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Using response probability and ratio imputation in the estimation under callbacks
Journal of the Korean Statistical Society 39 (2010) 511–521
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Using response probability and ratio imputation in the estimation under callbacks Hyeonah Park ∗ , Jongwoo Jeon Department of Statistics, Seoul National University, Seoul, 151-742, Republic of Korea
article
info
Article history: Received 1 March 2009 Accepted 4 November 2009 Available online 26 November 2009 AMS 2000 subject classifications: primary 62D05 secondary 62G09
abstract Although the response model has been frequently applied to nonresponse weighting adjustment or imputation, the estimation under callbacks has been relatively underdeveloped in the response model. We propose an estimator under callbacks using both the response probability and the ratio imputation and a replication variance estimator of the estimator. We also study the estimation of the response probability. A simulation study illustrates our technique. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
Keywords: Callbacks Ratio imputation Response probability Variance estimation
1. Introduction Generally, almost all surveys rely on callbacks to raise the response rate of persons who are not at home. The technique of callbacks and the estimation after callbacks have been considered numerous times by many survey researchers. A method of selecting subsamples from persons not at home and the estimation of double sampling were first considered by Hansen and Hurwitz (1946). Deming (1953) studied the estimation of population mean when responses are collected until the ith callback attempt. Groves (1989) provided an excellent summary of these approaches. Recently, Elliott, Little, and Lewitzky (2000) considered the subsampling callbacks, where an efficient subsampling strategy considering variance and cost from the repeated callback attempts was established. The estimators were relatively underdeveloped in the response model under callbacks though the response model has been applied to nonresponse weighting adjustment or imputation. There is a vast literature on nonresponse weighting adjustment or imputation under uniform or non-uniform response model. See, for example, Lipsitz, Ibrahim, and Zhao (1999), Rao and Sitter (1995), Rosenbaum (1987), Rao and Shao (1992), Robins, Rotnitzky, and Zhao (1994) and Shao and Steel (1999). In this article, we propose an estimator using the response probability and the ratio imputation in the response model under callbacks. We first prove the unbiasedness and the efficiency of the proposed estimator under the assumption that we know the true response probability. We also suggest a replication variance estimator of the estimator which satisfies the consistency under infinite sample size. Since the response probability is usually unknown, the estimated response probability can be used instead of the true response probability. For the estimation of the response probability, one can refer to Ekholm and Laaksonen (1991), and Iannacchione (2003). We also propose a consistent replication variance estimator of this estimator.
∗
Corresponding author. Tel.: +82 2 880 5718. E-mail address: [email protected] (H. Park).
1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.11.001
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This paper is organized as follows. In Section 2, we introduce an estimator using the true response probability and the ratio imputation under callbacks and calculate the expectation and the variance of the estimator. In Section 3, we propose a replication variance estimator of the estimator of Section 2. We consider the estimation of the response probability assuming the logistic response model in Section 4, where the estimator and its variance estimator corresponding to the estimated response probability are also given. Finally, the numerical evaluation of the estimators in this paper is performed through a simulation study in Section 5. 2. Estimation using response probability and ratio imputation Let the population total be Y =
∑N
i=1
yi and the population mean be Y¯ = N −1
∑N
i=1
yi , where N is the population size
and yi is the value of the target variable of unit i. Let Yˆn be an estimator of the population total Y defined by Yˆn = i∈A wi yi , where n is the sample size, wi is the sampling weight of unit i and A = {1, 2, . . . , n} is the set of indices of the sample. We define the response indicator function under the first survey as
∑
Ri =
1, 0,
if unit i responds otherwise
for i ∈ A. Let πi = P (Ri = 1|i ∈ A) be the response probability of sample unit i under the first survey. We consider only one-step callback in this paper. We write the response indicator function under callback as
Ti =
1, 0,
if unit i responds otherwise
for i ∈ ANR , where ANR = {i : Ri = 0, i ∈ A} is the set of the indices of nonresponding units. Let pi = Pr (Ti = 1|i ∈ ANR ) be the response probability of sample unit i for i ∈ ANR under callback. We assume that Ri and Ti are ignorable such that πi and pi depend on an auxiliary variable zi but not on yi . In this section we also assume that all πi and pi are known priori. Now we suppose that there is another auxiliary variable xi that is related with the study variable yi and can be observed ∑n ∑n −1 ˜ throughout the sample. We first introduce some preliminary estimators. Let Y˜R = i=1 wi πi Ri yi and YT = i=1 wi (1 − 1 πi )−1 p− − Ri )Ti yi . For the auxiliary variable xi , we define X˜ T = i (1∑ n −1 and X˜ NR = (1 − Ri )xi . We define i=1 wi (1 − πi )
∑n
i=1
1 ˜ wi (1 − πi )−1 p− i (1 − Ri )Ti xi , XR =
∑n
i=1
wi πi−1 Ri xi
Y˜J = (W1 X˜ T−1 Y˜T + W2 X˜ R−1 Y˜R )X˜ NR and
ψ˜ = [Var (Y˜R ) + Var (Y˜J ) − 2Cov(Y˜R , Y˜J )]−1 [Var (Y˜J ) − Cov(Y˜R , Y˜J )],
(1)
where W1 = 1/2 and W2 = 1/2 are used in this article. Using W1 = 1 and W2 = 0, we see that the estimator Y˜J is equal to the estimator which was suggested by Park, Na, and Jeon (2008). Ratio imputation proposed by Rao and Sitter (1995) and by Rao (1996) uses X˜ R−1 Y˜R xi under πi = π , wi = n−1 and the estimator under this ratio imputation is (X˜ R−1 Y˜R )Xˆ n . If Y˜J in the proposed estimator is explained, the estimator using an auxiliary variable under callback is X˜ T−1 Y˜T−1 X˜ NR and the estimator under ratio imputation is X˜ R−1 Y˜R−1 X˜ NR . If πi are equal and wi = n−1 , we can see that the ratio imputation proposed by Rao and Sitter (1995) and by Rao (1996) is used in the estimator under ratio imputation, that is, in X˜ R−1 Y˜R−1 X˜ NR . Then, our proposed estimator is defined as
˜ Y˜R + (1 − ψ) ˜ Y˜J . Y˜M = ψ ˜ . Considering the proposed estimator from a different Note that the variance of kY˜R + (1 − k)Y˜J is minimized at k = ψ standpoint, the estimator Y˜M could be considered as the improved version of estimator under ratio imputation using response probability which is estimated by both responders and callbacks. We adopt the extended definition of the response indicator function introduced by Fay (1991). Conceptually, the response indicator functions Ri and Ti can be extended to the entire population. In this section we assume the following conditions: (A1) A sequence of finite populations and samples are defined as in Isaki and Fuller (1982). The finite populations satisfy that for some τ > 0 N −1
N −
θi2+τ = O(1),
i =1
where θi represents yi , xi and zi . The sampling mechanism satisfies E (Yˆn ) = Y .
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
513
(A2) For nonnegative constants C1 , C2 and C3 , C1 < πi < C2 and pi > C3 . (A3) The response indicator functions Ri and Ti are mutually independent, respectively, such that P (Ri = 1, Rj = 1) = P (Ri = 1)P (Rj = 1) and P (Ti = 1, Tj = 1) = P (Ti = 1)P (Tj = 1) for different i and j. (A4) The sampling mechanism satisfies that for nonnegative constants D1 , D2 , D3 and D4 D1 < max (N −1 nwi ) < D2 1≤i≤N
and D3 < N −2 nVar (Yˆn ) < D4 , where the variance is calculated from the sampling mechanism. In the following theorem we deal with the expectation and the variance of our proposed estimator Y˜M . Theorem 1. Under the assumptions (A1)–(A4), E (Y˜M ) = Y + o(n−1/2 N )
(2)
Var (Y˜M ) = Var (Yˆn ) + (K1 + K2 + 2K3 )−1 (K1 K2 − K32 ) + o(n−1 N 2 ),
(3)
and
where
K1 = E
n −
w (πi
−1
2 i
− 1)
,
y2i
i=1
n −
wi2 [W2 πi−1 (yi − rxi ) − (1 − πi )−1 (W1 yi + W2 rxi )]2 πi (1 − πi ) i =1 n − 2 2 −1 −1 2 + W1 wi (1 − πi ) (pi − 1)(yi − rxi ) ,
K2 = E
i=1
n −
w πi yi [yi (πi − W2 ) + W2 rxi ] 2 i
−1
and r = X −1 Y for X =
∑N
K3 = E
i=1
i=1
xi .
Proof. Note that X˜ T−1 Y˜T X˜ NR = Y˜T + r˜1 (X˜ NR − X˜ T ) and X˜ R−1 Y˜R X˜ NR = Y˜R + r˜2 (X˜ NR − X˜ R ), where r˜1 = X˜ T−1 Y˜T and r˜2 = X˜ R−1 Y˜R . Under (A1), (A3) and (A4), E [(X˜ NR − X ) ] = Var (Xˆ n ) + E 2
n −
w (πi 2 i
−1
− 1) xi
−1 2
i=1
= O(n−1 N 2 ), E [(X˜ T − X )2 ] = Var (Xˆ n ) + E
n − i=1
= O(n−1 N 2 )
w (1 − πi ) (πi + pi − 1) 2 i
−1
−1
x2i
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and E [(X˜ R − X )2 ] = Var (Xˆ n ) + E
n −
w (πi 2 i
−1
− 1)
x2i
i=1
= O(n−1 N 2 ). where Xˆ n =
∑n
i=1
wi xi . Similarly, we also obtain that
Y˜T − Y = OP (n−1/2 N ) and
Y˜R − Y = OP (n−1/2 N ).
