廣義容許區間之研究

Abstract

本研究針對隨機變數2 1 ~ ( , ) q i i i W Nθ hσ = Σ ，利用Tsui and Weerahandi (1989)及 Weerahandi (1993)所提出的廣義檢定函數及廣義信賴區間的想法，來建立其合 理的雙尾容許區間。給定信賴係數(confidence level) γ ，構建一個隨機區間 2 2 2 2 1 1 [ (ˆ, , , ), (ˆ, , , )] L q U q TθSLS TθSLS 使其至少包含W 之分佈的特定比例(content)β ，其 數學式可以表示如下： θ θ β γ θ [ [ ( ˆ, , , ) ≤ ≤ ( ˆ, 2 , , 2 )] ≥ ] = 1 2 2 ( ˆ, 2 , , 2 ) 1 1 S S W L q U q P P T S S W T S S q Λ Λ Λ 其中2 1 ˆ~ ( , ) q i i i θ Nθ cσ = Σ ； 2 / 2 ~ 2i i i ni n S σ χ ； i i h , c 為已知的係數，i = 1,2,Λ , q。且由統 計模擬結果中顯示我們所提出的方法對於解決本問題，有其實際應用價值。另外 值得一提的是，此方法亦可應用到所有常態分配下的均衡混合線性模型之容許區 間建構。

A tolerance interval procedure is derived from the concept of generalized pivotal quantities which is usually used to obtain confidence intervals in situations where standard procedures do not lead to useful solutions. We apply the generalized confidence intervals approach and propose a two-sided tolerance interval for the distribution 2 1 ( , ) q i i i Nθ hσ = Σ based on mutually independent statistics ˆθ , 2 1 S , 2 2 S , …, 2 q S where ˆθ is distributed with 2 1 ( , ) q i i i Nθ cσ = Σ , hi and ci are known constants, and i i2/ i2 n S σ are independent chi-squared random variables with ni df, for i=1,2,..,q. Some practical examples are given to illustrate the applications of the proposed procedure. A simulation study is conducted to evaluate its frequentist coverage probability. The results indicate that the proposed method may be recommended for use in practical applications. The procedure provided in this paper can be applied to tolerance interval questions arising in arbitrary normal balanced mixed linear model situations.