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Table 1 The results of five different tests in Example 4

From: A Simple but Efficient Approach for Testing Fuzzy Hypotheses

Test H 0(μ) H 1(μ) p 01 p 10 Accepted CF
number      hypothesis  
1 \(\left \{ \begin {array}{lr} \mu -3 \ \ & \ \ 3<\mu \leq 4 \\ 5-\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) \(\left \{ \begin {array}{lr} \mu \ \ & \ \ 0<\mu \leq 1 \\ 2-\mu \ \ & \ \ 1<\mu \leq 2 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) 0.019 0.244 \(\tilde {H}_{1}\) 0.927
2 \(\left \{ \begin {array}{lr} \mu -3 \ \ & \ \ 3<\mu \leq 4 \\ 5-\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) \(\left \{ \begin {array}{lr} \mu +5 \ \ & \ \ -5<\mu \leq -4 \\ -3-\mu \ \ & \ \ -4<\mu \leq -3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) 0.019 0 \(\tilde {H}_{0}\) 0.99
3 \(\left \{ \begin {array}{lr} \mu -3 \ \ & \ \ 3<\mu \leq 4 \\ 5-\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) \(\left \{ \begin {array}{lr} \frac {\mu +1}{2} \ \ & \ \ -1<\mu \leq 1 \\ \frac {3-\mu }{2} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) 0.019 0.283 \(\tilde {H}_{1}\) 0.937
4 \(\left \{ \begin {array}{lr} \frac {\mu -2}{2} \ \ & \ \ 2<\mu \leq 4 \\ \frac {6-\mu }{2} \ \ & \ \ 4<\mu \leq 6 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) \(\left \{ \begin {array}{lr} \frac {\mu +1}{2} \ \ & \ \ -1<\mu \leq 1 \\ \frac {3-\mu }{2} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) 0.041 0.283 \(\tilde {H}_{1}\) 0.873
5 \(\left \{ \begin {array}{lr} \frac {\mu +4}{8} \ \ & \ \ -4<\mu \leq 4 \\ 5-\mu \ \ & \ \ 4<\mu \leq 5 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) \(\left \{ \begin {array}{lr} \mu \ \ & \ \ -1<\mu \leq 1 \\ \frac {5-\mu }{4} \ \ & \ \ 1<\mu \leq 3 \\ 0 \ \ &\ \ o.w. \end {array} \right.\) 0.527 0.452 \(\tilde {H}_{0}\) 0.538