Then, by Taylor’s expansion, we have r˜1 − r = X −1 [(Y˜T − Y ) − r (X˜ T − X )] + oP (n−1/2 ) = OP (n−1/2 ), r˜2 − r = X
−1
[(Y˜R − Y ) − r (X˜ R − X )] + oP (n−1/2 ) = OP (n−1/2 )
(4) (5)
and Y˜J = W1 [Y˜T + r (X˜ NR − X˜ T )] + W2 [Y˜R + r (X˜ NR − X˜ R )] + oP (n−1/2 N ). Observe that
˜ Y˜R − Yˆn ) + (1 − ψ){ ˜ W1 [(Y˜T − Yˆn ) + r (X˜ NR − X˜ T )] + W2 [(Y˜R − Yˆn ) + r (X˜ NR − X˜ R )]} + oP (n−1/2 N ). Y˜M − Yˆn = ψ( This, together with (A1), implies (2). ˜, Let Y˜J′ = W1 [Y˜T + r (X˜ NR − X˜ T )] + W2 [Y˜R + r (X˜ NR − X˜ R )]. By definition of ψ Var (Y˜M ) = [Var (Y˜R ) + Var (Y˜J′ ) − 2Cov(Y˜R , Y˜J′ )]−1 [Var (Y˜R )Var (Y˜J′ ) − Cov(Y˜R , Y˜J′ )2 ] + o(n−1 N 2 ) since Y˜J − Y˜J′ = oP (n−1/2 N ). Observe that from (A3), Var (Y˜R ) = Var (Yˆn ) + K1 , Var (Y˜J′ ) = Var (Yˆn ) + K2 , Cov(Y˜R , Y˜J′ ) = Var (Yˆn ) − K3 . Thus, the result (3) follows immediately.
3. Variance estimation using known response probability In this section we propose a method of estimating the variance of the estimator when we know the response probability. Note that the variance must be estimated to calculate the efficiency of the proposed estimator. We consider the replication method such as jackknife, since it is well known that the replication method is good to estimate the variances of complex estimators. First, we illustrate a replication variance estimator for Yˆn . Let an estimator of Var (Yˆn ) be Vˆ (Yˆn ) =
L −
ck (Yˆn(k) − Yˆn )2 ,
k=1
where L is the number of replications, ck is a factor associated with the kth replication determined by the replication method (k) and Yˆn is the kth estimator of Y based on the observations included in the kth replication, that is, Yˆn(k) =
n −
wi(k) yi ,
i =1
(k)
where wi
is the replication weight for the ith unit in the kth replication. For example, if the inclusion probability is N −1 n
and wi = n−1 N, then the standard jackknife variance estimator Vˆ (Yˆn ) is defined by L = n, ck = (1 − N −1 n)n−1 (n − 1), wi(k) = (n − 1)−1 nwi for i ̸= k and wk(k) = 0. We suggest a replication estimator for the variance of Y˜M by Vˆ (Y˜M ) =
L −
(k)
ck (Y˜M − Y˜M )2 ,
k=1
where (k)
(k)
˜ Y˜R Y˜M = ψ
˜ Y˜J(k) + (1 − ψ)
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
515
(k) −1 (k) (k) Ri yi and Y˜J = (W1 X˜ T(k)−1 Y˜T(k) + W2 X˜ R(k)−1 Y˜R(k) )X˜ NR . Here, analogously as before, Y˜T(k) = = i =1 i =1 w i π i ∑ ∑ ∑ ( k (k) ( k ) ( k ) n = ni=1 wi(k) wi (1 − πi )−1 pi −1 (1 − Ri )Ti yi , X˜ T = i=1 wi (1 − πi )−1 pi −1 (1 − Ri )Ti xi , X˜ R ) = ni=1 wi(k) πi−1 Ri xi and X˜ NR (1 − πi )−1 (1 − Ri )xi . Note that the superscript (k) denotes the kth replication. (k)
for Y˜R
∑n
∑n
(k)
We assume that the variance of a linear estimator of the total is a quadratic function of y, that is, N −2 nVar (Yˆn ) =
N − N −
Ωij yi yj ,
(6)
i=1 j=1
where the coefficients Ωij satisfy max Ωii = O(N −1 )
(7)
1≤i≤N
and N −
|Ωij | = O(N −1 ).
(8)
i =1
For example, the simple random sampling with wi = n−1 N satisfies (7) and (8) because
Ωij =
N − 1 ( 1 − N − 1 n) −N −1 (N − 1)−1 (1 − N −1 n)
if i = j if i ̸= j .
In order to establish the consistency of Vˆ (Y˜M ), we first prove the consistency of the variance estimator of Y˜R in the following lemma. Lemma 1. Suppose that the conditions (A1)–(A4) are satisfied. We assume that for any y with bounded fourth moment, E [Var (Yˆn )−1 Vˆ (Yˆn ) − 1]2 = o(1),
(9)
where the expectation is calculated under the given sampling mechanism. Assume also that N −1 n = o(1).
(10)
Then, Vˆ (Y˜R ) =
L −
(k)
ck (Y˜R
− Y˜R )2 = Var (Y˜R ) + oP (n−1 N 2 ).
(11)
k=1
Proof. From (A4) and (9), Vˆ (Y˜R ) = Var (Y˜R |R1 , . . . , RN ) + oP (n−1 N 2 ). From (10), Var [E (Y˜R |R1 , . . . , RN )] = o(n−1 N 2 ). Then, (11) follows immediately if we prove Var [N −2 nVar (Y˜R |R1 , . . . , RN )] = o(1).
(12)
From (6) and independence of Ri ’s, we can see that Var [N −2 nVar (Y˜R |R1 , . . . , RN )] =
N − N − (Ωij2 + Ωij Ωji )Var (yπ i yπ j ) i=1 j=1
≤ 2 max Var (yπ i yπ j ) max |Ωij | 1≤i,j≤N
1≤i,j≤N
N − N −
|Ωij |,
i =1 j =1
where yπ i = πi−1 Ri yi and the variances are taken with respect to Ri ’s. From (A1) with τ > 2 and (A2), max1≤i,j≤N Var (yπ i yπ j ) = O(1). By (7) and the nonnegative definiteness of Ω = (Ωij ), max1≤i,j≤N |Ωij | = O(N −1 ). These, together with (8), imply (12). Secondly, we deal with the variance estimator for Y˜J . Here, an additional condition is listed.
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
(A5) For the replication factors, max ck−1 = O(L)
1≤k≤L
and E [{ck (Yˆn(k) − Yˆn )2 }2 ] < Cy L−2 {Var (Yˆn )}2 for Yˆn corresponding to yi with finite fourth moment and some constant Cy . Lemma 2. Assume the conditions (A1)–(A5), (9) and (10). Then, Vˆ (Y˜J ) =
L −
(k)
ck (Y˜J
− Y˜J )2 = Var (Y˜J ) + oP (n−1 N 2 ).
(13)
k=1
(k)
Proof. Split Y˜J (k)
Y˜J
− Y˜J as
− Y˜J = (Y˜J(k) − Y˜J′(k) ) + (Y˜J′(k) − Y˜J′ ) + (Y˜J′ − Y˜J ),
(k) (k) − X˜ R(k) )]. Observe that for r˜1(k) = Y˜T(k) /X˜ T(k) and r˜2(k) = Y˜R(k) /X˜ R(k) , − X˜ T(k) )] + W2 [Y˜R(k) + r (X˜ NR = W1 [Y˜T(k) + r (X˜ NR (k) − X˜ NR ) − (X˜ T(k) − X˜ T )] (Y˜J(k) − Y˜J′(k) ) + (Y˜J′ − Y˜J ) = W1 (˜r1(k) − r )[(X˜ NR (k) − X˜ NR ) − (X˜ R(k) − X˜ R )] + (˜r2(k) − r˜2 )(X˜ NR − X˜ R ) +(˜r1(k) − r˜1 )(X˜ NR − X˜ T ) + W2 (˜r2(k) − r )[(X˜ NR ′(k)
where Y˜J
= oP (n−1/2 N ) since (k)
(k) − X˜ T = OP (n−1/2 N ), X˜ R(k) − X˜ R = OP (n−1/2 N ), X˜ NR − X˜ NR = OP (n−1/2 N ), (k) (k) (k) r˜1 − r˜1 = X˜ T−1 {(Y˜T − Y˜T ) − r˜1 (X˜ T − X˜ T )} + oP (n−1/2 ) −1/2 = OP (n )
X˜ T
and (k)
(k)
r˜2 − r˜2 = X˜ R−1 {(Y˜R
− Y˜R ) − r˜2 (X˜ R(k) − X˜ R )} + oP (n−1/2 )
= OP (n−1/2 ), which are consequence of (A3)–(A5), (4) and (5) and the fact that X˜ NR − X˜ T = Op (n−1/2 N ) and X˜ NR − X˜ R = Op (n−1/2 N ). Then, from (9) and (A5), Vˆ (Y˜J ) =
L −
′(k)
ck (Y˜J
− Y˜J′ )2 + oP (n−1 N 2 )
k=1
= Var (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) + oP (n−1 N 2 ). If we prove Var [E (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN )] = o(n−1 N 2 )
(14)
Var [N −2 nVar (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN )] = o(1),
(15)
and
then we have Var (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) = Var (Y˜J′ ) + oP (n−1 N 2 ), which implies (13) since Y˜J − Y˜J′ = oP (n−1/2 N ). If we use the same steps of Lemma 1, we can easily prove (13).
In the following theorem, the consistency of the variance estimator Vˆ (Y˜M ) is established. Theorem 2. Under the same conditions as in Lemma 2, Vˆ (Y˜M ) = Var (Y˜M ) + oP (n−1 N 2 ).
