or ‘Education’. For each survey data, we applied the PTFH model: hλ (yi ) = β0 + β1 log xi + vi + εi ,

i = 1, . . . , 47,

where vi ∼ N (0, A), εi ∼ N (0, Di ) and β0 , β1 , A and λ are model parameters. For comparison, we also applied the log-FH model corresponding λ = 0 in the above model, and the classical Fay–Herriot model. The model parameters were estimated

1 2 3 4 5 6 7 8 9

12 13 14 15 16 17 18 19 20 21 22 23 24

8

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

Table 3 The percentage RB values of the MSE estimator with known Di and unknown Di (Parenthesis) in each group.

λ max

Pattern (a)

G1 G2 G3 G4 G5

Mean

G1 G2 G3 G4 G5

min

G1 G2 G3 G4 G5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20

22 23 24 25 26

Pattern (c)

0.6

1.0

0.2

0.6

1.0

0.2

0.6

1.0

6.3 (28.1) 4.9 (44.6) 12.5 (32.9) 6.4 (33.8) 8.7 (15.9)

17.9 (44.3) 15.1 (69.2) 10.4 (40.8) 12.6 (28.3) 12.4 (39.1)

19.5 (51.4) 17.2 (78.8) 14.6 (47.0) 17.6 (40.2) 16.3 (52.1)

3.9 (98.4) 4.4 (46.4) −2.8 (24.2) −2.6 (12.1) −1.5 (6.2)

7.3 (80.8) 4.4 (107.2) 3.3 (12.5) 6.7 (8.8) 2.8 (43.9)

9.0 (82.6) 5.3 (99.8) 4.6 (10.6) 8.9 (8.0) 6.1 (43.6)

7.2 (113.5) 4.1 (37.3) −3.4 (11.3) −5.0 (16.1) −5.3 (4.6)

1.3 (47.1) −3.6 (29.8) −4.1 (37.5) −4.5 (84.4) −4.2 (26.0)

2.7 (44.1) −4.9 (20.8) −5.0 (29.7) −4.8 (75.7) −1.6 (19.5)

2.5 (22.4) −1.0 (21.9) 5.2 (17.7) 3.3 (20.4) 1.4 (11.1)

7.4 (31.4) 7.8 (33.4) 1.9 (21.1) 6.4 (17.7) 5.7 (22.1)

8.6 (36.9) 9.8 (39.9) 5.2 (28.7) 9.9 (26.9) 9.9 (32.8)

−1.9

1.4 (32.3) −2.0 (21.5) −1.7 (3.6) −0.4 (−2.8) −0.2 (6.4)

2.6 (33.9) −2.0 (18.8) −0.4 (2.4) 1.5 (−2.2) 2.4 (7.2)

−2.7 (47.6) −6.0 (20.2) −6.1 (−0.1) −7.4 (−5.3) −7.0 (−6.8)

−3.3

−3.1

−4.9

−3.2

−2.4

−6.9

(20.2) 2.8 (18.6) −6.7 (19.4) −0.2 (14.1) −2.4 (8.8)

(13.7) −7.5 (2.7) −8.7 (−5.0) −11.9 (−7.0) −7.9 (−7.3)

−4.5 (17.3) −7.5 (−4.7) −6.5 (−4.7) −3.9 (−9.5) −2.8 (−9.7)

−8.5 (21.5) −9.6 (8.1) −10.3 (−7.3) −9.1 (−12.6) −8.5 (−16.1)

−6.6

(15.0) −0.9 (14.2) −8.2 (13.2) −3.0 (6.9) −6.4 (−0.6)

−5.2 (16.1) −7.0 (−2.7) −6.6 (−4.4) −5.8 (−11.6) −5.4 (−10.4)

−6.9

(17.2) −5.8 (11.5) 0.9 (7.0) −1.9 (10.0) −3.4 (6.4)

