Fuzzy hypotheses tests for crisp data using non-asymptotic fuzzy estimators, fuzzy critical values and a degree of rejection or acceptance

Abstract

The fuzzy hypotheses testing is based on critical values and fuzzy test statistics produced by fuzzy estimators. This approach is particularly useful in critical situations, where subtle comparisons between almost equal statistical quantities have to be made. In such cases the fuzzy hypotheses tests give better results than the crisp ones, since they give us the possibility of partial rejection or acceptance of $H_0$ using a degree of rejection or acceptance obtained by ordering fuzzy numbers. In our approach we use non-asymptotic fuzzy estimators which are triangular shaped fuzzy numbers that generalize the fuzzy estimators based on confidence intervals in such a way that they are functions.

Appendix

Samples of humidity taken from Hellenic national meteorological service Hellenic (2018).

Sample 1

$$ \begin{array}{llllllllllll} 42.30 &{} 79.19 &{} 57.26 &{} 46.47 &{} 67.42 &{} 65.11 &{} 62.45 &{} 55.90 &{} 59.20 &{} 71.09&{}79.52 &{} 57.00 \\ 67.61 &{} 65.72 &{} 55.76 &{} 59.20 &{} 72.17&{} 53.81 &{} 65.72 &{} 54.18 &{} 55.10 &{} 70.89&{} 42.42 &{} 67.42 \\ 55.22 &{} 69.00 &{} 57.21 &{} 50.32 &{} 72.03&{} 54.64 &{} 65.72 &{} 55.76 &{} 59.20 &{} 74.74 &{} 46.75 &{} 77.71\\ 81.51 &{} 60.25 &{} 53.98 &{} 56.35 &{} 67.58 &{} 75.50 &{} 65.35 &{} 52.87 &{} 56.35 &{} 68.44 &{} 55.01 &{} 46.32 \\ 61.81 &{} 69.70 &{} 53.67 &{} 59.41 &{} 74.41 &{} 71.55 &{} 66.17 &{} 58.54 &{} 51.36 &{} 76.25 &{} 67.42 &{} 44.00\\ 71.35 &{} 65.35 &{} 54.76 &{} 56.35 &{} 73.83 &{} 76.27 &{} 63.51 &{} 50.18 &{} 56.35 &{} 70.62&{}83.69 &{} 60.46 \\ 80.08 &{} 61.69 &{} 49.69 &{} 56.35 &{} 68.02 &{} 82.29 &{} 59.35 &{} 48.78 &{} 66.18 &{} 59.45 &{} 46.47 &{} 67.42\\ 80.93 &{} 62.20 &{} 59.44 &{} 43.79 &{} 63.92&{} 81.99 &{} 61.69 &{} 52.02 &{} 50.07 &{} 61.57&{} 52.83 &{} 70.47 \\ 56.55 &{} 46.47 &{} 67.42 &{} 46.89 \end{array} $$

Sample 2

$$ \begin{array}{llllllllllll} 62.43 &{} 31.64 &{} 78.77 &{} 64.30 &{} 64.34&{} 44.90 &{} 54.63 &{} 72.23 &{} 68.60 &{} 70.52&{}32.08 &{} 69.76 \\ 45.84 &{} 49.65 &{} 59.86 &{} 63.45 &{} 61.54 &{} 48.54 &{} 50.26 &{} 65.21 &{} 67.61 &{} 62.60 &{} 65.19 &{} 68.76 \\ 50.90 &{} 50.59 &{} 70.24 &{} 66.25 &{} 65.78 &{} 52.52 &{} 45.66 &{} 86.93 &{} 60.67 &{} 63.00&{} 73.63 &{} 37.71 \\ 44.76 &{} 71.80 &{} 62.53 &{} 67.26 &{} 72.13 &{} 41.22 &{} 63.84 &{} 71.37 &{} 63.76 &{} 68.27 &{} 75.30 &{} 62.79 \\ 42.51 &{} 59.19 &{} 65.75 &{} 58.24 &{} 61.97 &{} 32.17 &{} 69.32 &{} 72.64 &{} 68.92 &{} 65.63 &{} 70.66 &{} 65.99 \\ 41.26 &{} 60.26 &{} 71.02 &{} 63.12 &{} 69.02&{} 35.88 &{} 67.43 &{} 71.84 &{} 63.66 &{} 67.48 &{} 37.42 &{} 70.67 \\ 31.01 &{} 69.81 &{} 61.23 &{} 60.43 &{} 69.88 &{} 32.11 &{} 74.74 &{} 73.18 &{} 58.83 &{} 71.28 &{} 61.97 &{} 66.21 \\ 54.88 &{} 74.29 &{} 69.53 &{} 61.99 &{} 74.05 &{} 37.36 &{} 79.22 &{} 72.35 &{} 61.02 &{} 74.41&{} 62.60 &{} 36.22 \\ 65.98 &{} 62.81 &{} 70.78 &{} 59.67 \end{array} $$