(16)
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
517
Proof. Observe that L −
(k) ˜2 ck (Y˜M − Y˜M )2 = ψ
L −
k=1
(k)
ck (Y˜R
˜ 2 − Y˜R )2 + (1 − ψ)
L −
k=1
(k)
ck (Y˜J
− Y˜J )2
k=1
˜ 1 − ψ) ˜ + 2ψ(
L −
(k)
ck (Y˜R
− Y˜R )(Y˜J(k) − Y˜J ).
k=1
Using the similar arguments in Lemma 2, we obtain that L −
(k)
ck (Y˜R
+ Y˜J(k) − Y˜R − Y˜J )2 =
k=1
L −
(k)
ck (Y˜R
+ Y˜J′(k) − Y˜R − Y˜J′ )2 + oP (n−1 N 2 ).
k=1
By (A4) and (9), L −
(k)
ck (Y˜R
+ Y˜J′(k) − Y˜R − Y˜J′ )2 = Var (Y˜R + Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) + oP (n−1 N 2 )
k=1
= Var (Y˜R + Y˜J ) + oP (n−1 N 2 ), where the same steps in Lemmas 1 and 2 and the fact that Y˜J − Y˜J′ = oP (n−1/2 N ) have been used for the last equality. This, together with (11) and (13), implies that L −
(k)
ck (Y˜R
− Y˜R )(Y˜J(k) − Y˜J ) = Cov(Y˜R , Y˜J ) + oP (n−1 N 2 ),
k=1
which implies (16).
4. Estimation using estimated response probability In many practical situations, it is impossible to know the response probability and the weight. In this section, we estimate the response probability and the weight. We assume the parametric logistic model for the response probability:
πi = π (zi ; α) = (1 + exp(−ziT α))−1 , where zi is the value of an auxiliary variable for unit i and α = (α1 , . . . , αp )T . The response probability pi under callback is assumed to be a constant p. Let πˆ i = π (zi ; α) ˆ be the estimated response probability, where αˆ satisfies n1/2 (αˆ − α) = n−1/2
n −
H (zi , Ri ; α) + oP (1),
(17)
i=1
where E [H (zi , Ri ; α)] = 0 and E H (zi , Ri ; α)H (zi , Ri ; α)T is positive definite (cf. (Kim & Park, 2006)). For example, the logistic regression model defined by πi = [1 + exp(−α1 − α2 zi )]−1 satisfies
H (Ri ; α) = n[I (α1 , α2 )]−1 (Ri − πi )(1, zi )T and I (α1 , α2 ) = E
n −
πi (1 − πi )(1, zi )T (1, zi ) .
i =1
∗ The response probability pi is estimated by pˆ = R∗−1 T ∗ , where R∗ = i=1 wi (1 − Ri ) and T = i=1 wi (1 − Ri )Ti . The estimators with the estimated response probabilities plugged-in are defined analogously as before. We define ∑n ∑n YˆR = ˆ i−1 Ri yi and YˆJ = (W1 Xˆ T−1 YˆT + W2 Xˆ R−1 YˆR )Xˆ NR , where Xˆ T = ˆ i )−1 pˆ −1 (1 − Ri )Ti xi , YˆT = i =1 w i π i =1 w i ( 1 − π
∑n
∑n
∑ wi πˆ i−1 Ri xi and Xˆ NR = ni=1 wi (1 − πˆ i )−1 (1 − Ri )xi . We discuss the asymptotic properties of YˆR and YˆJ in the following Lemma.
∑n
i =1
wi (1 − πˆ i )−1 pˆ −1 (1 − Ri )Ti yi , Xˆ R =
∑n
i=1
Lemma 3. Assume the same conditions as in Lemma 2. Let us define
ΓR =
N − i=1
ΓT =
πi (∂πi−1 /∂α)yi ,
ΓRX =
N −
πi (∂πi−1 /∂α)xi ,
i=1
ΓY =
N −
pyi ,
ΓX =
i=1
N N − − (1 − πi )(∂(1 − πi )−1 /∂α)yi and ΓTX = (1 − πi )(∂(1 − πi )−1 /∂α)xi . i=1
i=1
N − i=1
pxi ,
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
Then, YˆR − Y˜R = (αˆ − α)T ΓR + oP (n−1/2 N )
(18)
and YˆJ − Y˜J = W1 T¯ −1 [(R∗ − R¯ ) − p−1 (T ∗ − T¯ )](ΓY − r ΓX )
+ (αˆ − α)T [W1 ΓT + W2 (ΓR + r ΓTX − r ΓRX )] + oP (n−1/2 N ), ∑N ∑N ¯ where T¯ = i=1 (1 − πi )p, R = i=1 (1 − πi ).
(19)
Proof. By (17) and Taylor expansion,
πˆ i−1 − πi−1 = (αˆ − α)T (∂πi−1 /∂α) + oP (n−1/2 ). Then, using (A1), (A3) and (A4), we obtain (18). Observe that YˆJ − Y˜J = W1 {(YˆT − Y˜T ) + rˆ1 [(Xˆ NR − X˜ NR ) − (Xˆ T − X˜ T )] + (ˆr1 − r˜1 )(X˜ NR − X˜ T )} + W2 {(YˆR − Y˜R )
+ rˆ2 [(Xˆ NR − X˜ NR ) − (Xˆ R − X˜ R )] + (ˆr2 − r˜2 )(X˜ NR − X˜ R )}, where rˆ1 = YˆT /Xˆ T and rˆ2 = YˆR /Xˆ R . By Taylor’s expansion and the assumed conditions,
(1 − πˆ i )−1 − (1 − πi )−1 = (αˆ − α)T (∂(1 − πi )−1 /∂α) + oP (n−1/2 ) and −1 pˆ −1 − p−1 = T¯ ′ [R∗ − R¯ − p−1 (T ∗ − T¯ )] + oP (n−1/2 ).
Then, we have YˆT − Y˜T = T¯ −1 [R∗ − R¯ − p−1 (T ∗ − T¯ )]ΓY + (αˆ − α)T ΓT + oP (n−1/2 N ), Xˆ T − X˜ T = T¯ −1 [R∗ − R¯ − p−1 (T ∗ − T¯ )]ΓX + (αˆ − α)T ΓTX + oP (n−1/2 N ) and Xˆ R − X˜ R = (αˆ − α)T ΓRX + oP (n−1/2 N ). We also obtain that Xˆ NR − X˜ NR = (αˆ − α)T ΓTX + oP (n−1/2 N ), rˆ1 − r˜1 = X˜ T−1 [(YˆT − Y˜T ) − r˜1 (Xˆ T − X˜ T )] + oP (n−1/2 ) and rˆ2 − r˜2 = X˜ R−1 [(YˆR − Y˜R ) − r˜2 (Xˆ R − X˜ R )] + oP (n−1/2 ). These, together with (4), (5) and the fact that X˜ NR − X˜ T = OP (n−1/2 N ) and X˜ NR − X˜ R = OP (n−1/2 N ), immediately imply (19). (k)
Let πˆ i
L −
= π (zi ; αˆ (k) ) be the kth replication of πˆ i , where αˆ (k) = (αˆ 1(k) , . . . , αˆ p(k) )T is the kth replication of αˆ satisfying ck αˆ (k) − αˆ
(k) T αˆ − αˆ = Σα + oP n−1
(20)
k=1
for Σα = E [E {(αˆ − E [E (α)| ˆ A])(αˆ − E [E (α)| ˆ A])T }|A]. In jackknife method and logistic regression model, a method to fine αˆ (k) is to look for a maximum likelihood estimator of α after the kth observation is deleted. Note that the replication variance estimators can be defined analogously as before, where the response probabilities are replaced by the kth replicates of the estimated ones. For example, Vˆ (YˆR ) =
L −
(k)
ck (YˆR
− YˆR )2 ,
k=1
(k) where YˆR =
∑n
i=1
Cˆ (YˆR , YˆJ ) =
wi(k) πˆ i(k)−1 Ri yi . The covariance of YˆR and YˆJ is estimated by
L − k=1
(k)
ck (YˆR
− YˆR )(YˆJ(k) − YˆJ ),
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
(k)
(k)−1 (k)
(k)−1 (k)
(k)
(k)
(k)
(k)
519
(k)
where YˆJ = (W1 Xˆ T YˆT + W2 Xˆ R YˆR )Xˆ NR with YˆT , Xˆ T ,Xˆ R and Xˆ NR analogously defined using the kth replicates of the estimated response probabilities. Finally, the proposed estimator is given by
ˆ YˆR + (1 − ψ) ˆ YˆJ , YˆM = ψ ˆ = [Vˆ (YˆR ) + Vˆ (YˆJ ) − 2Cˆ (YˆR , YˆJ )]−1 [Vˆ (YˆJ ) − Cˆ (YˆR , YˆJ )]. where ψ We first discuss the consistency of Vˆ (YˆR ) in the following Lemma. Lemma 4. Assume the conditions of Lemma 2. Assume also that L −
(k)
ck (Yˆn(k) − Yˆn )(αˆ l
− αˆ l ) = Cov(Yˆn , αˆ l ) + oP (n−1 N )
(21)
k=1
for 1 ≤ l ≤ p. Then, (k)
YˆR
− YˆR = Y˜R(k) − Y˜R + (αˆ (k) − α) ˆ T ΓR + oP (n−1/2 N )
(22)
and Vˆ (YˆR ) = Var (YˆR ) + oP (n−1 N 2 ). (k)
Proof. Write YˆR (k)
YˆR
(23)
− YˆR as
− YˆR = (YˆR(k) − Y˜R(k) ) + (Y˜R(k) − Y˜R ) + (Y˜R − YˆR ).
From (17) and (20),
πˆ i(k)−1 − πˆ i−1 = (αˆ (k) − α) ˆ T (∂πi−1 /∂α) + oP (n−1/2 ) which, together with the assumed conditions, implies that
ˆ T ΓR + oP (n−1/2 N ) (YˆR(k) − Y˜R(k) ) + (Y˜R − YˆR ) = (αˆ (k) − α) and, hence, (22). Observe that by (A5) L −
(k)
ck (YˆR
− YˆR )2 =
L −
(k)
ck (Y˜R
− Y˜R )2 +
+2
ck [(αˆ (k) − α) ˆ T ΓR ]2
k=1
k=1
k=1
L −
L −
(k)
ck (Y˜R
− Y˜R )[(αˆ (k) − α) ˆ T ΓR ] + oP (n−1 N 2 ).
k=1
From (17) and (21), L −
˜ (k)
ck (YR
(k)
− Y˜R )(αˆ l − αˆ l )ΓRl = Cov Y˜R , n ΓRl −1
k=1
n −
Hli |R1 , . . . , RN
+ oP (n−1 N 2 )
i =1
where ΓRl is the lth element of ΓR for l = 1, . . . , p. From (10), we have
Cov E (Y˜R |R1 , . . . , RN ), E
n
−1
ΓRl
n −
Hli |R1 , . . . , RN
= o(n−1 N 2 ).
i =1
The similar arguments as in Lemma 1 lead to L −
(k)
ck (Y˜R
− Y˜R )[(αˆ (k) − α) ˆ T ΓR ] = Cov(Y˜R , (αˆ − α)T ΓR ) + oP (n−1 N 2 ),
k=1
which, together with (11), (18) and (20), implies (23).