(17.1) −7.8 (−5.2) −10.4 (−9.4) −11.2 (−16.1) −9.3 (−18.4)

−9.9 (−8.8) −11.3 (−13.2) −11.9 (−18.2) −9.8 (−20.0)

(33.5) −3.4 (15.5) −6.3 (11.2) −8.0 (2.2) −6.0 (−3.0)

(27.7) −5.9 (4.8) −8.4 (10.4) −7.5 (15.6) −6.8 (−1.2)

(27.0)

−7.3 (−0.6) −9.1 (4.6)

−7.9 (10.9)

−6.4 (−4.4) (17.0)

by the maximum likelihood method in all models. For computing Di in each model, we used the past data for consecutive eight years from 2006 to 2013, which are denoted by zit for t = 1, . . . , 8. In the FH and log-FH models, we simply calculated the sampling variance of {zi1 , . . . , zi8 } and {log zi1 , . . . , log zi8 }, respectively. In the PTFH model, similarly to Section 3.1, we first maximize (2.4) with Di = Di (λ) and let Di = Di ( λ). In the PTFH model, we have  λ = 0.59 in ‘‘Education’’ and  λ = 0.86 in ‘‘Health’’. Moreover, based on 1000 parametric bootstrap samples, we obtained 95% confidence intervals of λ, (0.20, 1.16) in ‘‘Education’’ and (0.18, 1.99) in ‘‘Health’’, which indicate the log-transformation might not be appropriate. In Fig. 1, we present the estimated regression lines of the −1  three models, noting that y = h (β0 +  β1 x) in PTFH, and y = exp( β0 +  β1 x) in log-FH. From the figure, it is observed λ that all the regression lines are similar. For assessing the suitability of normality assumptions of error terms, we computed 

  the standardized residuals: ei = ri / Aˆ + Di , where ri is the estimates of vi + εi , so that ri = h λ (yi ) − β0 − β1 xi in PTFH,

ri = log yi −  β0 −  β1 xi in log-FH and ri = yi −  β0 −  β1 xi in FH, noting that ei asymptotically follows the standard normal distribution if the model specification is correct. In Fig. 2, we give the estimated density of ei in each model, which does not strongly supports the normality assumption of three models, but all the estimated densities are close to symmetric. Hence, the normality assumption might be plausible. In fact, we calculated the p-value of the Kolmogorov–Smirnov test for normality of ei , presented in Table 5, and found that the normality assumption was not rejected in the three models in both items. Moreover, in Table 5, we provide AICs based on the maximum marginal likelihood for the three models. It can be seen that AICs of PTFH and log-FH are similar and smaller that of FH in ‘‘Education’’ while that of PTFH is the smallest in ‘‘Health’’. For investigation of goodness-of-fit of the PTFH model, we set zi = h λ (yi ) and wi = log xi , and applied the penalized spline model used in Opsomer et al. (2008): zi = β0 +

p 

ℓ=1 21

Pattern (b)

0.2

βℓ wiℓ +

K 

γℓ (wi − κℓ )p+ + vi + εi ,

i = 1, . . . , m,

(4.1)

ℓ=1 p

where vi ∼ (0, A), εi ∼ N (0, Di ) and (γ1 , . . . , γK )t ∼ N (0, α IK ), (x)+ denotes the function xp I (x > 0), and κ1 < · · · < κK is a set of fixed knots which determine the flexibility of splines. We set K = 20 and took κ1 and κK as 10% and 90% quantiles of wi , respectively, and set κ2 , . . . , κK −1 at regular interval. For the degree of splines, we considered three cases: p = 1, 2, 3. We estimated model parameters β0 , . . . , βp , A and α by the maximum likelihood method. In Fig. 3, we present the estimated regression lines of three penalized spline models (p = 1, 2, 3) as well as that of PTFH, which shows that the linear parametric structure in the PTFH model seems plausible and PTFH would fit well in both items.

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

9

Table 4 The CV values of the MSE estimator with known Di and unknown Di (Parenthesis) in each group.

