# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Classical Hypothesis Testing

## 3. Fuzzy Estimation

#### 3.1. Estimation of the Mean of a Normal Variable with Known Variance

#### 3.2. Estimation of the Mean of a Random Variable from a Large Sample (with Unknown Variance)

#### 3.3. Estimation of the Variance of a Normal Variable

## 4. Possibilistic Statistical Tests of Crisp Hypotheses

## 5. Tests on the Mean of a Normal Distribution with Known Variance or of Any Distribution from a Large Sample

**Example**

**1.**

**Example**

**2.**

## 6. Hypotheses Tests for the Variance of a Normal Distribution

**Example**

**3.**

- (1)
- The former are based on the correct implementation of $\tilde{{\sigma}_{u}^{2}}$, according to the theory, whereas the latter contain the error of using $\lambda $-cuts instead of $\alpha $-cuts.
- (2)
- The tests of [29] use fuzzy critical values, which are created using probabilistic concepts.

**Example**

**4.**

## 7. Hypothesis Tests for the Mean of a Normal Random Variable with Unknown Variance

**Example**

**5.**

**Example**

**6.**

## 8. Conclusions

- (1)
- The use of the fuzzy critical values of [29,30] in hypothesis tests leads to wrong results, since they are not in good agreement with the crisp tests, which have been known for nearly one hundred years (Examples 2, 3 and 5). Meanwhile, the possibilistic fuzzy hypothesis tests developed in this paper give results which are in much better agreement with the crisp tests.
- (2)
- The use of the implementation of $\tilde{{\sigma}_{u}^{2}}$ of [29] in the hypothesis tests leads, also, to wrong results since they are not in good agreement with the crisp tests (Example 3). Meanwhile, the fuzzy hypothesis tests, which are based on test statistics produced by the correct implementation of $\tilde{{\sigma}_{u}^{2}}$ (taking into consideration the relation $a\left(\lambda \right)$ between the significance level a and the parameter $\lambda $), give results which are in better agreement with the crisp tests.

- (a)
- (b)
- The ratio of variances of two normal random variables, using test statistics constructed by the correct implementation of $\tilde{{\sigma}_{u}^{2}}$, correcting the respective results produced by tests which use the wrong implementation of $\tilde{{\sigma}_{u}^{2}}$ of [29] or the biased estimator $\tilde{{\sigma}_{b}^{2}}$ of [30].
- (c)
- The regression coefficient $\beta $ and the predicted value ${y}_{p}\left({x}_{0}\right)$ of Y for a given ${x}_{0}$ in a linear regression model $Y=a+\beta x$, using fuzzy test statistics constructed by the fuzzy estimators $\tilde{\beta}$ and $\tilde{{y}_{p}}\left({x}_{0}\right)$ produced by the known classical statistics confidence intervals, $[\widehat{\beta}-{t}_{a/2;n-1}{s}_{\widehat{\beta}},\phantom{\rule{0.277778em}{0ex}}\widehat{\beta}-{t}_{a/2;n-1}{s}_{\widehat{\beta}}]$ of $\beta $ ($\widehat{\beta}$ and ${s}_{\widehat{\beta}}$ the sample regression coefficient and its standard deviation) and $[\widehat{y}-{t}_{a/2;n-1}{s}_{\widehat{y}},\phantom{\rule{0.277778em}{0ex}}\widehat{y}-{t}_{a/2;n-1}{s}_{\widehat{y}}]$ of ${y}_{p}$ ($\widehat{y}$ and ${s}_{\widehat{y}}$ the predicted value of Y for $x={x}_{0}$ from the given sample and its standard deviation).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Sample 1 |

$\begin{array}{cccccccccc}12.14\hfill & 21.48\hfill & 20.82\hfill & 20.96\hfill & 22.66\hfill & 20.61\hfill & 21.5\hfill & 27.86\hfill & 27.33\hfill & 25.75\hfill \\ 27.8\hfill & 29.32\hfill & 28.52\hfill & 28.76\hfill & 28.31\hfill & 28.1\hfill & 27.99\hfill & 27.43\hfill & 28.8\hfill & 28.17\hfill \\ 29.02\hfill & 24.55\hfill & 22.29\hfill & 23.35\hfill & 23.37\hfill & 23.96\hfill & 5.96\hfill & 16.54\hfill & 17.25\hfill & 17.98\hfill \\ 18.78\hfill & 13.59\hfill & 10.85\hfill & 10.82\hfill & 13.42\hfill & 15.19\hfill & 8\hfill & 5.56\hfill & 8.66\hfill & 6.5\hfill \\ 10.21\hfill & 4.87\hfill & 5.66\hfill & 3.2\hfill & 6.2\hfill & 2.97\hfill & 4.4\hfill & 10.19\hfill & 6.2\hfill & 6.9\hfill \end{array}$ |