We state the following lemma regarding the consistency of Vˆ (YˆJ ) without proof since the technical details resembles that in Lemma 4.
520
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
Lemma 5. Assume the same conditions as in Lemma 4. Then, (k)
YˆJ
− YˆJ = (Y˜J(k) − Y˜J ) + W1 T¯ −1 [(R∗(k) − R∗ ) − p−1 (T ∗(k) − T ∗ )](ΓY − r ΓX ) + (αˆ (k) − α) ˆ T [W1 ΓT + W2 (ΓR + r ΓTX − r ΓRX )] + oP (n−1/2 N )
(24)
and Vˆ (YˆJ ) =
L −
(k)
ck (YˆJ
− YˆJ )2 = Var (YˆJ ) + oP (n−1 N 2 ),
(25)
k=1
where R∗(k) =
∑n
i =1
wi(k) (1 − Ri ) and T ∗(k) =
∑n
i =1
wi(k) (1 − Ri )Ti .
The variance estimator for YˆM is defined by Vˆ (YˆM ) =
L −
(k)
ck (YˆM − YˆM )2 ,
k=1
(k) ˆ YˆR(k) + (1 − ψ) ˆ YˆJ(k) . In the following theorem, we deal with the asymptotic unbiasedness of YˆM and the where YˆM = ψ consistency of the variance estimator Vˆ (YˆM ).
Theorem 3. Under the same conditions as in Lemma 4, E (YˆM ) = Y + o(n−1/2 N )
(26)
Vˆ (YˆC ) = Var (YˆC ) + oP (n−1 N 2 ).
(27)
and
Proof. Using the same techniques of Theorem 2 and Lemma 4, we can obtain that L −
(k)
ck (YˆR
+ YˆJ(k) − YˆR − YˆJ )2 = Var (YˆR + YˆJ ) + oP (n−1 N 2 ),
k=1
which, together with (23) and (25), implies that Cˆ (YˆR , YˆJ ) = Cov(YˆR , YˆJ ) + oP (n−1 N 2 ) and, hence, that
ψˆ − ψ = oP (1) for ψ = [Var (YˆR ) + Var (YˆJ ) − 2Cov(YˆR , YˆJ )]−1 [Var (YˆJ ) − Cov(YˆR , YˆJ )]. Write
ˆ − ψ)(YˆR − Y ) + [(1 − ψ) ˆ − (1 − ψ)](YˆJ − Y ) + ψ YˆR + (1 − ψ)YˆI . YˆM = (ψ Then, using (18) and (19), we obtain that YˆM = ψ YˆR + (1 − ψ)YˆJ + oP (n−1/2 N ), which implies (26). Furthermore, it is easily obtained from (22) and (24) that (k)
(k)
YˆM − YˆM = ψ(YˆR
− YˆR ) + (1 − ψ)(YˆJ(k) − YˆJ ) + oP (n−1/2 N ).
Finally, using (23) and (25), we obtain (27).
5. Simulation results We show the results of a limited simulation study to test our theory. In the simulation study, B = 1000 samples of size n = 100 are generated by yi = 4xi +
√
xi ϵi ,
where xi ∼ Uniform(0, 1), ϵi ∼ N (0, 1) for i = 1, . . . , n, and xi and ϵi are independent. For a parameter model of the response probability under the first survey, we suggest the logistic model πi = [1 + exp(−α1 − α2 zi )]−1 , where zi ∼ Uniform(0, 1) and the value of α is assumed to be (α1 , α2 ) = (0, −2), (−0.5, 0.5) and (−0.5,1). Consequently, the overall response rate becomes 0.28, 0.44 and 0.50, respectively. We use constants 0.3, 0.4 and 0.5 for the response probability p under callback. We consider the maximum likelihood estimator to estimate α and calculate the value iteratively using the Newton–Raphson method. The response probability p under callback is estimated as the response rate among nonresponses. For the variance estimator, we use the standard jackknife method under simple random sampling, where ck is n−1 (n − 1) (k) and wi is defined as (n − 1)−1 nwi for i ̸= k and 0 for i = k.
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
521
Table 1 MSE (Y¯D )/MSE (Y˜M ) and MES (Y¯D )/MSE (YˆM ).
(α1 , α2 )
0.3
0.4
0 .5
(0.0, −2.0)
1.300 1.309 1.458 1.381 1.343 1.302
1.178 1.190 1.355 1.283 1.253 1.223
1.088 1.089 1.274 1.208 1.200 1.173
(−0.5, 0.5) (−0.5, 1.0)
Table 2 Relative means (t-statistics, in parentheses) of Vˆ (Y˜M ) and Vˆ (YˆM ).
(α1 , α2 )
0.3
0.4
0.5
(0.0, −2.0)
1.020(0.462) 0.948(−1.096) 1.051(1.101) 1.004(0.084) 1.050(1.063) 0.996(−0.098)
1.004(0.095) 0.931(−1.433) 1.036(0.785) 0.991(−0.207) 1.033(0.706) 0.988(−0.259)
0.995(−0.119) 0.910(−1.886) 1.019(0.432) 0.976(−0.556) 1.019(0.410) 0.976(−0.536)
(−0.5, 0.5) (−0.5, 1.0)
To know the properties of the variance estimator, we compute the relative mean and t-statistic. The relative mean of the variance estimator is that the empirical mean of the variance estimator divided by the empirical variance of the point estimator. The t-statistic for the variance estimator is the empirical bias of the variance estimator divided by the empirical standard error of the empirical bias, which was provided by Kim (2004). Using B samples of (yi , xi , ϵi , Ri , Ti ), i = 1, . . . , n, wi = n−1 and W1 = 0.5, we compute the empirical values of MSE (Y¯D )/MSE (Y˜M ) and MSE (Y¯D )/MSE (YˆM ), where MSE (Y¯D ) is the mean square error of the estimator suggested by Deming (1953). We also compute the relative means and t-statistics of Vˆ (Y˜M ) and Vˆ (YˆM ). Each cell in Table 1 contains MSE (Y¯D )/MSE (Y˜M ) and MSE (Y¯D )/MSE (YˆM ) in this order for changing response probabilities πi and p. Each cell in Table 2 shows the relative means (t-statistics, in parentheses) of Vˆ (Y˜M ) and Vˆ (YˆM ) in this order for changing response probabilities. From observing the simulation results in Table 1 it can be judged that MSE (Y˜M ) and MSE (YˆM ) are smaller than MSE (Y¯D ). From Table 2 we can see that Vˆ (Y˜M ) and Vˆ (YˆM ) are consistent estimators for Var (Y˜M ) and Var (YˆM ), respectively. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0063964). References Deming, W. E. (1953). On a probability mechanism to attain an economic balance between the resultant error of response and the bias of nonresponse. Journal of the American Statistical Association, 48, 743–772. Ekholm, A., & Laaksonen, S. (1991). Weighting via response modeling in the finnish household budget survey. Journal of Official Statistics, 7, 325–337. Elliott, M. R., Little, R. J. A., & Lewitzky, S. (2000). Subsampling callbacks to improve survey efficiency. Journal of the American Statistical Association, 95, 730–738. Fay, R. E. (1991). A design-based perspective on missing data variance. In The ASA proceedings of bureau of the census annual research conference (pp. 429–440). Washington (DC): US Bureau of the Census. Groves, R. M. (1989). Survey errors and survey costs. New York: Wiley. Hansen, M. H., & Hurwitz, W. N. (1946). The problem of nonresponse in sample surveys. Journal of the American Statistical Association, 41, 517–529. Isaki, C. T., & Fuller, W. A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77, 89–96. Iannacchione, V. G. (2003). Sequential weight adjustment for location and cooperation propensity for the 1995 national survey of family growth. Journal of Official Statistics, 19, 31–43. Kim, J. K. (2004). Finite sample properties of multiple imputation estimators. The Annals of Statistics, 32, 766–783. Kim, J. K., & Park, H. (2006). Imputation using response probability. Canadian Journal of Statistics, 34, 171–182. Lipsitz, S. R., Ibrahim, J. G., & Zhao, L. P. (1999). A weighted estimating equation for missing covariate data with properties similar to maximum likelihood. Journal of the American Statistical Association, 94, 1147–1160. Park, H., Na, S., & Jeon, J. (2008). Estimation using response probability under callbacks. Statistics and probability letters, 78, 1735–1741. Rao, J. N. K. (1996). On variance estimation with imputed survey data. Journal of the American Statistical Association, 91, 499–506. Rao, J. N. K., & Sitter, R. R. (1995). Variance estimation under two-phase sampling with application to imputation for missing data. Biometrika, 82, 453–460. Rao, J. N. K., & Shao, J. (1992). Jackknife variance estimation with survey data under hot deck imputation. Biometrika, 79, 811–822. Rosenbaum, P. R. (1987). Model-based direct adjustment. Journal of the American Statistical Association, 82, 387–394. Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89, 846–866. Shao, J., & Steel, P. (1999). Variance estimation for survey data with composite imputation and non-negligible sampling fractions. Journal of the American Statistical Association, 94, 254–265.
