TY - JOUR

T1 - Bayes and empirical Bayes estimation with errors in variables

AU - Zhang, Shunpu

AU - Karunamuni, Rohana J.

N1 - Funding Information:
where the random disturbance or the random error e is independent of X. Assume that e has a known distribution F~. We investigate the problem of estimation of 0 based on Y with the squared error loss. In this paper, it is of our interest to develop both Bayes (in the case when G is known) and empirical Bayes (in the case when G is unknown) estimators for the preceding problem. In Section 2 below, we obtain the Bayes * Corresponding author. I Research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

PY - 1997/4/15

Y1 - 1997/4/15

N2 - Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter θ. Assume that the parameter θ has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + ε, where the measurement error ε is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of θ based on the contaminated observation Y. We obtain an expression for the Bayes estimator when the prior G is known. For the case G is completely unknown, an empirical Bayes estimator is proposed based on a sequence of observations Y1,Y2,...,Yn, where Yi's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.

AB - Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter θ. Assume that the parameter θ has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + ε, where the measurement error ε is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of θ based on the contaminated observation Y. We obtain an expression for the Bayes estimator when the prior G is known. For the case G is completely unknown, an empirical Bayes estimator is proposed based on a sequence of observations Y1,Y2,...,Yn, where Yi's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.

KW - Asymptotically optimal

KW - Bayes

KW - Empirical Bayes

KW - Kernel density estimates

KW - Squared error loss estimation

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U2 - 10.1016/S0167-7152(96)00106-X

DO - 10.1016/S0167-7152(96)00106-X

M3 - Article

AN - SCOPUS:0031569671

VL - 33

SP - 23

EP - 34

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 1

ER -