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(13)
Asymptotic Optimality
Bayes Risk
Bayes Rule
Convergence Rate
Decision Procedure
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Empirical Bayes
Exponential Family
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Convergence Rates for Empirical Bayes TwoAction Problems II. Continuous Case
Convergence Rates for Empirical Bayes TwoAction Problems II. Continuous Case,10.1214/aoms/1177692557,The Annals of Mathematical Statistics,M. V. John
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Convergence Rates for Empirical Bayes TwoAction Problems II. Continuous Case
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Citations: 24
)
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M. V. Johns
,
J. Van Ryzin
For a general discussion of
empirical Bayes
problems and motivation of the present paper see Section 1 of the previous paper [1]. In that paper we studied the convergence to Bayes optimality and its rate properties for
empirical Bayes
twoaction problems in certain discrete exponential families. This paper continues that investigation for the continuous case. Under appropriate conditions, Theorems 3 and 4 yield convergence rates to
Bayes risk
of $O(n^{\beta})$ for $0 < \beta < 1$, for the $(n + 1)\mathrm{st}$ stage risk of the continuous case
empirical Bayes
procedures of Section 2. These theorems provide, for the continuous case,
convergence rate
results for the
empirical Bayes
procedures of the general type considered by Robbins [5] and Samuel [6] for two different parameterizations of a model. The rate results given here in the continuous case involve upper bounds and are weaker than the discrete case results in [1] wherein exact rates are reported. Specifically, in Section 2 we present the two cases to be considered and define the appropriate
empirical Bayes
procedures for each. Section 3 gives some technical lemmas and Section 4 establishes the
asymptotic optimality
(the asymptotic Bayes property) of the procedures introduced. The main results on rates, Theorems 3 and 4, are given in Section 5. Section 6 examines in detail two specific examplesthe negative exponential and the normal distributionsand gives corollaries to Theorems 3 and 4 which state convergence rates depending on moment properties of the unknown
prior distribution
of the parameters. Section 7 gives an example with $\beta$ arbitrarily close to 1 in the rate $O(n^{\beta})$. The model we consider is the following. Let $f_\lambda(x)$ be a family of Lebesgue densities indexed by a parameter $\lambda$ in an interval of the real line. As in [1], we wish to test the hypothesis $H_1: \lambda \leqq c \mathrm{vs}. H_2: \lambda > c$ with the
loss function
being \begin{align*}L_1(\lambda) &= 0 \quad\text{if}\quad \lambda \leqq c \\ &= b(\lambda  c) \quad\text{if}\quad \lambda > c \\ L_2(\lambda) &= b(c  \lambda) \quad\text{if}\quad \lambda \leqq c \\ &= 0 \quad\text{if}\quad \lambda > c\end{align*} where $L_i(\lambda)$ indicates the loss when action $i$ (deciding in favor of $H_i$) is taken, $i = 1, 2$ and $b$ is a positive constant. Let $\delta(x) = \mathrm{Pr}\{\text{accepting} H_1 \mid X = x\}$ be a randomized
decision rule
for the above twoaction problem. If $G = G(\lambda)$ is a
prior distribution
on $\lambda$, then the risk of the (randomized)
decision procedure
$\delta$ under
prior distribution
$G$ is given as in [1] by, \begin{align*}\tag{1}r(\delta, G) &= \int\int \{L_1(\lambda)f_\lambda(x)\delta(x) + L_2(\lambda)f_\lambda(x)(1  \delta(x))\} dx dG(\lambda) \\ &= b \int \alpha(x)\delta(x) dx + C_G\end{align*} where $C_G = \int L_2(\lambda) dG(\lambda)$ and \begin{equation*}\tag{2}\alpha(x) = \int \lambda f_\lambda(x) dG(\lambda)  cf(x)\end{equation*} with \begin{equation*}\tag{3}f(x) = \int f_\lambda(x) dG(\lambda).\end{equation*} From (1) it is clear that a
Bayes rule
(the minimizer of (1) given $G$) is \begin{align*}\tag{4}\delta_G(x) &= 1\quad\text{if} \alpha(x) \leqq 0 \\ &= 0\quad\text{if} \alpha(x) > 0.\end{align*} Hence, the minimal attainable risk knowing $G$ (the Bayes risk) is \begin{equation*}\tag{5}r^\ast(G) = \inf_\delta r(\delta, G) = r(\delta_G, G).\end{equation*}
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 934947, 1972
DOI:
10.1214/aoms/1177692557
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Citation Context
(8)
...For example, Johns & VR [
1
] have constructed asymptotically optimal EBT for the continuous oneparameter exponential family; Karunamuni[2] and Liang [3] had improved results of [1] by considering the monotonic of the testing problem in the construction of their proposed EBT...