Contents lists available at ScienceDirect
Journal of the Korean Statistical Society journal homepage: www.elsevier.com/locate/jkss
Using response probability and ratio imputation in the estimation under callbacks Hyeonah Park ∗ , Jongwoo Jeon Department of Statistics, Seoul National University, Seoul, 151-742, Republic of Korea
article
info
Article history: Received 1 March 2009 Accepted 4 November 2009 Available online 26 November 2009 AMS 2000 subject classifications: primary 62D05 secondary 62G09
abstract Although the response model has been frequently applied to nonresponse weighting adjustment or imputation, the estimation under callbacks has been relatively underdeveloped in the response model. We propose an estimator under callbacks using both the response probability and the ratio imputation and a replication variance estimator of the estimator. We also study the estimation of the response probability. A simulation study illustrates our technique. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
Keywords: Callbacks Ratio imputation Response probability Variance estimation
1. Introduction Generally, almost all surveys rely on callbacks to raise the response rate of persons who are not at home. The technique of callbacks and the estimation after callbacks have been considered numerous times by many survey researchers. A method of selecting subsamples from persons not at home and the estimation of double sampling were first considered by Hansen and Hurwitz (1946). Deming (1953) studied the estimation of population mean when responses are collected until the ith callback attempt. Groves (1989) provided an excellent summary of these approaches. Recently, Elliott, Little, and Lewitzky (2000) considered the subsampling callbacks, where an efficient subsampling strategy considering variance and cost from the repeated callback attempts was established. The estimators were relatively underdeveloped in the response model under callbacks though the response model has been applied to nonresponse weighting adjustment or imputation. There is a vast literature on nonresponse weighting adjustment or imputation under uniform or non-uniform response model. See, for example, Lipsitz, Ibrahim, and Zhao (1999), Rao and Sitter (1995), Rosenbaum (1987), Rao and Shao (1992), Robins, Rotnitzky, and Zhao (1994) and Shao and Steel (1999). In this article, we propose an estimator using the response probability and the ratio imputation in the response model under callbacks. We first prove the unbiasedness and the efficiency of the proposed estimator under the assumption that we know the true response probability. We also suggest a replication variance estimator of the estimator which satisfies the consistency under infinite sample size. Since the response probability is usually unknown, the estimated response probability can be used instead of the true response probability. For the estimation of the response probability, one can refer to Ekholm and Laaksonen (1991), and Iannacchione (2003). We also propose a consistent replication variance estimator of this estimator.
∗
Corresponding author. Tel.: +82 2 880 5718. E-mail address: [email protected] (H. Park).
1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.11.001
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
This paper is organized as follows. In Section 2, we introduce an estimator using the true response probability and the ratio imputation under callbacks and calculate the expectation and the variance of the estimator. In Section 3, we propose a replication variance estimator of the estimator of Section 2. We consider the estimation of the response probability assuming the logistic response model in Section 4, where the estimator and its variance estimator corresponding to the estimated response probability are also given. Finally, the numerical evaluation of the estimators in this paper is performed through a simulation study in Section 5. 2. Estimation using response probability and ratio imputation Let the population total be Y =
∑N
i=1
yi and the population mean be Y¯ = N −1
∑N
i=1
yi , where N is the population size
and yi is the value of the target variable of unit i. Let Yˆn be an estimator of the population total Y defined by Yˆn = i∈A wi yi , where n is the sample size, wi is the sampling weight of unit i and A = {1, 2, . . . , n} is the set of indices of the sample. We define the response indicator function under the first survey as
∑
Ri =
1, 0,
if unit i responds otherwise
for i ∈ A. Let πi = P (Ri = 1|i ∈ A) be the response probability of sample unit i under the first survey. We consider only one-step callback in this paper. We write the response indicator function under callback as
Ti =
1, 0,
if unit i responds otherwise
for i ∈ ANR , where ANR = {i : Ri = 0, i ∈ A} is the set of the indices of nonresponding units. Let pi = Pr (Ti = 1|i ∈ ANR ) be the response probability of sample unit i for i ∈ ANR under callback. We assume that Ri and Ti are ignorable such that πi and pi depend on an auxiliary variable zi but not on yi . In this section we also assume that all πi and pi are known priori. Now we suppose that there is another auxiliary variable xi that is related with the study variable yi and can be observed ∑n ∑n −1 ˜ throughout the sample. We first introduce some preliminary estimators. Let Y˜R = i=1 wi πi Ri yi and YT = i=1 wi (1 − 1 πi )−1 p− − Ri )Ti yi . For the auxiliary variable xi , we define X˜ T = i (1∑ n −1 and X˜ NR = (1 − Ri )xi . We define i=1 wi (1 − πi )
∑n
i=1
1 ˜ wi (1 − πi )−1 p− i (1 − Ri )Ti xi , XR =
∑n
i=1
wi πi−1 Ri xi
Y˜J = (W1 X˜ T−1 Y˜T + W2 X˜ R−1 Y˜R )X˜ NR and
ψ˜ = [Var (Y˜R ) + Var (Y˜J ) − 2Cov(Y˜R , Y˜J )]−1 [Var (Y˜J ) − Cov(Y˜R , Y˜J )],
(1)
where W1 = 1/2 and W2 = 1/2 are used in this article. Using W1 = 1 and W2 = 0, we see that the estimator Y˜J is equal to the estimator which was suggested by Park, Na, and Jeon (2008). Ratio imputation proposed by Rao and Sitter (1995) and by Rao (1996) uses X˜ R−1 Y˜R xi under πi = π , wi = n−1 and the estimator under this ratio imputation is (X˜ R−1 Y˜R )Xˆ n . If Y˜J in the proposed estimator is explained, the estimator using an auxiliary variable under callback is X˜ T−1 Y˜T−1 X˜ NR and the estimator under ratio imputation is X˜ R−1 Y˜R−1 X˜ NR . If πi are equal and wi = n−1 , we can see that the ratio imputation proposed by Rao and Sitter (1995) and by Rao (1996) is used in the estimator under ratio imputation, that is, in X˜ R−1 Y˜R−1 X˜ NR . Then, our proposed estimator is defined as
˜ Y˜R + (1 − ψ) ˜ Y˜J . Y˜M = ψ ˜ . Considering the proposed estimator from a different Note that the variance of kY˜R + (1 − k)Y˜J is minimized at k = ψ standpoint, the estimator Y˜M could be considered as the improved version of estimator under ratio imputation using response probability which is estimated by both responders and callbacks. We adopt the extended definition of the response indicator function introduced by Fay (1991). Conceptually, the response indicator functions Ri and Ti can be extended to the entire population. In this section we assume the following conditions: (A1) A sequence of finite populations and samples are defined as in Isaki and Fuller (1982). The finite populations satisfy that for some τ > 0 N −1
N −
θi2+τ = O(1),
i =1
where θi represents yi , xi and zi . The sampling mechanism satisfies E (Yˆn ) = Y .
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
513
(A2) For nonnegative constants C1 , C2 and C3 , C1 < πi < C2 and pi > C3 . (A3) The response indicator functions Ri and Ti are mutually independent, respectively, such that P (Ri = 1, Rj = 1) = P (Ri = 1)P (Rj = 1) and P (Ti = 1, Tj = 1) = P (Ti = 1)P (Tj = 1) for different i and j. (A4) The sampling mechanism satisfies that for nonnegative constants D1 , D2 , D3 and D4 D1 < max (N −1 nwi ) < D2 1≤i≤N
and D3 < N −2 nVar (Yˆn ) < D4 , where the variance is calculated from the sampling mechanism. In the following theorem we deal with the expectation and the variance of our proposed estimator Y˜M . Theorem 1. Under the assumptions (A1)–(A4), E (Y˜M ) = Y + o(n−1/2 N )
(2)
Var (Y˜M ) = Var (Yˆn ) + (K1 + K2 + 2K3 )−1 (K1 K2 − K32 ) + o(n−1 N 2 ),
(3)
and
where
K1 = E
n −
w (πi
−1
2 i
− 1)
,
y2i
i=1
n −
wi2 [W2 πi−1 (yi − rxi ) − (1 − πi )−1 (W1 yi + W2 rxi )]2 πi (1 − πi ) i =1 n − 2 2 −1 −1 2 + W1 wi (1 − πi ) (pi − 1)(yi − rxi ) ,
K2 = E
i=1
n −
w πi yi [yi (πi − W2 ) + W2 rxi ] 2 i
−1
and r = X −1 Y for X =
∑N
K3 = E
i=1
i=1
xi .
Proof. Note that X˜ T−1 Y˜T X˜ NR = Y˜T + r˜1 (X˜ NR − X˜ T ) and X˜ R−1 Y˜R X˜ NR = Y˜R + r˜2 (X˜ NR − X˜ R ), where r˜1 = X˜ T−1 Y˜T and r˜2 = X˜ R−1 Y˜R . Under (A1), (A3) and (A4), E [(X˜ NR − X ) ] = Var (Xˆ n ) + E 2
n −
w (πi 2 i
−1
− 1) xi
−1 2
i=1
= O(n−1 N 2 ), E [(X˜ T − X )2 ] = Var (Xˆ n ) + E
n − i=1
= O(n−1 N 2 )
w (1 − πi ) (πi + pi − 1) 2 i
−1
−1
x2i
514
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
and E [(X˜ R − X )2 ] = Var (Xˆ n ) + E
n −
w (πi 2 i
−1
− 1)
x2i
i=1
= O(n−1 N 2 ). where Xˆ n =
∑n
i=1
wi xi . Similarly, we also obtain that
Y˜T − Y = OP (n−1/2 N ) and
Y˜R − Y = OP (n−1/2 N ).