Sample 2 |

$\begin{array}{cccccccccc}26.4\hfill & 27.7\hfill & 28.7\hfill & 29.63\hfill & 29.44\hfill & 29.89\hfill & 28.15\hfill & 29.15\hfill & 29.05\hfill & 29.34\hfill \\ 29.08\hfill & 29.94\hfill & 25.66\hfill & 26.85\hfill & 24.55\hfill & 26.09\hfill & 25.16\hfill & 22.33\hfill & 20.11\hfill & 19.8\hfill \\ 20.41\hfill & 21.97\hfill & 18.44\hfill & 17.01\hfill & 14.54\hfill & 16.72\hfill & 18.34\hfill & 10.88\hfill & 7.63\hfill & 12.99\hfill \\ 10.35\hfill & 12.54\hfill & 6.6\hfill & 6.61\hfill & 0.83\hfill & 6.68\hfill & 4.43\hfill & 6.59\hfill & 11.52\hfill & 10.18\hfill \\ 8.82\hfill & 8.68\hfill & 9.1\hfill & 12.16\hfill & 12.87\hfill & 12.43\hfill & 12.32\hfill & 15.3\hfill & 17.47\hfill & 15.86\hfill \end{array}$ |

## References

- Hogg, R.V.; Tanis, E.A. Probability and Statistical Inference, 6th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2001. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Taheri, S.M. Trends in fuzzy statistics. Aust. J. Stat.
**2003**, 32, 239–257. [Google Scholar] [CrossRef] - Tanaka, H.; Okuda, T.; Asai, K. Fuzzy information and decision in a statistical model. In Advances in Fuzzy Set Theory and Applications; Gupta, M.M., Ragade, R.K., Yager R., R., Eds.; North-Holland: Amsterdam, The Netherlands, 1979; pp. 303–320. [Google Scholar]
- Casals, M.R.; Gil, M.A.; Gil, P. On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information. Fuzzy Sets Syst.
**1986**, 20, 175–190. [Google Scholar] [CrossRef] - Casals, M.R. Bayesian testing of fuzzy parametric hypotheses from fuzzy information. Oper. Res.
**1993**, 27, 189–199. [Google Scholar] [CrossRef] [Green Version] - Watanabe, N.; Imaizumi, T. A fuzzy statistical test of fuzzy hypotheses. Fuzzy Sets Syst.
**1993**, 53, 167–178. [Google Scholar] [CrossRef] - Arnold, B.F. An approach to fuzzy hypothesis testing. Metrika
**1996**, 44, 119–126. [Google Scholar] [CrossRef] - Arnold, B.F. Testing fuzzy hypotheses with crisp data. Fuzzy Sets Syst.
**1998**, 94, 323–333. [Google Scholar] [CrossRef] - Taheri, S.M.; Behboodian, J. Neyman-Pearson Lemma for fuzzy hypotheses testing. Metrika
**1999**, 49, 3–17. [Google Scholar] [CrossRef] - Taheri, S.M.; Behboodian, J. ABayesian approach to fuzzy hypotheses testing. Fuzzy Sets Syst.
**2001**, 123, 39–48. [Google Scholar] [CrossRef] - Chachi, J.; Taheri, S.M. Optimal statistical tests based on fuzzy random variables. Iran. J. Fuzzy Syst.
**2018**, 15, 27–45. [Google Scholar] - Akbari, M.G.; Hesamian, G. Testing statistical hypotheses for intuitionistic fuzzy data. Soft Comput.
**2019**, 23, 10385–10392. [Google Scholar] [CrossRef] - Arefi, M. Testing statistical hypotheses under fuzzy data and based on a new signed distance. Iran. J. Fuzzy Syst.
**2018**, 15, 153–176. [Google Scholar] - Hung, J.-L.; Chen, C.-C.; Lai, C.-M. Possibility Measure of Accepting Statistical Hypothesis. Mathematics
**2020**, 8, 551. [Google Scholar] [CrossRef] [Green Version] - Chukhrova, N.; Johannssen, A. Fuzzy hypothesis testing for a population proportion based on set-valued information. Fuzzy Sets Syst.
**2020**, 387, 127–157. [Google Scholar] [CrossRef] - Chukhrova, N.; Johannssen, A. Randomized versus non-randomized hypergeometric hypothesis testing with crisp and fuzzy hypotheses. Stat. Pap.
**2018**, 61, 1–37. [Google Scholar] [CrossRef] - Filzmoser, P.; Viertl, R. Testing hypotheses with fuzzy data: The fuzzy p-value. Metrika
**2004**, 59, 21–29. [Google Scholar] [CrossRef] - Parchami, A.; Taheri, S.M.; Mashinchi, M. Fuzzy p-value in testing fuzzy hypotheses with crisp data. Stat Pap.