λ max

G1 G2 G3 G4 G5

Mean

G1 G2 G3 G4 G5

min

G1 G2 G3 G4 G5

Pattern (a)

Pattern (b)

Pattern (c)

0.2

0.6

1.0

0.2

0.6

1.0

0.2

0.6

1.0

1.15 (2.72) 1.17 (2.25) 2.25 (1.34) 1.02 (1.44) 0.72 (1.10)

1.03 (1.65) 1.33 (2.50) 0.98 (1.80) 1.26 (1.40) 1.01 (1.59)

1.14 (1.96) 1.50 (3.42) 1.16 (2.16) 1.54 (2.08) 1.23 (2.21)

0.78 (2.13) 0.55 (1.20) 0.59 (0.88) 0.46 (0.70) 0.67 (0.91)

0.60 (1.78) 0.73 (2.44) 0.87 (1.02) 0.84 (0.94) 0.73 (1.60)

0.66 (1.89) 0.71 (2.49) 1.00 (1.1) 0.98 (1.06) 0.88 (1.87)

0.77 (2.01) 0.54 (0.91) 0.52 (0.68) 0.51 (0.66) 0.47 (0.56)

0.62 (1.19) 0.50 (0.84) 0.46 (0.98) 0.50 (1.72) 0.66 (0.90)

0.67 (1.29) 0.54 (0.82) 0.53 (0.97) 0.57 (1.70) 0.77 (0.91)

0.99 (1.05) 0.78 (1.23) 1.02 (1.08) 0.75 (1.10) 0.71 (0.99)

0.74 (1.33) 0.91 (1.46) 0.86 (1.27) 0.93 (1.24) 0.92 (1.36)

0.82 (1.57) 1.06 (1.83) 1.03 (1.60) 1.13 (1.65) 1.12 (1.83)

0.51 (1.07) 0.46 (0.80) 0.46 (0.76) 0.43 (0.64) 0.52 (0.70)

0.50 (1.07) 0.56 (1.05) 0.61 (0.81) 0.66 (0.80) 0.66 (0.94)

0.55 (1.15) 0.63 (1.11) 0.71 (0.88) 0.78 (0.89) 0.77 (1.05)

0.45 (1.10) 0.40 (0.73) 0.42 (0.60) 0.41 (0.56) 0.41 (0.53)

0.41 (0.86) 0.42 (0.66) 0.41 (0.70) 0.44 (0.85) 0.49 (0.69)

0.43 (0.90) 0.47 (0.68) 0.45 (0.72) 0.50 (0.87) 0.56 (0.72)

0.61 (0.95) 0.57 (0.89) 0.63 (0.95) 0.66 (0.97) 0.68 (0.92)

0.58 (0.93) 0.73 (1.10) 0.79 (1.11) 0.81 (1.07) 0.80 (1.06)

0.65 (1.12) 0.86 (1.28) 0.88 (1.35) 0.99 (1.33) 1.00 (1.40)

0.37 (0.69) 0.39 (0.64) 0.40 (0.61) 0.41 (0.61) 0.43 (0.59)

0.41 (0.86) 0.49 (0.75) 0.52 (0.72) 0.59 (0.69) 0.58 (0.74)

0.45 (0.91) 0.56 (0.80) 0.60 (0.80) 0.69 (0.76) 0.69 (0.83)

0.35 (0.71) 0.34 (0.57) 0.37 (0.54) 0.37 (0.49) 0.38 (0.50)

0.35 (0.73) 0.39 (0.58) 0.39 (0.55) 0.42 (0.55) 0.42 (0.55)

0.36 (0.76) 0.41 (0.59) 0.43 (0.58) 0.45 (0.59) 0.47 (0.59)

Table 5 AIC and p-value of Kolmogorov–Smirnov (KS) test for normality of standardized residuals. Data