Then, by Taylor’s expansion, we have r˜1 − r = X −1 [(Y˜T − Y ) − r (X˜ T − X )] + oP (n−1/2 ) = OP (n−1/2 ), r˜2 − r = X
−1
[(Y˜R − Y ) − r (X˜ R − X )] + oP (n−1/2 ) = OP (n−1/2 )
(4) (5)
and Y˜J = W1 [Y˜T + r (X˜ NR − X˜ T )] + W2 [Y˜R + r (X˜ NR − X˜ R )] + oP (n−1/2 N ). Observe that
˜ Y˜R − Yˆn ) + (1 − ψ){ ˜ W1 [(Y˜T − Yˆn ) + r (X˜ NR − X˜ T )] + W2 [(Y˜R − Yˆn ) + r (X˜ NR − X˜ R )]} + oP (n−1/2 N ). Y˜M − Yˆn = ψ( This, together with (A1), implies (2). ˜, Let Y˜J′ = W1 [Y˜T + r (X˜ NR − X˜ T )] + W2 [Y˜R + r (X˜ NR − X˜ R )]. By definition of ψ Var (Y˜M ) = [Var (Y˜R ) + Var (Y˜J′ ) − 2Cov(Y˜R , Y˜J′ )]−1 [Var (Y˜R )Var (Y˜J′ ) − Cov(Y˜R , Y˜J′ )2 ] + o(n−1 N 2 ) since Y˜J − Y˜J′ = oP (n−1/2 N ). Observe that from (A3), Var (Y˜R ) = Var (Yˆn ) + K1 , Var (Y˜J′ ) = Var (Yˆn ) + K2 , Cov(Y˜R , Y˜J′ ) = Var (Yˆn ) − K3 . Thus, the result (3) follows immediately.
3. Variance estimation using known response probability In this section we propose a method of estimating the variance of the estimator when we know the response probability. Note that the variance must be estimated to calculate the efficiency of the proposed estimator. We consider the replication method such as jackknife, since it is well known that the replication method is good to estimate the variances of complex estimators. First, we illustrate a replication variance estimator for Yˆn . Let an estimator of Var (Yˆn ) be Vˆ (Yˆn ) =
L −
ck (Yˆn(k) − Yˆn )2 ,
k=1
where L is the number of replications, ck is a factor associated with the kth replication determined by the replication method (k) and Yˆn is the kth estimator of Y based on the observations included in the kth replication, that is, Yˆn(k) =
n −
wi(k) yi ,
i =1
(k)
where wi
is the replication weight for the ith unit in the kth replication. For example, if the inclusion probability is N −1 n
and wi = n−1 N, then the standard jackknife variance estimator Vˆ (Yˆn ) is defined by L = n, ck = (1 − N −1 n)n−1 (n − 1), wi(k) = (n − 1)−1 nwi for i ̸= k and wk(k) = 0. We suggest a replication estimator for the variance of Y˜M by Vˆ (Y˜M ) =
L −
(k)
ck (Y˜M − Y˜M )2 ,
k=1
where (k)
(k)
˜ Y˜R Y˜M = ψ
˜ Y˜J(k) + (1 − ψ)
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
515
(k) −1 (k) (k) Ri yi and Y˜J = (W1 X˜ T(k)−1 Y˜T(k) + W2 X˜ R(k)−1 Y˜R(k) )X˜ NR . Here, analogously as before, Y˜T(k) = = i =1 i =1 w i π i ∑ ∑ ∑ ( k (k) ( k ) ( k ) n = ni=1 wi(k) wi (1 − πi )−1 pi −1 (1 − Ri )Ti yi , X˜ T = i=1 wi (1 − πi )−1 pi −1 (1 − Ri )Ti xi , X˜ R ) = ni=1 wi(k) πi−1 Ri xi and X˜ NR (1 − πi )−1 (1 − Ri )xi . Note that the superscript (k) denotes the kth replication. (k)
for Y˜R
∑n
∑n
(k)
We assume that the variance of a linear estimator of the total is a quadratic function of y, that is, N −2 nVar (Yˆn ) =
N − N −
Ωij yi yj ,
(6)
i=1 j=1
where the coefficients Ωij satisfy max Ωii = O(N −1 )
(7)
1≤i≤N
and N −
|Ωij | = O(N −1 ).
(8)
i =1
For example, the simple random sampling with wi = n−1 N satisfies (7) and (8) because
Ωij =
N − 1 ( 1 − N − 1 n) −N −1 (N − 1)−1 (1 − N −1 n)
if i = j if i ̸= j .
In order to establish the consistency of Vˆ (Y˜M ), we first prove the consistency of the variance estimator of Y˜R in the following lemma. Lemma 1. Suppose that the conditions (A1)–(A4) are satisfied. We assume that for any y with bounded fourth moment, E [Var (Yˆn )−1 Vˆ (Yˆn ) − 1]2 = o(1),
(9)
where the expectation is calculated under the given sampling mechanism. Assume also that N −1 n = o(1).
(10)
Then, Vˆ (Y˜R ) =
L −
(k)
ck (Y˜R
− Y˜R )2 = Var (Y˜R ) + oP (n−1 N 2 ).
(11)
k=1
Proof. From (A4) and (9), Vˆ (Y˜R ) = Var (Y˜R |R1 , . . . , RN ) + oP (n−1 N 2 ). From (10), Var [E (Y˜R |R1 , . . . , RN )] = o(n−1 N 2 ). Then, (11) follows immediately if we prove Var [N −2 nVar (Y˜R |R1 , . . . , RN )] = o(1).
(12)
From (6) and independence of Ri ’s, we can see that Var [N −2 nVar (Y˜R |R1 , . . . , RN )] =
N − N − (Ωij2 + Ωij Ωji )Var (yπ i yπ j ) i=1 j=1
≤ 2 max Var (yπ i yπ j ) max |Ωij | 1≤i,j≤N
1≤i,j≤N
N − N −
|Ωij |,
i =1 j =1
where yπ i = πi−1 Ri yi and the variances are taken with respect to Ri ’s. From (A1) with τ > 2 and (A2), max1≤i,j≤N Var (yπ i yπ j ) = O(1). By (7) and the nonnegative definiteness of Ω = (Ωij ), max1≤i,j≤N |Ωij | = O(N −1 ). These, together with (8), imply (12). Secondly, we deal with the variance estimator for Y˜J . Here, an additional condition is listed.
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
(A5) For the replication factors, max ck−1 = O(L)
1≤k≤L
and E [{ck (Yˆn(k) − Yˆn )2 }2 ] < Cy L−2 {Var (Yˆn )}2 for Yˆn corresponding to yi with finite fourth moment and some constant Cy . Lemma 2. Assume the conditions (A1)–(A5), (9) and (10). Then, Vˆ (Y˜J ) =
L −
(k)
ck (Y˜J
− Y˜J )2 = Var (Y˜J ) + oP (n−1 N 2 ).
(13)
k=1
(k)
Proof. Split Y˜J (k)
Y˜J
− Y˜J as
− Y˜J = (Y˜J(k) − Y˜J′(k) ) + (Y˜J′(k) − Y˜J′ ) + (Y˜J′ − Y˜J ),
(k) (k) − X˜ R(k) )]. Observe that for r˜1(k) = Y˜T(k) /X˜ T(k) and r˜2(k) = Y˜R(k) /X˜ R(k) , − X˜ T(k) )] + W2 [Y˜R(k) + r (X˜ NR = W1 [Y˜T(k) + r (X˜ NR (k) − X˜ NR ) − (X˜ T(k) − X˜ T )] (Y˜J(k) − Y˜J′(k) ) + (Y˜J′ − Y˜J ) = W1 (˜r1(k) − r )[(X˜ NR (k) − X˜ NR ) − (X˜ R(k) − X˜ R )] + (˜r2(k) − r˜2 )(X˜ NR − X˜ R ) +(˜r1(k) − r˜1 )(X˜ NR − X˜ T ) + W2 (˜r2(k) − r )[(X˜ NR ′(k)
where Y˜J
= oP (n−1/2 N ) since (k)
(k) − X˜ T = OP (n−1/2 N ), X˜ R(k) − X˜ R = OP (n−1/2 N ), X˜ NR − X˜ NR = OP (n−1/2 N ), (k) (k) (k) r˜1 − r˜1 = X˜ T−1 {(Y˜T − Y˜T ) − r˜1 (X˜ T − X˜ T )} + oP (n−1/2 ) −1/2 = OP (n )
X˜ T
and (k)
(k)
r˜2 − r˜2 = X˜ R−1 {(Y˜R
− Y˜R ) − r˜2 (X˜ R(k) − X˜ R )} + oP (n−1/2 )
= OP (n−1/2 ), which are consequence of (A3)–(A5), (4) and (5) and the fact that X˜ NR − X˜ T = Op (n−1/2 N ) and X˜ NR − X˜ R = Op (n−1/2 N ). Then, from (9) and (A5), Vˆ (Y˜J ) =
L −
′(k)
ck (Y˜J
− Y˜J′ )2 + oP (n−1 N 2 )
k=1
= Var (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) + oP (n−1 N 2 ). If we prove Var [E (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN )] = o(n−1 N 2 )
(14)
Var [N −2 nVar (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN )] = o(1),
(15)
and
then we have Var (Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) = Var (Y˜J′ ) + oP (n−1 N 2 ), which implies (13) since Y˜J − Y˜J′ = oP (n−1/2 N ). If we use the same steps of Lemma 1, we can easily prove (13).
In the following theorem, the consistency of the variance estimator Vˆ (Y˜M ) is established. Theorem 2. Under the same conditions as in Lemma 2, Vˆ (Y˜M ) = Var (Y˜M ) + oP (n−1 N 2 ).
