Empirical Bayes estimation of the scale parameter in a Pareto distribution

Tiwari, Ram C.; Zalkikar, Jyoti

doi:10.1016/0167-9473(90)90006-4

Cited by 18 publications

(6 citation statements)

References 8 publications

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“…Pareto mixing of form (20) was considered in Tiwari and Zalkikar (1990) where, in the spirit of Nogami (1988), a kernel-based empirical Bayes approach was proposed to estimate θ. Observe that by (9), we have…”

Section: Examples and Derivation Of The Functions U K (X)mentioning

confidence: 99%

“…On the other hand, in nonparametric empirical Bayes methods, the prior distribution and its parametric form are not specified at all. One of the techniques to nonparametric empirical Bayes methods is to estimate the top Ψ(y) and bottom p(y) of estimator (1) separately and then compute the ratio, this includes Singh (1976Singh ( , 1979, Nogami (1988), Tiwari and Zalkikar (1990), Ma and Balalrishnan (2000) among others. The other set of methods rely on estimating the ratio in estimator (1) directly and this one includes Robbins (1983), Pensky and Ni (2000), Pensky and Alotaibi (2005) and…”

Section: Introductionmentioning

confidence: 99%

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Nonparametric empirical Bayes estimation based on generalized Laguerre series

Benhaddou¹,

Connell²

2021

Preprint

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In this work, we delve into the nonparametric empirical Bayes theory and approximate the classical Bayes estimator by a truncation of the generalized Laguerre series and then estimate its coefficients by minimizing the prior risk of the estimator. The minimization process yields a system of linear equations the size of which is equal to the truncation level. We focus on the empirical Bayes estimation problem when the mixing distribution, and therefore the prior distribution, has a support on the positive real half-line or a subinterval of it. By investigating several common mixing distributions, we develop a strategy on how to select the parameter of the generalized Laguerre function basis so that our estimator possesses a finite variance. We show that our generalized Laguerre empirical Bayes approach is asymptotically optimal in the minimax sense. Finally, our convergence rate is compared and contrasted with several results from the literature.

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Section: Examples and Derivation Of The Functions U K (X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonparametric empirical Bayes estimation based on generalized Laguerre series

Benhaddou¹,

Connell²

2021

Preprint

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“…For this purpose the lower bounds for the mean squared error in the class of all possible empirical Bayes estimators are derived. Singh [23], Singh & Wei [24] and Tiwari L· Zalkikar [25] investigated the lower bounds for the mean squared risk. However, the shortcoming of the approach used by these authors is that the lower bounds are obtained for the selected method of empirical Bayes estimation only.…”

Section: Introductionmentioning

confidence: 99%

“…After Robbins [22] formulated the empirical Bayes estimation problem many papers were devoted to the empirical Bayes method. Empirical Bayes estimators were constructed for many familiar probability distributions (see for instance, [1], [2], [5], [7], [8], [12], [13], [14], [21], [23], [24], [25], [26]). In some papers (see [10], [11], [14], [21], [23], [24]) the estimators are given along with the upper bounds for the risk function which leads to the following question: what is the best convergence rate of empirical Bayes estimators for a given family of conditional distributions?…”

Section: Introductionmentioning

confidence: 99%

Empirical Bayes Estimation of a Location Parameter

Pensky¹

1997

Statistics &Amp; Risk Modeling

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Assume that in independent two-dimensional random vectors (Λι, θ\) ,..., (ΧΝ,ΘΝ), each Θϊ is distributed according to some unknown prior distribution density g, and that , given θ,, Xi has the conditional density function q(x -Θ,), i = 1,..., TV. In each pair the first component is observable, but the second is not. After the (JV+l)-th pair (X^r+i, is obtained, the objective is to construct the empirical Bayes estimator of a polynomial f(0w + 1) = with given coefficients bj. In the paper we derive the empirical Bayes estimator of b(9) without any parametric assumptions on g. The upper bound for the mean squared error is obtained. The lower bound for the mean squared error over the class of all possible empirical Bayes estimators is also derived. It is shown that the estimators constructed in the paper have the optimal or nearly optimal convergence rates. Examples for familiar families of conditional distributions are considered.

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“…Articles [7,11] and [10] study empirical Bayes procedures for truncated exponential distribution. Articles [12] and [4] investigate empirical Bayes estimation for a Pareto distribution. Articles [2] and [3] study empirical Bayes procedures in some non-exponential families using asymmetric linex loss functions among others.…”

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confidence: 99%

Testing in two parameter exponential distributions via empirical Bayes method

Huang

Shih

2008

J. Appl. Math. Comput.

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A monotone empirical Bayes test δ * n for testing the hypotheses H 0 : θ ≤ θ 0 versus H 1 : θ > θ 0 using a linear error loss in the two-parameter exponential distribution having probability density p(x|θ, β) = 1 β exp(−(x − θ)/β), x > θ > 0, with unknown scale parameter β is considered. The asymptotic optimality of δ * n is studied and under certain regularity conditions, it is shown that the regret of δ * n converges to zero at a rate n −1+ 1 2(r+α)+1 , where n is the number of past data available and r is some related positive integer with 0 ≤ α ≤ 1. The rate improves significantly the result of [10] under unknown β.

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Empirical Bayes estimation of the scale parameter in a Pareto distribution

Cited by 18 publications

References 8 publications

Nonparametric empirical Bayes estimation based on generalized Laguerre series

Nonparametric empirical Bayes estimation based on generalized Laguerre series

Empirical Bayes Estimation of a Location Parameter

Testing in two parameter exponential distributions via empirical Bayes method

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