Education Health

AIC

p-value of KS test

PTFH

log-FH

FH

PTFH

log-FH

FH

313.1 172.9

312.9 180.7

314.5 183.4

0.577 0.519

0.469 0.440

0.848 0.375

Finally, we computed the MSE estimates of the small area estimators for the three models. In the PTFH model, we used the estimator given in Theorem 2.1 with 1000 bootstrap samples and 5000 Monte Carlo samples of vi and εi for numerical evaluation of g1i . For the MSE estimates in the log-FH and FH models, we used the estimator given in Slud and Maiti (2006) and Datta and Lahiri (2000), respectively. We report the small area estimates and MSE estimates in seven prefectures around Tokyo in Table 6. It can be seen that log-FH and FH produce relatively similar estimates of area means while the estimates from PTFH are not similar to those models. Regarding MSE estimates, we can observe that the values in PTFH are smaller than the other two models, but we cannot directly compare these results since each MSE estimates are calculated based on the different sampling variances Di . 5. Concluding remarks We proposed the use of the dual power transformation in the classical Fay–Herriot model for estimating positive valued small area means, which we call parametric transformed Fay–Herriot (PTFH) model. We derived the best predictor (BP) and obtained empirical best predictor by replacing unknown model parameters in BP with their maximum likelihood estimator. For risk evaluation of the predictor, we derived the second unbiased estimator of mean squared errors of the predictor based on the parametric bootstrap. In the simulation study, we have numerically evaluated the prediction errors of the proposed method together with some existing methods. In empirical study, we applied the method to the Japanese survey data. For checking goodness-of-fit of PTFH, we suggested fitting the penalized spline model for the transformed data, as presented in the application. While we did not consider estimating the penalized spline and the transformation parameter simultaneously, estimation of PTFH model with penalized spline could be useful. However, the detailed discussion seems exceed the scope of this paper, and is left to a valuable future study. Concerning the calculation of the sampling variance Di , we only considered the case where some auxiliary observations are available as considered in Sections 3 and 4. However,

1 2 3 4 5 6 7 8

9

10 11 12 13 14 15 16 17 18 19 20

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S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

Fig. 1. The scatter plots of (xi , yi ) with estimated regression lines in the parametric transformed Fay–Herriot (PTFH) model, the log-transformed Fay–Herriot (log-FH) model and the classical Fay–Herriot (FH) model.

Fig. 2. The estimated density of standardized residuals in the parametric transformed Fay–Herriot (PTFH) model, the log-transformed Fay–Herriot (log-FH) model and the classical Fay–Herriot (FH) model.

8

due to highly variable survey weights and small samples, the sampling variance Di could be badly estimated in areas with small samples, and nothing can reassure practitioners about what effects might have on the results from the PTFH model. Regarding model companion in Section 4, we used the AIC based on the maximum marginal likelihood (marginal AIC) for model comparison. However, the use of the marginal AIC in the Fay–Herriot model is known to have some drawbacks, and Han (2013) and Marhuenda et al. (2014) proposed alternative information criteria. However, the use of these criteria in the transformation model would not be straightforward due to the variability of estimating the additional transformation parameter, thereby the detailed discussion is left to a future study.

9

Uncited references

1 2 3 4 5 6 7

10

Chambers et al., 2014, Molina and Rao, 2010 and Ruppert et al., 2003.

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

11

Fig. 3. The scatter plots of (log xi , h λ (yi )) with estimated regression lines in the parametric transformed Fay–Herriot (PTFH) model and the nonparametric (NP) model based on the penalized spline with three orders (p = 1, 2, 3). Table 6 The small area estimates and the root of MSE (RMSE) estimates in seven prefectures around Tokyo from four models, the parametric transformed Fay–Herriot (PTFH) model, the log-transformed Fay–Herriot (log-FH) model and the classical Fay–Herriot (FH) model. Data