(16)
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
517
Proof. Observe that L −
(k) ˜2 ck (Y˜M − Y˜M )2 = ψ
L −
k=1
(k)
ck (Y˜R
˜ 2 − Y˜R )2 + (1 − ψ)
L −
k=1
(k)
ck (Y˜J
− Y˜J )2
k=1
˜ 1 − ψ) ˜ + 2ψ(
L −
(k)
ck (Y˜R
− Y˜R )(Y˜J(k) − Y˜J ).
k=1
Using the similar arguments in Lemma 2, we obtain that L −
(k)
ck (Y˜R
+ Y˜J(k) − Y˜R − Y˜J )2 =
k=1
L −
(k)
ck (Y˜R
+ Y˜J′(k) − Y˜R − Y˜J′ )2 + oP (n−1 N 2 ).
k=1
By (A4) and (9), L −
(k)
ck (Y˜R
+ Y˜J′(k) − Y˜R − Y˜J′ )2 = Var (Y˜R + Y˜J′ |T1 , . . . , TN , R1 , . . . , RN ) + oP (n−1 N 2 )
k=1
= Var (Y˜R + Y˜J ) + oP (n−1 N 2 ), where the same steps in Lemmas 1 and 2 and the fact that Y˜J − Y˜J′ = oP (n−1/2 N ) have been used for the last equality. This, together with (11) and (13), implies that L −
(k)
ck (Y˜R
− Y˜R )(Y˜J(k) − Y˜J ) = Cov(Y˜R , Y˜J ) + oP (n−1 N 2 ),
k=1
which implies (16).
4. Estimation using estimated response probability In many practical situations, it is impossible to know the response probability and the weight. In this section, we estimate the response probability and the weight. We assume the parametric logistic model for the response probability:
πi = π (zi ; α) = (1 + exp(−ziT α))−1 , where zi is the value of an auxiliary variable for unit i and α = (α1 , . . . , αp )T . The response probability pi under callback is assumed to be a constant p. Let πˆ i = π (zi ; α) ˆ be the estimated response probability, where αˆ satisfies n1/2 (αˆ − α) = n−1/2
n −
H (zi , Ri ; α) + oP (1),
(17)
i=1
where E [H (zi , Ri ; α)] = 0 and E H (zi , Ri ; α)H (zi , Ri ; α)T is positive definite (cf. (Kim & Park, 2006)). For example, the logistic regression model defined by πi = [1 + exp(−α1 − α2 zi )]−1 satisfies
H (Ri ; α) = n[I (α1 , α2 )]−1 (Ri − πi )(1, zi )T and I (α1 , α2 ) = E
n −
πi (1 − πi )(1, zi )T (1, zi ) .
i =1
∗ The response probability pi is estimated by pˆ = R∗−1 T ∗ , where R∗ = i=1 wi (1 − Ri ) and T = i=1 wi (1 − Ri )Ti . The estimators with the estimated response probabilities plugged-in are defined analogously as before. We define ∑n ∑n YˆR = ˆ i−1 Ri yi and YˆJ = (W1 Xˆ T−1 YˆT + W2 Xˆ R−1 YˆR )Xˆ NR , where Xˆ T = ˆ i )−1 pˆ −1 (1 − Ri )Ti xi , YˆT = i =1 w i π i =1 w i ( 1 − π
∑n
∑n
∑ wi πˆ i−1 Ri xi and Xˆ NR = ni=1 wi (1 − πˆ i )−1 (1 − Ri )xi . We discuss the asymptotic properties of YˆR and YˆJ in the following Lemma.
∑n
i =1
wi (1 − πˆ i )−1 pˆ −1 (1 − Ri )Ti yi , Xˆ R =
∑n
i=1
Lemma 3. Assume the same conditions as in Lemma 2. Let us define
ΓR =
N − i=1
ΓT =
πi (∂πi−1 /∂α)yi ,
ΓRX =
N −
πi (∂πi−1 /∂α)xi ,
i=1
ΓY =
N −
pyi ,
ΓX =
i=1
N N − − (1 − πi )(∂(1 − πi )−1 /∂α)yi and ΓTX = (1 − πi )(∂(1 − πi )−1 /∂α)xi . i=1
i=1
N − i=1
pxi ,
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
Then, YˆR − Y˜R = (αˆ − α)T ΓR + oP (n−1/2 N )
(18)
and YˆJ − Y˜J = W1 T¯ −1 [(R∗ − R¯ ) − p−1 (T ∗ − T¯ )](ΓY − r ΓX )
+ (αˆ − α)T [W1 ΓT + W2 (ΓR + r ΓTX − r ΓRX )] + oP (n−1/2 N ), ∑N ∑N ¯ where T¯ = i=1 (1 − πi )p, R = i=1 (1 − πi ).
(19)
Proof. By (17) and Taylor expansion,
πˆ i−1 − πi−1 = (αˆ − α)T (∂πi−1 /∂α) + oP (n−1/2 ). Then, using (A1), (A3) and (A4), we obtain (18). Observe that YˆJ − Y˜J = W1 {(YˆT − Y˜T ) + rˆ1 [(Xˆ NR − X˜ NR ) − (Xˆ T − X˜ T )] + (ˆr1 − r˜1 )(X˜ NR − X˜ T )} + W2 {(YˆR − Y˜R )
+ rˆ2 [(Xˆ NR − X˜ NR ) − (Xˆ R − X˜ R )] + (ˆr2 − r˜2 )(X˜ NR − X˜ R )}, where rˆ1 = YˆT /Xˆ T and rˆ2 = YˆR /Xˆ R . By Taylor’s expansion and the assumed conditions,
(1 − πˆ i )−1 − (1 − πi )−1 = (αˆ − α)T (∂(1 − πi )−1 /∂α) + oP (n−1/2 ) and −1 pˆ −1 − p−1 = T¯ ′ [R∗ − R¯ − p−1 (T ∗ − T¯ )] + oP (n−1/2 ).
Then, we have YˆT − Y˜T = T¯ −1 [R∗ − R¯ − p−1 (T ∗ − T¯ )]ΓY + (αˆ − α)T ΓT + oP (n−1/2 N ), Xˆ T − X˜ T = T¯ −1 [R∗ − R¯ − p−1 (T ∗ − T¯ )]ΓX + (αˆ − α)T ΓTX + oP (n−1/2 N ) and Xˆ R − X˜ R = (αˆ − α)T ΓRX + oP (n−1/2 N ). We also obtain that Xˆ NR − X˜ NR = (αˆ − α)T ΓTX + oP (n−1/2 N ), rˆ1 − r˜1 = X˜ T−1 [(YˆT − Y˜T ) − r˜1 (Xˆ T − X˜ T )] + oP (n−1/2 ) and rˆ2 − r˜2 = X˜ R−1 [(YˆR − Y˜R ) − r˜2 (Xˆ R − X˜ R )] + oP (n−1/2 ). These, together with (4), (5) and the fact that X˜ NR − X˜ T = OP (n−1/2 N ) and X˜ NR − X˜ R = OP (n−1/2 N ), immediately imply (19). (k)
Let πˆ i
L −
= π (zi ; αˆ (k) ) be the kth replication of πˆ i , where αˆ (k) = (αˆ 1(k) , . . . , αˆ p(k) )T is the kth replication of αˆ satisfying ck αˆ (k) − αˆ
(k) T αˆ − αˆ = Σα + oP n−1
(20)
k=1
for Σα = E [E {(αˆ − E [E (α)| ˆ A])(αˆ − E [E (α)| ˆ A])T }|A]. In jackknife method and logistic regression model, a method to fine αˆ (k) is to look for a maximum likelihood estimator of α after the kth observation is deleted. Note that the replication variance estimators can be defined analogously as before, where the response probabilities are replaced by the kth replicates of the estimated ones. For example, Vˆ (YˆR ) =
L −
(k)
ck (YˆR
− YˆR )2 ,
k=1
(k) where YˆR =
∑n
i=1
Cˆ (YˆR , YˆJ ) =
wi(k) πˆ i(k)−1 Ri yi . The covariance of YˆR and YˆJ is estimated by
L − k=1
(k)
ck (YˆR
− YˆR )(YˆJ(k) − YˆJ ),
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
(k)
(k)−1 (k)
(k)−1 (k)
(k)
(k)
(k)
(k)
519
(k)
where YˆJ = (W1 Xˆ T YˆT + W2 Xˆ R YˆR )Xˆ NR with YˆT , Xˆ T ,Xˆ R and Xˆ NR analogously defined using the kth replicates of the estimated response probabilities. Finally, the proposed estimator is given by
ˆ YˆR + (1 − ψ) ˆ YˆJ , YˆM = ψ ˆ = [Vˆ (YˆR ) + Vˆ (YˆJ ) − 2Cˆ (YˆR , YˆJ )]−1 [Vˆ (YˆJ ) − Cˆ (YˆR , YˆJ )]. where ψ We first discuss the consistency of Vˆ (YˆR ) in the following Lemma. Lemma 4. Assume the conditions of Lemma 2. Assume also that L −
(k)
ck (Yˆn(k) − Yˆn )(αˆ l
− αˆ l ) = Cov(Yˆn , αˆ l ) + oP (n−1 N )
(21)
k=1
for 1 ≤ l ≤ p. Then, (k)
YˆR
− YˆR = Y˜R(k) − Y˜R + (αˆ (k) − α) ˆ T ΓR + oP (n−1/2 N )
(22)
and Vˆ (YˆR ) = Var (YˆR ) + oP (n−1 N 2 ). (k)
Proof. Write YˆR (k)
YˆR
(23)
− YˆR as
− YˆR = (YˆR(k) − Y˜R(k) ) + (Y˜R(k) − Y˜R ) + (Y˜R − YˆR ).
From (17) and (20),
πˆ i(k)−1 − πˆ i−1 = (αˆ (k) − α) ˆ T (∂πi−1 /∂α) + oP (n−1/2 ) which, together with the assumed conditions, implies that
ˆ T ΓR + oP (n−1/2 N ) (YˆR(k) − Y˜R(k) ) + (Y˜R − YˆR ) = (αˆ (k) − α) and, hence, (22). Observe that by (A5) L −
(k)
ck (YˆR
− YˆR )2 =
L −
(k)
ck (Y˜R
− Y˜R )2 +
+2
ck [(αˆ (k) − α) ˆ T ΓR ]2
k=1
k=1
k=1
L −
L −
(k)
ck (Y˜R
− Y˜R )[(αˆ (k) − α) ˆ T ΓR ] + oP (n−1 N 2 ).
k=1
From (17) and (21), L −
˜ (k)
ck (YR
(k)
− Y˜R )(αˆ l − αˆ l )ΓRl = Cov Y˜R , n ΓRl −1
k=1
n −
Hli |R1 , . . . , RN
+ oP (n−1 N 2 )
i =1
where ΓRl is the lth element of ΓR for l = 1, . . . , p. From (10), we have
Cov E (Y˜R |R1 , . . . , RN ), E
n
−1
ΓRl
n −
Hli |R1 , . . . , RN
= o(n−1 N 2 ).
i =1
The similar arguments as in Lemma 1 lead to L −
(k)
ck (Y˜R
− Y˜R )[(αˆ (k) − α) ˆ T ΓR ] = Cov(Y˜R , (αˆ − α)T ΓR ) + oP (n−1 N 2 ),
k=1
which, together with (11), (18) and (20), implies (23).
