Bayesian Estimation of a Truncation Parameter for a One-Sided TEF
For a one-sided truncated exponential family (oTEF) of distributions with a truncation parameter \(\gamma \) and a natural parameter \(\theta \) as a nuisance parameter, the asymptotic behavior of the Bayes estimator of \(\gamma \) is discussed.
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Bayesian Estimation of a Truncation Parameter for a One-Sided TEF
For a one-sided truncated exponential family (oTEF) of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the asymptotic behavior of the Bayes estimator of γ is discussed.
6.1 Introduction In this chapter, following mostly the paper by Akahira (2016), the estimation problem on γ for a oTEF of distributions is considered from the Bayesian viewpoint. Under a quadratic loss and a smooth prior on γ , the Bayes estimator of γ is well known to be expressed as a form of the posterior mean. In Sect. 6.3, when θ is known, the stochastic expansion of the Bayes estimator γˆB,θ of γ is derived, and the second order asymptotic mean and asymptotic variance of γˆB,θ are given. In Sect. 6.4, when θ is unknown, the stochastic expansion of the Bayes estimator γˆB,θˆM L plugging the MLE θˆM L in θ of γˆB,θ is derived, and the second-order asymptotic mean and asymptotic variance of γˆB,θˆM L are given. In Sect. 6.5, several examples for a lower-truncated exponential, a lower-truncated normal, Pareto, a lower-truncated beta, and a lowertruncated Erlang distributions are given. In Appendix E, the proofs of Theorems 6.3.1 and 6.4.1 are given.
6.2 Formulation and Assumptions Suppose that X 1 , X 2 , . . . , X n , . . . is a sequence of i.i.d. random variables according to Pθ,γ , having the density (1.7), which belongs to a oTEF of distributions. Let π(γ ) be a prior density with respect to the Lebesgue measure over the open interval © The Author(s) 2017 M. Akahira, Statistical Estimation for Truncated Exponential Families, JSS Research Series in Statistics, DOI 10.1007/978-981-10-5296-5_6
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6 Bayesian Estimation of a Truncation Parameter for a One-Sided TEF
(c, d), and L(γˆ , γ ) the quadratic loss (γˆ − γ )2 of any estimator γˆ = γˆ (X) based on X := (X 1 , . . . , X n ). Suppose that θ is known. Then, it is easily seen that the Bayes estimator of γ w.r.t. L and π is given by γˆB,θ (X) :=
X (1)
c
tπ(t) dt bn (θ, t)
π(t)
X (1)
bn (θ, t)
c
dt,
(6.1)
where X (1) := min1≤i≤n X i . In what follows, we always assume that a(·) and u(·) are functions of class C 3 and π(·) is a function of class C 2 on the interval (c, d), where C k is the class of all k times continuously differentiable functions for any positive integer k.
6.3 Bayes Estimator γˆ B,θ of γ When θ is Known Letting u = n(t − γ ), we have from (6.1) 1 γˆB,θ (X) = γ + n
T(1)
τn
uπ (γ + (u/n)) du bn (θ, γ + (u/n))
T(1)
τn
π (γ + (u/n)) du , bn (θ, γ + (u/n)) (6.2)
where τn := n(c − γ ) and T(1) := n(X (1) − γ ). Let b( j) (θ, γ ) := π( j) (γ ) :=
∂j log b(θ, γ ) ∂γ j
∂j log π(γ ) ∂γ j
( j = 1, 2, . . . ),
(6.3)
( j = 1, 2, . . . ).
(6.4)
It is noted from (1.5) that k(θ, γ ) :=
a(γ )eθu(γ ) = −b(1) (θ, γ ). b(θ, γ )
(6.5)
Then, we have the following. Theorem 6.3.1 For a oTEF Po of distributions having densities of the form (1.7) with a truncation parameter γ and a natural parameter θ , let γˆB,θ be the Bayes estimator (6
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