Prefecture

DE

Estimates

RMSE

PTFH

log-FH

FH

PTFH

log-FH

FH

Education

Ibaraki Tochigi Gunma Saitama Chiba Tokyo Kanagawa

21.97 21.88 14.12 32.61 21.55 22.04 22.32

21.80 21.63 14.76 27.41 20.92 21.84 21.87

21.54 21.21 15.49 23.81 20.19 21.55 21.32

21.44 21.30 15.17 22.78 20.08 21.06 20.86

1.12 1.65 2.74 4.49 3.31 1.73 2.37

1.99 2.41 3.68 4.68 3.90 2.16 2.93

1.95 2.10 2.93 4.83 3.87 3.12 3.34

Health

Ibaraki Tochigi Gunma Saitama Chiba Tokyo Kanagawa

10.35 11.76 8.74 11.13 12.81 13.80 14.50

10.37 11.71 8.88 11.13 12.64 13.73 14.45

10.67 11.34 10.00 11.21 11.64 12.66 13.55

10.70 11.33 10.12 11.22 11.64 12.53 13.61

0.25 0.61 0.51 0.19 0.60 0.18 0.10

0.83 1.08 1.23 0.77 1.06 0.79 0.63

0.85 1.08 1.25 0.79 1.07 0.85 0.62

Acknowledgments

1

We would like to thank the Associate Editor and the four reviewers for many valuable comments and helpful suggestions which led to an improved version of this paper. We would also like to thank professor J. N. K. Rao for helpful comments and suggestions and also thank attendees at the seminar in Statistics Canada. This work was supported by JSPS KAKENHI Grant Numbers, JP16H07406, JP15H01943 and JP26330036.

2 3 4 5

Appendix

6

A.1. Derivation of (2.5)

7

Note that E[( µi − µi )2 ] = E[µ2i ] − E[ µ2i ]. From (2.3), it follows that E[ µ2i ] =

 R3

1 −1 2 2 t   h− λ (s)hλ (t )φ(s; θi (u), σi )φ(t ; θi (u), σi )φ(u; xi β, A + Di )dsdtdu,

where  θi (u) = ai u + (1 − ai )xti β with ai = A/(A + Di ). Let S and T be random variables mutually independently distributed  as N (θi (U ), σi2 ) under given U = u, and let U be a random variable distributed as N (xti β, A + Di ). The marginal distribution

8

9

10 11

12

1

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

of the vector (S , T )t is xti β 1 ,A ai xti β

 N2

2

3 4 5





ai 1



.

1 −1 t Then, we have E[ µ2i ] = E[h− λ (S )hλ (T )], where the expectation is taken with respect to the marginal distribution of (S , T ) . t Introducing random variables z1 and z2 mutually independently distributed as N (0, A), we can express S = xi β+ c1i z1 + c2i z2 and T = xti β + c1i z1 − c2i z2 , thereby we obtain the expression 1 t −1 t E[ µ2i ] = E[h− λ (xi β + c1i z1 + c2i z2 )hλ (xi β + c1i z1 − c2i z2 )].

6 7

1 t 2 Since E[µ2i ] can be expressed as E[µ2i ] = E[{h− λ (xi β + z1 )} ], we obtain (2.5).

8

A.2. Proof of Lemma 2.2

9

For notational simplicity, we define  µi(φ) = ∂ µi /∂φ and  µi(φφ) = ∂ 2 µi /∂φ∂φt . Expanding  µi around  µi , we get 1  µi −  µi =  µti(φ) ( φ − φ) + ( φ − φ)t  µi(φφ) (yi ; φ∗ )( φ − φ),