...For example, Johns & VR [1] have constructed asymptotically optimal EBT for the continuous oneparameter exponential family; Karunamuni[2] and Liang [3] had improved results of [
1
] by considering the monotonic of the testing problem in the construction of their proposed EBT...
...Kx r = is a BorelMeasurable bounded function vanishing outside the interval [0,
1
] ,...
Chen Jiaqing
,
et al.
TwoSided Empirical Bayes Test for the Location Parameter of Lognormal...
...Johns & VR[2] have constructed asymptotically optimal EBT for the continuous oneparameter exponential family; Karunamuni[3] and Liang[4] had improved results of [
2
] by considering the monotonic of the testing problem in the construction of their proposed EBT...
Chen Jiaqing
,
et al.
Optimal rates of empirical Bayes tests for a positive exponential fami...
...In the literature, Johns and van Ryzin (
1972
) and van Houwelingen (
1976
), among many others, have studied empirical Bayes tests for the continuous exponential family...
Tachen Liang
.
Empirical Bayes Testing for Double Exponential Distributions
...Johns and van Ryzin
5
and van Houwelingen
6
studied empirical Bayes tests for oneparameter continuous exponential family; Johns and van Ryzin
7
and Liang
8
considered empirical Bayes tests for discrete exponential families...
Tachen Liang
.
On empirical Bayes twotail tests in a positive exponential family
...For example, Johns and Van Ryzin [3,
4
] discussed the EB test problem in discrete and continuous oneparameter exponential families with a linear loss function; Van Houwelingen [5] and Liang [6] studied the monotonic EB test problems for the above exponential families; Hu and Pan [7] considered the EB onetailed test problem for the scale exponential family with a linear loss function; Singh and Wei [8] studied the EB twotailed test ...
...Lemma 1 [
4
] Let R(G) and Rn(δn ,G ) be given by (8) and (18), respectively...
Li Wei
,
et al.
Empirical Bayes test for scale exponential family
Sort by:
Citations
(24)
Robust empirical Bayes tests for continuous distributions
(
Citations: 3
)
R. J. Karunamuni
,
J. Li
,
J. Wu
Journal:
Journal of Statistical Planning and Inference  J STATIST PLAN INFER
, vol. 140, no. 1, pp. 268282, 2010
TwoSided Empirical Bayes Test for the Location Parameter of Lognormal Distribution
Chen Jiaqing
,
Liu Cihua
Conference:
International Conference on Information Engineering and Computer Science  ICIECS
, pp. 15, 2010
Optimal rates of empirical Bayes tests for a positive exponential family in the case of negatively associated samples
Chen Jiaqing
,
Liu Cihua
Conference:
International Conference on Information Science and Engineering  ICISE
, 2010
Empirical Bayes Testing for Double Exponential Distributions
(
Citations: 1
)
Tachen Liang
Journal:
Communications in Statisticstheory and Methods  COMMUN STATISTTHEOR METHOD
, vol. 36, no. 8, pp. 15431553, 2007
On empirical Bayes twotail tests in a positive exponential family
(
Citations: 2
)
Tachen Liang
Journal:
Journal of Nonparametric Statistics  J NONPARAMETR STAT
, vol. 18, no. 78, pp. 449464, 2006