Randomized versus non-randomized hypergeometric hypothesis testing with crisp and fuzzy hypotheses

Abstract

This paper is concerned with fuzzy hypothesis testing in the framework of the randomized and non-randomized hypergeometric test for a proportion. Moreover, we differentiate between a test of significance and an alternative test to control the type I error or both error types simultaneously. In contrast to classical (non-)randomized hypothesis testing, fuzzy hypothesis testing provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. In particular, various types of hypotheses with user-specified membership functions can be formulated. Additionally, the proposed test methods are compared via a comprehensive case study, which demonstrates the high flexibility of fuzzy hypothesis testing in practical applications.

Appendix A: Proofs

1.1 A.1 Monotonicity of Q(x) regarding left-tailed crisp case

The probability mass function of a discrete sample random variable X is defined by:

$$\begin{aligned} P(X=x)&=\frac{\left( {\begin{array}{c}pN\\ x\end{array}}\right) \left( {\begin{array}{c}N-pN\\ n-x\end{array}}\right) }{\left( {\begin{array}{c}N\\ n\end{array}}\right) }\\ {}&=\left( {\begin{array}{c}n\\ x\end{array}}\right) \frac{\big [p(p-\frac{1}{N})\cdots (p-\frac{x-1}{N})\big ]\big [ (1-p)\cdots (1-p-\frac{n-x-1}{N})\big ] }{1(1-\frac{1}{N})\cdots (1-\frac{n-1}{N})} \end{aligned}$$

Further, the likelihood ratio can be represented as:

$$\begin{aligned} Q(x)&=\frac{\left( {\begin{array}{c}n\\ x\end{array}}\right) \frac{[\widetilde{p}_{11}(\widetilde{p}_{11}-\frac{1}{N})\cdots (\widetilde{p}_{11}-\frac{x-1}{N})][(1-\widetilde{p}_{11}) \cdots (1-\widetilde{p}_{11}- \frac{n-x-1}{N})]}{1(1-\frac{1}{N})\cdots (1-\frac{n-1}{N})}}{\left( {\begin{array}{c}n\\ x\end{array}}\right) \frac{[\widetilde{p}_{01}(\widetilde{p}_{01}-\frac{1}{N})\cdots (\widetilde{p}_{01}-\frac{x-1}{N})][(1-\widetilde{p}_{01})\cdots (1-\widetilde{p}_{01}- \frac{n-x-1}{N})]}{1(1-\frac{1}{N})\cdots (1-\frac{n-1}{N})}}\\&=\frac{[\widetilde{p}_{11}(\widetilde{p}_{11}- \frac{1}{N})\cdots (\widetilde{p}_{11}- \frac{x-1}{N})][(1-\widetilde{p}_{11})\cdots (1-\widetilde{p}_{11}-\frac{n-x-1}{N})]}{[\widetilde{p}_{01}(\widetilde{p}_{01}- \frac{1}{N})\cdots (\widetilde{p}_{01}- \frac{x-1}{N})][(1-\widetilde{p}_{01})\cdots (1-\widetilde{p}_{01}-\frac{n-x-1}{N})]} \end{aligned}$$

Considering the limit of the likelihood ratio as the population size N approaches infinity we obtain:

$$\begin{aligned} \lim _{N\rightarrow \infty }Q(x)&=\lim _{N\rightarrow \infty }\frac{\overbrace{\left[ \widetilde{p}_{11}\left( \widetilde{p}_{11}-\frac{1}{N}\right) \cdots \left( \widetilde{p}_{11}-\frac{x-1}{N}\right) \right] }^{\rightarrow (\widetilde{p}_{11})^{x}}\overbrace{\left[ \left( 1-\widetilde{p}_{11}\right) \cdots \left( 1-\widetilde{p}_{11}- \frac{n-x-1}{N}\right) \right] }^{\rightarrow (1-\widetilde{p}_{11})^{n-x}}}{\underbrace{\left[ \widetilde{p}_{01}\left( \widetilde{p}_{01}-\frac{1}{N}\right) \cdots \left( \widetilde{p}_{01}-\frac{x-1}{N}\right) \right] }_{\rightarrow (\widetilde{p}_{01})^{x}}\underbrace{\left[ (1-\widetilde{p}_{01})\cdots \left( 1-\widetilde{p}_{01}-\frac{n-x-1}{N}\right) \right] }_{\rightarrow (1-\widetilde{p}_{01})^{n-x}}}\nonumber \\&=\frac{(\widetilde{p}_{11})^{x}(1-\widetilde{p}_{11})^{n-x}}{(\widetilde{p}_{01})^{x}(1-\widetilde{p}_{01})^{n-x}}=\left( \frac{1-\widetilde{p}_{11}}{1-\widetilde{p}_{01}}\right) ^{n}\left[ \frac{\widetilde{p}_{11}(1-\widetilde{p}_{01})}{\widetilde{p}_{01}(1-\widetilde{p}_{11})}\right] ^{x} \end{aligned}$$

(A.1)

Since the second term $\left[ \frac{\widetilde{p}_{11}(1-\widetilde{p}_{01})}{\widetilde{p}_{01}(1-\widetilde{p}_{11})}\right] ^{x}$ in (A.1) only depends on the realization x of the H(N, M, n)-distributed test statistic X and it holds $0<\widetilde{p}_{11}<\widetilde{p}_{01}<1$, the likelihood ratio is strictly monotonically decreasing in x. $\square$

1.2 A.2 Monotonicity of Q(x) regarding left-tailed fuzzy case

The weighted probability mass function of a discrete sample random variable X is defined for $p\in \text {supp}(\widetilde{H}_{0})$ and for $p\in \text {supp}(\widetilde{H}_{1})$ as follows:

$$\begin{aligned}&\max \lbrace 0, w_0(p)\rbrace P(X=x)=\max \lbrace 0, w_0(p)\rbrace \left( {\begin{array}{c}pN\\ x\end{array}}\right) \left( {\begin{array}{c}N-pN\\ n-x\end{array}}\right) \bigg /\left( {\begin{array}{c}N\\ n\end{array}}\right) \\&\max \lbrace 0, w_1(p)\rbrace P(X=x)=\max \lbrace 0, w_1(p)\rbrace \left( {\begin{array}{c}pN\\ x\end{array}}\right) \left( {\begin{array}{c}N-pN\\ n-x\end{array}}\right) \bigg /\left( {\begin{array}{c}N\\ n\end{array}}\right) \end{aligned}$$

Further, the likelihood ratio has the following representation (compare also the proof in Appendix A.1):

Considering the limit of the likelihood ratio as the population size N approaches infinity we obtain

Since $z_{\alpha }$, $z_{\beta }$ as also $w_0(\widetilde{p}_0^*)$, $w_1(\widetilde{p}_1^*)$ are independent from the realization x, the function Q(x) is in turn monotonically decreasing in x (compare the crisp case). $\square$