### Abstract

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 Y_{1},Y_{2},...,Y_{n}, where Y_{i}'s are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.

Original language | English (US) |
---|---|

Pages (from-to) | 23-34 |

Number of pages | 12 |

Journal | Statistics and Probability Letters |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Apr 15 1997 |

### Fingerprint

### Keywords

- Asymptotically optimal
- Bayes
- Empirical Bayes
- Kernel density estimates
- Squared error loss estimation

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Statistics and Probability Letters*,

*33*(1), 23-34. https://doi.org/10.1016/S0167-7152(96)00106-X

**Bayes and empirical Bayes estimation with errors in variables.** / Zhang, Shunpu; Karunamuni, Rohana J.

Research output: Contribution to journal › Article

*Statistics and Probability Letters*, vol. 33, no. 1, pp. 23-34. https://doi.org/10.1016/S0167-7152(96)00106-X

}

TY - JOUR

T1 - Bayes and empirical Bayes estimation with errors in variables

AU - Zhang, Shunpu

AU - Karunamuni, Rohana J.

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

UR - http://www.scopus.com/inward/record.url?scp=0031569671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031569671&partnerID=8YFLogxK

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 -