本研究論文主要探討伽瑪尺度參數在兩個檢定問題下之概似比改進之研究。
本研究論文假設獨立隨機變數Xij,i=1,…,p,j=1,…,ni,服從伽瑪分配,其中包含未知尺度參數θi和已知形狀參數τi。首先考慮在符號檢定下之概似比改進之研究,檢定H0:θi≤θi0對某些i=1,…,p和H1:θi>θi0 對所有i=1,…,p,其中θi0均為大於零之常數,當p=2時,對任意之α值,0<α<0.4,建構兩個較強檢力檢定 ϕ_{k_1,k_2}^A 和 ϕ_{k_1,k_2}^B$如同 Liu 和 Berger (1995)和 Saikali 和 Berger (2002)所建構檢定一樣,每一檢定之拒絕域均包含最大概似比之拒絕域和額外之拒絕域,也因此新檢定之檢力比尺度為$alpha$的最大概似比檢定強,當p>2時,可藉由ϕ_{k_1,k_2}^A或ϕ_{k_1,k_2}^B$建構一個齊一較強檢力檢定。之後,考慮簡單秩序檢定之概似比改進之研究,檢定H0:θi+1≤θi對某些i=1,…,p-1和H1:θ1<⋯<θp,針對其檢定問題,當p=3時,對任意之α值,0<α<0.4,建構兩齊一較強檢力檢定ϕ_{k_3^*,k_2^*,k_1^*}^A和ϕ_{k_3^*,k_2^*,k_1^*}^B$,同樣地,每一檢定之拒絕域均包含最大概似比之拒絕域和額外之拒絕域,因此,其檢力比最大概似比檢定強,當p>3時,也可藉由ϕ_{k_3^*,k_2^*,k_1^*}^A或ϕ_{k_3^*,k_2^*,k_1^*}^B建構一個齊一較強檢力檢定。值得一提的是此建構方法和之前有關齊一較強檢力檢定的建構方法均不同,且到目前為止針對簡單秩序檢定問題無人成功的建構齊一較強檢力檢定。針對此兩檢定問題,本研究論文所建構之齊一較強檢力檢定均是交集-聯集式檢定,且本研究論文成果也可應用在雙參數指數分配中之尺度參數和常態變異數下之檢定問題,呈現一組資料在兩檢定問題之下之檢定結果。

In this study, we consider the sign testing and simple-order testing problems for gamma scale parameters. For each testing problem, we propose two new tests that are uniformly more powerful than the likelihood ratio tests (LRT). For 1,…,p, let Xij, j=1,…,ni, denote independent random samples that have gamma distributions with a unknown scale parameter θi and a known shape parameter τi. Let θ10,…,θp0 be positive constants. We consider the sign testing problem of testing H0:θi≤θi0 for some i=1,…,p versus H1:θi>θi0 for all i=1,…,p. For p=2 and any 0<α<0.4, we propose two new tests, ϕ_{k_1,k_2}^A and ϕ_{k_1,k_2}^B, which are similar to the tests proposed by Liu and Berger (1995) and Saikali and Berger (2002) in that they contain the rejection region for the LRT and the additional sets
that touch the LRT rejection region at only a single point. However, the sizes of ϕ_{k_1,k_2}^A and ϕ_{k_1,k_2}^B are still α. Therefore, ϕ_{k_1,k_2}^A and ϕ_{k_1,k_2}^B are uniformly more powerful than the LRT. Then, we propose a uniformly more powerful test constructed by either ϕ_{k_1,k_2}^A or ϕ_{k_1,k_2}^B for a general testing problem (p>2). Subsequently, we consider the simple-order testing problem of testing H_0:θi+1≤θi for some i=1,…,p-1 versus
H1:θ1<⋯<θp. For p=3 and any 0<α<0.4, we propose two tests, ϕ_{k_3^*,k_2^*,k_1^*}^A and ϕ_{k_3^*,k_2^*,k_1^*}^B. Again, these new tests contain the rejection region for the LRT and the additional region. Therefore, the new tests are uniformly more powerful than the LRT. Analogically, we propose a uniformly more powerful test constructed by either ϕ_{k_3^*,k_2^*,k_1^*}^A or ϕ_{k_3^*,k_2^*,k_1^*}^B for a general testing problem (p>3). To our knowledge, no one has obtained a uniformly more powerful test as yet. In summary, these new tests contain the rejection region for the LRT and the additional region. These proposed tests are intersection-union tests (IUTs). We apply our results to test the scale parameters of two-parameter exponential distributions and normal variances for the sign testing problem and simple-order testing problem, respectively. Finally, we illustrate the proposed tests with appropriate data.

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