10

11

12

2

where φ∗ is on the line connecting φ and  φ. Then, it holds that   1 g2i (φ) = E ( φ − φ)t  µi(φ) µti(φ) ( φ − φ) + R1 + R2 , 4 where R1 = E[ µti(φ) ( φ − φ)( φ − φ)t  µi(φφ) (yi ; φ∗ )( φ − φ)] and R2 = E[{( φ − φ)t  µi(φφ) (yi ; φ∗ )( φ − φ)}]. We first show that R1 = o(m−1 ) and R2 = o(m−1 ). We only prove R1 = o(m−1 ) since the evaluation of R2 is quite similar. In what follows, we define ∂ 2 µ∗i /∂φk ∂φℓ = ∂ 2 µi (yi ; φ∗ )/∂φk ∂φℓ . It follows that R1 =

≡

 2 ∗    p+2  p+2  p+2  ∂  µi ∂ µi ( φj − φj )( φk − φk )( φℓ − φℓ ) E ∂φj ∂φk ∂φℓ j=1 k=1 ℓ=1 p+2  p+2  p+2 

U1jkℓ ,

j=1 k=1 ℓ=1

and

    2 ∗  4  14   4  43  ∂  ∂  µi µ i 3     φ − φ )( φ − φ )( φ − φ ) E |U1jkl | ≤ E  ( j j k k ℓ ℓ  ∂φj ∂φk ∂φℓ   8  18  2 ∗ 8  18   4  14  ∂  ∂   µ µi  i      ≤E  E  E φa − φa  ∂φj  ∂φk ∂φℓ  a∈{j,k,ℓ} 13 14

15

from Hölder’s inequality. From the asymptotic normality of  φ given in Lemma 2.1, it follows E[| φ − φ|r ] = O(m−r /2 ) for arbitrary r > 0. Then, we have

 a∈{j,k,ℓ}

 4 1/4   E φa − φa  = o(m−1 ).

Noting that 1 1/λ  ∂ h− ∂  λ ( x) ≡ λx + 1 + λ2 x2 ∂λ ∂λ   1  h− x 1 λ (x) 2 2 = − log(λx + 1 + λ x ) , √ λ λ 1 + λ 2 x2

the straightforward calculation shows that

 µi(λ)

    ∞ 1 ∂ h− ∂φ(t ;  θi , σi2 ) 1 λ (t ) = φ(t ;  θi , σi2 )dt + h− ( t ) dt λ ∂λ ∂λ −∞ −∞             −1 1 θ h (θ ) 1 1 ∂ hλ (yi ) i i    1 −1 2 θ 2 )y  (θ ) = E  λ (θ ) log (λθ + 1 + λ + E (θ − θ ) h yi − 2 E h−  yi i i i i i i λ λ i λ λ Di ∂λ 2 2 1+λ θ 

∞



i

≡ E[f1 (θi )|yi ] + E[f2 (θi )|yi ] + E[f3 (θi , yi )|yi ],

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

13

where

1

∂ hλ (yi ) log x λ 1 = x − log x + hλ (x). ∂λ λ λ 



2

Note that E



3

E[f (θi , yi )|yi ]

a 

  ≤ E E[f (θi , yi )a |yi ] = E[f (θi , yi )a ]

4

for a > 0 from Jensen’s inequality. Since E[f1 (θi )a ] < ∞, E[f2 (θi )a ] < ∞, E[f3 (θi , yi )a ] < ∞ for a > 0, it follows that

    ∂ µi 8  < ∞. E  ∂λ 

5

6

Similarly, we have

 µi(β) =  µi(A) =

Di xi

(A + Di )σ

2 i



  

E

     5/2 1 (θi −  θi )2 − σi h− λ (θi )yi ,

1 E (θi −  θi )h− λ (θi )yi

D2i 2σ (A + Di )2 4 i

which leads to

7

 8   ∂  µ i  < ∞, E  ∂φk 

8

for k = 1, . . . , p + 1. Moreover, straightforward but considerable calculations show that E |∂ 2 µ∗i /∂φk ∂φℓ |8 < ∞. Hence, −1 −1 we have R1 = o(m ). A quite similar evaluation shows that R2 = o(m ), which leads to





g2i (φ) = E ( φ − φ)t  µi(φ) µti(φ) ( φ − φ) + o(m−1 ).