**2010**, 51, 209. [Google Scholar] [CrossRef] - Mylonas, N.; Papadopoulos, B. Fuzzy p-value of Hypotheses Tests with Crisp Data Using Non-Asymptotic Fuzzy Estimators. J. Stoch. Anal.
**2021**, 2, 1. [Google Scholar] [CrossRef] - Couso, I.; Sanchez, L. Mark-recapture techniques in statistical tests for imprecise data. Internat. J. Approx. Reason.
**2011**, 52, 240–260. [Google Scholar] [CrossRef] [Green Version] - Grzegorzewski, P.; Hryniewicz, O. Soft methods in hypotheses testing. In Soft Computing for Risk Evaluation and Management: Applications in Technology, Environment and Finance, in Studies in Fuzziness and Soft Computing; Ruan, D., Kacprzyk, J., Fedrizzi, M., Eds.; Physica-Verlag: Heidelberg, Germany, 2001; Volume 76, pp. 55–72. [Google Scholar]
- Hryniewicz, O. Possibilistic decisions and fuzzy statistical tests. Fuzzy Sets Syst.
**2006**, 157, 2665–2673. [Google Scholar] [CrossRef] - Hryniewicz, O. Statistical properties of the fuzzy p-value. Internat. J. Approx. Reason.
**2018**, 93, 544–560. [Google Scholar] [CrossRef] - Berkachy, R.; Donze, L. Testing hypotheses by fuzzy methods: A comparison with the classical approach. In Applying Fuzzy Logic for the Digital Economy and Society, in Fuzzy Management Methods; Meier, A., Portmann, E., Teran, L., Eds.; Springer: Cham, Switzerland, 2019; pp. 1–22. [Google Scholar]
- Parchami, A. Fuzzy decision making in testing hypotheses: An introduction to the packages FPV and fuzzy.p.value with practical examples. Iran. J. Fuzzy Syst.
**2020**, 17, 67–77. [Google Scholar] - Chukhrova, N.; Johannssen, A. Fuzzy hypothesis testing: Systematic review and bibliography. Appl. Soft Comput.
**2021**, 106, 107331. [Google Scholar] [CrossRef] - Buckley, J.J. Fuzzy statistics: Hypotheses testing. Soft Comput.
**2005**, 9, 512–518. [Google Scholar] [CrossRef] - Buckley, J.J. Fuzzy Statistics; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Mylonas, N.; Papadopoulos, B. Fuzzy hypotheses tests for crisp data using non-asymptotic fuzzy estimators and a degree of rejection or acceptance. Evol. Syst.
**2021**, 1–18. [Google Scholar] [CrossRef] - Sfiris, D.; Papadopoulos, B. Non-asymptotic fuzzy estimators based on confidence intervals. Inf. Sci.
**2014**, 279, 446–459. [Google Scholar] [CrossRef] - Dubois, D.; Foulloy, L.; Mauris, G.; Prade, H. Probability-possibility transformations, triangular fuzzy-sets and probabilistic inequalities. Reliab. Comput.
**2004**, 10, 273–297. [Google Scholar] [CrossRef] - Kaya, I.; Kahraman, C. A new perspective on fuzzy process capability indices: Robustness. Expert Syst. Appl.
**2010**, 37, 4593–4600. [Google Scholar] [CrossRef] - Kaya, I.; Kahraman, C. Fuzzy Process Capability Analysis and Applications. In Production Engineering and Management under Fuzziness, Studies in Fuzziness and Soft Computing; Kahraman, C., Yavuz, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; Volume 252, pp. 483–513. [Google Scholar]
- Senvar, O.; Tozan, H. Process Capability and Six Sigma Methodology Including Fuzzy and Lean Approaches. In Products and Services from R and D to Final Solutions; Fuerstner, I., Ed.; IntechOpen, 2010; Available online: https://www.intechopen.com/books/products-and-services–from-r-d-to-final-solutions/process-capability-and-six-sigma-methodology-including-fuzzy-and-lean-approaches (accessed on 1 January 2020). [CrossRef] [Green Version]
- Dubois, D.; Prade, H. Ranking of fuzzy numbers in the setting of possibility theory. Inf. Sci.
**1983**, 30, 183–224. [Google Scholar] [CrossRef] - Hellenic National Meteorological Service. Available online: http://www.hnms.gr/emy/el/climatology/climatologymonth (accessed on 1 January 2020).

