On the estimation of a normal precision and a normal variance ratio

Bobotas, Panayiotis; Kourouklis, Stavros

doi:10.1016/j.stamet.2010.01.001

Cited by 14 publications

(11 citation statements)

References 17 publications

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“…For estimating σ 2 , Maruyama and Strawderman (2006) derived the improved estimator δ φ 2 = (1 + W 2 )S 2 /[(m + 2)(r 2 + 1 + W 2 )], whereas Bobotas and Kourouklis (2010) obtained the improved estimator δ φ 1 = (n − 4)(r 1 + 1 + W 1 )S −1 1 /(1 + W 1 ) of 1/σ 1 , for 0 < r i < r 0i where r 0i are specified constants, i = 1, 2. Since L(t) = (t − 1) 2 is convex in t k , 1 k 2, Corollary 2 provides the Strawderman-type estimator of ρ δ = n − 4 m + 2…”

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

“…Thus, not even in the case k = 1, the dominance of δ in (10) can be established by Kubokawa (1994b, Theorems 2.2 and 2.4) despite the fact that it has then the form of Kubokawa (1994b) double-adjustment improved estimators. For k = 1, δ in (10) is also given in Bobotas and Kourouklis (2010).…”

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

“…From Maruyama and Strawderman (2006) and Bobotas and Kourouklis (2010) we use the improved estimators of σ 2 and 1/σ 1 , δ φ 2 = (1 + W 2 )S 2 /[m(r 2 + 1 + W 2 )] and δ φ 1 = (n − 2)(r 1 + 1 + W 1 )S −1 1 /(1 + W 1 ), respectively, where 0 < r i < r 0i and r 0i are specified constants, i = 1, 2. Since L(t) = t − ln t − 1 is convex in ln t and t k , 0 < k 1, by Corollaries 1, 2, we obtain the improved Strawderman-type estimators…”

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

“…The estimator in (12) for k = 1 and δ φ 1 δ φ 2 in (11) were also derived by Bobotas and Kourouklis (2010).…”

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

“…Following Shinozaki (1995) approach, we can modify a class of Strawderman-type estimators of a normal precision given by Bobotas and Kourouklis (2010) and obtain the improved estimator of 1/σ 1 ,…”

Section: Improved Estimators Of the Ratio Of Two Normal Variances Basmentioning

confidence: 99%

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Estimating the ratio of two scale parameters: a simple approach

2010

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We describe a simple approach for estimating the ratio ρ = σ 2 /σ 1 of the scale parameters of two populations from a decision theoretic point of view. We show that if the loss function satisfies a certain condition, then the estimation of ρ reduces to separately estimating σ 2 and 1/σ 1 . This implies that the standard estimator of ρ can be improved by just employing an improved estimator of σ 2 or 1/σ 1 . Moreover, in the case where the loss function is convex in some function of its argument, we prove that such improved estimators of ρ are further dominated by corresponding ones that use all the available data. Using this result, we construct new classes of double-adjustment improved estimators for several well-known convex as well as non-convex loss functions. In particular, Strawderman-type estimators of ρ in general models are given whereas Shinozaki-type estimators of the ratio of two normal variances are briefly treated.Keywords Decision theory · Improved estimation of a scale parameter · Improved estimation of ratio of scale parameters · Stein's and Brewster and Zidek's techniques · Strawderman's technique · Kubokawa's approach

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Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

“…The estimator in (12) for k = 1 and δ φ 1 δ φ 2 in (11) were also derived by Bobotas and Kourouklis (2010).…”

Section: Improved Estimators Of ρ Based On Strawderman's Methodsmentioning

confidence: 99%

Section: Improved Estimators Of the Ratio Of Two Normal Variances Basmentioning

confidence: 99%

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Estimating the ratio of two scale parameters: a simple approach

2010

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Improved estimation of the scale parameter, the hazard rate parameter and the ratio of the scale parameters in exponential distributions: An integrated approach

Bobotas

Kourouklis

2011

Journal of Statistical Planning and Inference

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General dominance properties of double shrinkage estimators for ratio of positive parameters

Kubokawa

2014

Journal of Statistical Planning and Inference

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In estimation of ratio of variances in two normal distributions with unknown means, it has been shown in the literature that simple and crude ratio estimators based on two sample variances are dominated by shrinkage estimators using information contained in sample means. Of these, a natural double shrinkage estimator is the ratio of shrinkage estimators of variances, but its improvement over the crude ratio estimator depends on loss functions, namely, the improvement has not been established except the Stein loss function. In this paper, this dominance property is shown for some convex loss functions including the Stein and quadratic loss functions in the general framework of distributions with positive parameters and shrinkage estimators. The resulting new finding is that the generalized Bayes estimator of the ratio of variances dominates the crude ratio estimator relative to the quadratic loss. The paper also shows that the dominance property of the double shrinkage estimator holds for estimation of the difference of variances, but it does not hold for estimation of the product and sum of variances. Finally, it is demonstrated that the double shrinkage estimators for the ratio, product, sum and differences of variances are connected to estimation of linear combinations of the normal positive means, and the dominance and non-dominance results of the double shrinkage estimators coincide with the corresponding dominance results in estimation of linear combinations of means.

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On the estimation of a normal precision and a normal variance ratio

Cited by 14 publications

References 17 publications

Estimating the ratio of two scale parameters: a simple approach

Estimating the ratio of two scale parameters: a simple approach

Improved estimation of the scale parameter, the hazard rate parameter and the ratio of the scale parameters in exponential distributions: An integrated approach

General dominance properties of double shrinkage estimators for ratio of positive parameters

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