9 10

11

Finally, using the similar argument given in the proof of Theorem 3 in Kubokawa et al. (2016), we have E ( φ − φ)t  µi(φ) µti(φ) ( φ − φ) = tr E  µi(φ) µti(φ) E[( φ − φ)( φ − φ)t ] + o(m−1 )





 

=

1 m



tr V (φ)E  µi(φ) µti(φ)









+ o(m−1 ),

which completes the proof.

12

A.3. Proof of Theorem 2.1

13

Taylor series expansion of g1i ( φ) around φ gives E[g1i ( φ)] = g1i (φ) +

 2  ∂ g1i (φ)  1 ∂ g1i (φ)  t  E [ φ − φ] + tr E [( φ − φ)( φ − φ) ] + R3 , 2 ∂φt ∂φ∂φt

where R3 =

14

p+2 p+2 p+2 1    ∂ 3 g1i (φ) 

6 j=1 k=1 ℓ=1

  ( φj − φj )( φk − φk )( φℓ − φℓ ). ∂φj ∂φk ∂φℓ φ=φ∗

Since E[( φj − φj )( φk − φk )( φℓ − φℓ )] = o(m−1 ), it holds that E[R3 ] = o(m−1 ). Moreover, from Lemma 2.1, we have

 1 ∂ g1i (φ) 1 ∂ 2 g1i (φ)  b(φ) + tr V (φ) + o(m−1 ), E[g1i (φ) − g1i (φ)] = m ∂φt 2m ∂φ∂φt

15

16



thereby, we have E[g1i ( φ) − g1i (φ)] = m−1 c2 (φ) + o(m−1 ) with the smooth function c2 (φ). Hence, from Lemma 2.2 and Butar and Lahiri (2003), we obtain the second order unbiasedness of (2.7).

