# Optimal Concordant Tests
## 2 Methods
### Concordant Tests Based on Gamma Distribution  
The above [PERSON]'s concordant test can be extended to concordant tests based on gamma distribution. For this purpose, we first study a class of combination tests based on gamma distribution [10]. For given shape parameter \(\alpha\) and scale parameter \(\beta\), a \(p\)-value combination test statistic based on the gamma distribution can be constructed as follows:  
\[T_{G(\alpha,\beta)}(P_{1},P_{2},\cdots,P_{n})=\sum_{i=1}^{n}F_{G(\alpha,\beta)} ^{-1}(1-P_{i}), \tag{2}\]  
where \(F_{G(\alpha,\beta)}^{-1}(y)\) is the inverse function of the cumulative distribution function (CDF), \(F_{G(\alpha,\beta)}(x)\), of the random variable \(Gamma(\alpha,\beta)\), whose probability density function (PDF) is \(f_{G(\alpha,\beta)}(x)=\beta^{\alpha}x^{\alpha-1}\text{exp}(-\beta x)/\Gamma(\alpha)\) for \(x>0\), where the gamma function \(\Gamma(z)=\int_{0}^{\infty}x^{z-1}e^{-x}dx\). The right-sided \(p\)-value of the test defined in (2) can be defined as follows:  
\[P=P(Gamma(n\alpha,\beta)>t)=1-F_{G(n\alpha,\beta)}(t)=S_{G(n\alpha,\beta)}(t), \tag{3}\]  
where \(Gamma(n\alpha,\beta)\) is the random variable having gamma distribution with the shape parameter, \(n\alpha\), and the scale parameter, \(\beta\); and \(F_{G(n\alpha,\beta)}\) and \(S_{G(n\alpha,\beta)}\) are the CDF and the survival function of \(Gamma(n\alpha,\beta)\). It can be shown that, due to the property of gamma distribution, parameter \(\beta\) has no effect on the test defined in (2) and (3) [10]. Therefore, for simplicity, we use \(\beta=1\) in the above test and denote \(T_{G(\alpha,1)}\) as \(T_{G(\alpha)}\) hereafter. Furthermore, we can prove the following properties for the above class of gamma distribution-based tests [10]: (i) \(T_{G(0)}\triangleq\lim\limits_{\alpha\to 0+}T_{G(\alpha)}\) is equivalent to the [PERSON]'s minimal-p test; (ii) \(T_{G(0,5)}\) is the same as the chi-square test with degrees of freedom equal to \(n\); (iii) \(T_{G(1)}\) is equavalent to [PERSON] test; and (iv) \(T_{G(\infty)}\triangleq\lim\limits_{\alpha\to\infty}T_{G(\alpha)}\) is the same as the z-test. Therefore, the class of gamma distribution-based tests, \(T_{G(\alpha)}\), includes many popular tests as special cases. This observation motivates us to generalize the [PERSON]'s concordant test. We propose the gamma distribution-based concordant test with the following test statistic:  
\[T_{G(\alpha)}^{C}=\text{max}\Big{(}T_{G(\alpha)}^{L},T_{G(\alpha)}^{R}\Big{)}, \tag{4}\]  
where \(T_{G(\alpha)}^{L}=T_{G(\alpha)}(P_{1},P_{2},\cdots,P_{n})=\sum_{i=1}^{n}F_{G( \alpha)}^{-1}(1-P_{i})\) and \(T_{G(\alpha)}^{R}=T_{G(\alpha)}(1-P_{1},1-P_{2},\cdots,1-P_{n})=\sum_{i=1}^{n }F_{G(\alpha)}^{-1}(P_{i})\) are the gamma distribution-based tests using the left- and right-sided \(p\)-values, respectively. It is easy to see that when \(\alpha=1\), \(T_{G(\alpha)}^{C}\) is equavalent to the Pearson's concordant test \(Q^{C}\). Therefore, \(T_{G(\alpha)}^{C}\) is a generalization of the Pearson's concordant test. We show that the \(p\)-value for the proposed concordant test, \(T_{G(\alpha)}^{C}\), can also be estimated in a similar way as \(Q^{C}\): \(P\Big{[}T_{G(\alpha)}^{C}>A\Big{]}\approx 2P\Big{[}T_{G(\alpha)}^{L}>A\Big{]}\). Now, we study the properties of \(T_{G(\alpha)}^{C}\). First we need the following definitions and lemmas [13]. **Definition 1**. : _A function \(f\) on \(\mathbb{R}^{n}\) is nondecreasing if it is nondecreasing in each of its \(n\) arguments when the other \(n-1\) values are held fixed._  
**Definition 2**. : _Let \(X_{1},\ \cdots,X_{n}\) be random variables with a joint distribution. These random variables are associated if the covariance \(Cov(f(X),g(X))\geq 0\) holds for all nondecreasing functions, \(f\) and \(g\), for which the covariance is defined._  
**Lemma 1**. : _(Theorem 2.1 of [PERSON], [PERSON], and [PERSON] 1967 [13]) Independent random variables are associated._  
**Lemma 2**.