We state the following lemma regarding the consistency of Vˆ (YˆJ ) without proof since the technical details resembles that in Lemma 4.
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H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
Lemma 5. Assume the same conditions as in Lemma 4. Then, (k)
YˆJ
− YˆJ = (Y˜J(k) − Y˜J ) + W1 T¯ −1 [(R∗(k) − R∗ ) − p−1 (T ∗(k) − T ∗ )](ΓY − r ΓX ) + (αˆ (k) − α) ˆ T [W1 ΓT + W2 (ΓR + r ΓTX − r ΓRX )] + oP (n−1/2 N )
(24)
and Vˆ (YˆJ ) =
L −
(k)
ck (YˆJ
− YˆJ )2 = Var (YˆJ ) + oP (n−1 N 2 ),
(25)
k=1
where R∗(k) =
∑n
i =1
wi(k) (1 − Ri ) and T ∗(k) =
∑n
i =1
wi(k) (1 − Ri )Ti .
The variance estimator for YˆM is defined by Vˆ (YˆM ) =
L −
(k)
ck (YˆM − YˆM )2 ,
k=1
(k) ˆ YˆR(k) + (1 − ψ) ˆ YˆJ(k) . In the following theorem, we deal with the asymptotic unbiasedness of YˆM and the where YˆM = ψ consistency of the variance estimator Vˆ (YˆM ).
Theorem 3. Under the same conditions as in Lemma 4, E (YˆM ) = Y + o(n−1/2 N )
(26)
Vˆ (YˆC ) = Var (YˆC ) + oP (n−1 N 2 ).
(27)
and
Proof. Using the same techniques of Theorem 2 and Lemma 4, we can obtain that L −
(k)
ck (YˆR
+ YˆJ(k) − YˆR − YˆJ )2 = Var (YˆR + YˆJ ) + oP (n−1 N 2 ),
k=1
which, together with (23) and (25), implies that Cˆ (YˆR , YˆJ ) = Cov(YˆR , YˆJ ) + oP (n−1 N 2 ) and, hence, that
ψˆ − ψ = oP (1) for ψ = [Var (YˆR ) + Var (YˆJ ) − 2Cov(YˆR , YˆJ )]−1 [Var (YˆJ ) − Cov(YˆR , YˆJ )]. Write
ˆ − ψ)(YˆR − Y ) + [(1 − ψ) ˆ − (1 − ψ)](YˆJ − Y ) + ψ YˆR + (1 − ψ)YˆI . YˆM = (ψ Then, using (18) and (19), we obtain that YˆM = ψ YˆR + (1 − ψ)YˆJ + oP (n−1/2 N ), which implies (26). Furthermore, it is easily obtained from (22) and (24) that (k)
(k)
YˆM − YˆM = ψ(YˆR
− YˆR ) + (1 − ψ)(YˆJ(k) − YˆJ ) + oP (n−1/2 N ).
Finally, using (23) and (25), we obtain (27).
5. Simulation results We show the results of a limited simulation study to test our theory. In the simulation study, B = 1000 samples of size n = 100 are generated by yi = 4xi +
√
xi ϵi ,
where xi ∼ Uniform(0, 1), ϵi ∼ N (0, 1) for i = 1, . . . , n, and xi and ϵi are independent. For a parameter model of the response probability under the first survey, we suggest the logistic model πi = [1 + exp(−α1 − α2 zi )]−1 , where zi ∼ Uniform(0, 1) and the value of α is assumed to be (α1 , α2 ) = (0, −2), (−0.5, 0.5) and (−0.5,1). Consequently, the overall response rate becomes 0.28, 0.44 and 0.50, respectively. We use constants 0.3, 0.4 and 0.5 for the response probability p under callback. We consider the maximum likelihood estimator to estimate α and calculate the value iteratively using the Newton–Raphson method. The response probability p under callback is estimated as the response rate among nonresponses. For the variance estimator, we use the standard jackknife method under simple random sampling, where ck is n−1 (n − 1) (k) and wi is defined as (n − 1)−1 nwi for i ̸= k and 0 for i = k.
H. Park, J. Jeon / Journal of the Korean Statistical Society 39 (2010) 511–521
521
Table 1 MSE (Y¯D )/MSE (Y˜M ) and MES (Y¯D )/MSE (YˆM ).
(α1 , α2 )
0.3
0.4
0 .5
(0.0, −2.0)
1.300 1.309 1.458 1.381 1.343 1.302
1.178 1.190 1.355 1.283 1.253 1.223
1.088 1.089 1.274 1.208 1.200 1.173
(−0.5, 0.5) (−0.5, 1.0)
Table 2 Relative means (t-statistics, in parentheses) of Vˆ (Y˜M ) and Vˆ (YˆM ).
(α1 , α2 )
0.3
0.4
0.5
(0.0, −2.0)
1.020(0.462) 0.948(−1.096) 1.051(1.101) 1.004(0.084) 1.050(1.063) 0.996(−0.098)
1.004(0.095) 0.931(−1.433) 1.036(0.785) 0.991(−0.207) 1.033(0.706) 0.988(−0.259)
0.995(−0.119) 0.910(−1.886) 1.019(0.432) 0.976(−0.556) 1.019(0.410) 0.976(−0.536)
(−0.5, 0.5) (−0.5, 1.0)
To know the properties of the variance estimator, we compute the relative mean and t-statistic. The relative mean of the variance estimator is that the empirical mean of the variance estimator divided by the empirical variance of the point estimator. The t-statistic for the variance estimator is the empirical bias of the variance estimator divided by the empirical standard error of the empirical bias, which was provided by Kim (2004). Using B samples of (yi , xi , ϵi , Ri , Ti ), i = 1, . . . , n, wi = n−1 and W1 = 0.5, we compute the empirical values of MSE (Y¯D )/MSE (Y˜M ) and MSE (Y¯D )/MSE (YˆM ), where MSE (Y¯D ) is the mean square error of the estimator suggested by Deming (1953). We also compute the relative means and t-statistics of Vˆ (Y˜M ) and Vˆ (YˆM ). Each cell in Table 1 contains MSE (Y¯D )/MSE (Y˜M ) and MSE (Y¯D )/MSE (YˆM ) in this order for changing response probabilities πi and p. Each cell in Table 2 shows the relative means (t-statistics, in parentheses) of Vˆ (Y˜M ) and Vˆ (YˆM ) in this order for changing response probabilities. From observing the simulation results in Table 1 it can be judged that MSE (Y˜M ) and MSE (YˆM ) are smaller than MSE (Y¯D ). From Table 2 we can see that Vˆ (Y˜M ) and Vˆ (YˆM ) are consistent estimators for Var (Y˜M ) and Var (YˆM ), respectively. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0063964). References Deming, W. E. (1953). On a probability mechanism to attain an economic balance between the resultant error of response and the bias of nonresponse. Journal of the American Statistical Association, 48, 743–772. Ekholm, A., & Laaksonen, S. (1991). Weighting via response modeling in the finnish household budget survey. Journal of Official Statistics, 7, 325–337. Elliott, M. R., Little, R. J. A., & Lewitzky, S. (2000). Subsampling callbacks to improve survey efficiency. Journal of the American Statistical Association, 95, 730–738. Fay, R. E. (1991). A design-based perspective on missing data variance. In The ASA proceedings of bureau of the census annual research conference (pp. 429–440). Washington (DC): US Bureau of the Census. Groves, R. M. (1989). Survey errors and survey costs. New York: Wiley. Hansen, M. H., & Hurwitz, W. N. (1946). The problem of nonresponse in sample surveys. Journal of the American Statistical Association, 41, 517–529. Isaki, C. T., & Fuller, W. A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77, 89–96. Iannacchione, V. G. (2003). Sequential weight adjustment for location and cooperation propensity for the 1995 national survey of family growth. Journal of Official Statistics, 19, 31–43. Kim, J. K. (2004). Finite sample properties of multiple imputation estimators. The Annals of Statistics, 32, 766–783. Kim, J. K., & Park, H. (2006). Imputation using response probability. Canadian Journal of Statistics, 34, 171–182. Lipsitz, S. R., Ibrahim, J. G., & Zhao, L. P. (1999). A weighted estimating equation for missing covariate data with properties similar to maximum likelihood. Journal of the American Statistical Association, 94, 1147–1160. Park, H., Na, S., & Jeon, J. (2008). Estimation using response probability under callbacks. Statistics and probability letters, 78, 1735–1741. Rao, J. N. K. (1996). On variance estimation with imputed survey data. Journal of the American Statistical Association, 91, 499–506. Rao, J. N. K., & Sitter, R. R. (1995). Variance estimation under two-phase sampling with application to imputation for missing data. Biometrika, 82, 453–460. Rao, J. N. K., & Shao, J. (1992). Jackknife variance estimation with survey data under hot deck imputation. Biometrika, 79, 811–822. Rosenbaum, P. R. (1987). Model-based direct adjustment. Journal of the American Statistical Association, 82, 387–394. Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89, 846–866. Shao, J., & Steel, P. (1999). Variance estimation for survey data with composite imputation and non-negligible sampling fractions. Journal of the American Statistical Association, 94, 254–265.