17

18 19

14

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

S. Sugasawa, T. Kubokawa / Computational Statistics and Data Analysis xx (xxxx) xxx–xxx

References Arima, S., Datta, G.S., Liseo, B., 2015. Bayesian estimators for small area models when auxiliary information is measured with error. Scand. J. Stat. 42, 518–529. Bell, W., 2008. Examining sensitivity of small area inferences to uncertainty about sampling error variances. In: Proceedings of American Statistical Association. JSM, Section on Survey Research Methods. Benavent, R., Morales, D., 2016. Multivariate Fay–Herriot models for small area estimation. Comput. Statist. Data Anal. 94, 372–390. Brent, R., 1973. Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs N.J. Butar, F.B., Lahiri, P., 2003. On measures of uncertainty of empirical Bayes small-area estimators. J. Statist. Plann. Inference 112, 63–76. Box, G.E.P, Cox, D.R., 1964. An analysis of transformation (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 26, 211–252. Chambers, R.L., Chandra, H., Salvati, N., Tzavidis, N., 2014. Outlier robust small area estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 76, 47–69. Chatterjee, S., Lahiri, P., Li, H., 2008. Parametric bootstrap approximation to the distribution of EBLUP, and related prediction intervals in linear mixed models. Ann. Statist. 36, 1221–1245. Datta, G.S., Hall, P., Mandal, A., 2011. Model selection by testing for the presence of small-area effects, and application to area-level data. J. Amer. Statist. Assoc. 106, 362–374. Datta, G.S., Lahiri, P., 2000. A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems. Statist. Sinica 10, 613–627. Datta, G.S., Mandal, A., 2015. Small area estimation with uncertain random effects. J. Amer. Statist. Assoc. 110, 1735–1744. Datta, G.S., Rao, J.N.K., Smith, D.D., 2005. On measuring the variability of small area estimators under a basic area level model. Biometrika 92, 183–196. Diao, L., Smith, D.D., Datta, G.A., Maiti, T., Opsomer, J.D., 2014. Accurate confidence interval estimation of small area parameters under the Fay–Herriot model. Scand. J. Stat. 41, 497–515. Fay, R.E., Herriot, R.A., 1979. Estimates of income for small places: an application of James–Stein procedures to census data. J. Amer. Statist. Assoc. 74, 269–277. Ghosh, M., Maiti, T., Roy, A., 2008. Influence functions and robust Bayes and empirical Bayes small area estimation. Biometrika 95, 573–585. Gonzàlez-Manteiga, Z., Lombardia, M.J., Molina, I., Morales, D., Santamaria, L., 2010. Small area estimation under Fay–Herriot models with nonparametric estimation of heteroscedasticity. Stat. Model. 10, 215–239. Hall, P., Maiti, T., 2006. On parametric bootstrap methods for small area prediction. J. R. Stat. Soc. Ser. B Stat. Methodol. 68, 221–238. Han, B., 2013. Conditional Akaike information criterion in Fay–Herriot model. Stat. Methodol. 11, 53–67. Jiang, J., 1996. REML estimation: asymptotic behavior and related topics. Ann. Statist. 24, 255–286. Kubokawa, T., Sugasawa, S., Ghosh, M., Chaudhuri, S., 2016. Prediction in heteroscedastic nested error regression models with random dispersions. Statist. Sinica 26, 465–492. Lahiri, P., Rao, J.N.K., 1995. Robust estimation of mean squared error of small area estimators. J. Amer. Statist. Assoc. 90, 758–766. Li, H., Lahiri, P., 2010. An adjusted maximum likelihood method for solving small area estimation problems. J. Multivariate Anal. 101, 882–892. Maiti, T., Ren, H., Sinha, A., 2014. Prediction error of small area predictors shrinking both means and variances. Scand. J. Stat. 41, 775–790. Marhuenda, Y., Morales, D., Pardo, M.C., 2014. Information criteria for Fay–Herriot model selection. Comput. Statist. Data Anal. 70, 268–280. Molina, I., Rao, J.N.K., 2010. Small area estimation of poverty indicators. Canad. J. Statist. 38, 369–385. Molina, I., Rao, J.N.K., Datta, G.S., 2015. Small area estimation under a Fay–Herriot model with preliminary testing for the presence of random area effects. Surv. Methodol. 41, 1–19. Opsomer, J.D., Claeskens, G., Ranalli, M.G., Kauermann, G., Breidt, F.J., 2008. Non-parametric small area estimation using penalized spline regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 70, 265–286. Pfeffermann, D., 2014. New important developments in small area estimation. Statist. Sci. 28, 40–68. Prasad, N., Rao, J.N.K., 1990. The estimation of mean-squared errors of small-area estimators. J. Amer. Statist. Assoc. 90, 758–766. Rao, J.N.K., Molina, I., 2015. Small Area Estimation. Wiley. Ruppert, D., Wand, M.P., Carroll, R.J., 2003. Semiparametric Regression. Cambridge University Press, Cambridge. Sinha, S.K., Rao, J.N.K., 2009. Robust small area estimation. Canad. J. Statist. 37, 381–399. Slud, E.V., Maiti, T., 2006. Mean-squared error estimation in transformed Fay–Herriot models. J. R. Stat. Soc. Ser. B Stat. Methodol. 68, 239–257. Sugasawa, S., Kubokawa, T., 2015. Parametric transformed Fay–Herriot model for small area estimation. J. Multivariate Anal. 139, 295–311. Yang, Z.L., 2006. A modified family of power transformations. Econom. Lett. 92, 14–19. Ybarra, L.M.R., Lohr, S.L., 2008. Small area estimation when auxiliary information is measured with error. Biometrika 95, 919–931. Yoshimori, M., Lahiri, P., 2014. A new adjusted maximum likelihood method for the Fay–Herriot small area model. J. Multivariate Anal. 124, 281–294. You, Y., Chapman, B., 2006. Small area estimation using area level models and estimated sampling variances. Surv. Methodol. 32, 97–103.