**Figure 1.**Buckley’s fuzzy estimator of the mean $\mu $ of a random variable X, which follows normal distribution with variance ${\sigma}^{2}=2.3$, derived from a sample of $n=100$ observations with sample mean $\overline{x}=5$.

**Figure 2.**The possibility distribution of the fuzzy estimator $\tilde{\mu}$ of the mean $\mu $ of a random variable X, which follows normal distribution with variance ${\sigma}^{2}=2.3$ derived from a sample of $n=100$ observations with sample mean $\overline{x}=5$.

**Figure 3.**Non-asymptotic fuzzy estimators of the variance ${\sigma}^{2}$: correct unbiased (blue line), biased (green line), unbiased used in [29] (red line).

**Figure 4.**The possibility distribution of the fuzzy statistic Z for the test of Example 1 from a sample with ${\overline{x}}_{1}=17.6$.

**Figure 5.**The possibility distribution of the fuzzy statistic Z for the test of Example 1 from a sample with ${\overline{x}}_{2}=17.95$.

**Figure 6.**The possibility distribution of the fuzzy statistic $\tilde{Z}$ for the two-sided test of Example 2.

**Figure 7.**Fuzzy statistic $\tilde{Z}$ and critical values ${\tilde{CV}}_{i}$ for the respective test of Example 2 in [30].

**Figure 8.**The possibility distribution of the fuzzy statistic $\tilde{{\chi}^{2}}\left[\alpha \right]$ for the test of ${H}_{0}$ of Example 3.

**Figure 9.**The possibility distribution of the fuzzy statistic $\tilde{{\chi}^{2}}\left[\alpha \right]$ and the fuzzy critical values for the respective test of ${H}_{0}$ of Example 3 in [29].

**Figure 10.**The possibility distribution of the fuzzy statistic $\tilde{{\chi}^{2}}\left[\alpha \right]$ and critical values for the respective test of ${H}_{0}$ of Example 3 in [30].

**Figure 11.**The fuzzy statistic ${\tilde{\chi}}^{2}$ for the test of Example 4 for a sample with variance ${s}^{2}=82.28$.

**Figure 12.**The possibility distribution of the fuzzy statistic ${\tilde{\chi}}^{2}$ for the test of Example 4 for a sample with variance ${s}^{2}=72.42$.

**Figure 13.**The possibility distribution of the fuzzy statistic $\overline{T}$ for the test of ${H}_{0}$ of Example 5.

**Figure 14.**The possibility distribution of the fuzzy statistic $\overline{T}$ and the fuzzy critical values for the respective test of [15] of ${H}_{0}$ of Example 5.

**Figure 16.**The fuzzy statistic $\overline{T}$ and the fuzzy critical values for the test of ${H}_{0}$ of [30] for Example 6.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mylonas, N.; Papadopoulos, B.
Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing. *Algorithms* **2021**, *14*, 185.
https://doi.org/10.3390/a14060185

**AMA Style**

Mylonas N, Papadopoulos B.
Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing. *Algorithms*. 2021; 14(6):185.
https://doi.org/10.3390/a14060185

**Chicago/Turabian Style**

Mylonas, Nikos, and Basil Papadopoulos.
2021. "Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing" *Algorithms* 14, no. 6: 185.
https://doi.org/10.3390/a14060185