^{1}

^{1}

^{2}

^{1}

^{2}

The Hotelling T-squared statistic has been widely used for the testing of differences in means for the multivariate data. The existing statistic under classical statistics is applied when observations in multivariate data are determined, precise, and exact. In practice, it is not necessary that all observations in the data are determined and precise due to measurement in complex situations and under uncertainty environment. In this paper, we will introduce the Hotelling T-squared statistic under neutrosophic statistics (NS) which is the generalization of classical statistics and applied under uncertainty environment. We will discuss the application and advantage of the neutrosophic Hotelling T-squared statistic with the aid of data. From the comparison, we will conclude that the proposed statistic is more adequate and effective in uncertainty.

In classical statistics (CS), the univariate analysis is the technique to analyze the single-variable data. The multivariate analysis has been widely used to analyze data having more than one variable. In the multivariate technique under the CS, the Hotelling T-squared statistic has been widely applied in the variety of fields (see, for example, [

The Hotelling T-squared statistic derived under the CS can be only applied for the analysis when all observations in the multivariate data are determined, precise, and certain. In practice, the data under study are not always precise but linguistic. For example, the temperature of a certain city may be high, low, and medium or the measurement of variable data in a complex system may lead to being in an interval rather than the determined values. In such situations, the Hotelling T-squared statistic under the CS cannot be used for the analysis of the data. When observations are uncertain or fuzzy, the fuzzy Hotelling T-squared statistic can be applied for the testing of means of multivariate populations. Taleb et al. [

Recently, the neutrosophic logic, which is the extension of the fuzzy logic, attracted many researchers due to its applications in the variety of fields. The neutrosophic logic considered the measure of indeterminacy which fuzzy logic does not consider (see [

Aslam and Smarandache [

Let

The neutrosophic form of

Note here that

The neutrosophic sample mean and neutrosophic sample variance from

The neutrosophic form of

Note here that

The neutrosophic form of

Note here that

The neutrosophic sample covariance between two neutrosophic variables are given by

The neutrosophic form of

Note here that

Finally, neutrosophic sample correlation between the

The neutrosophic form of

Note here that

The neutrosophic descriptive statistics for

The neutrosophic sample mean variance and covariance and correlation are presented by the array

In this section, we discuss the proposed neutrosophic Hotelling

The neutrosophic form of

Note here that

For the given values of

The generalization of equations (

The neutrosophic form of

The statistic is given in equation (

The neutrosophic Hotelling

The software provides the

Now, we discuss the application of the proposed neutrosophic Hotelling

Step 1:

Step 2: some basic calculations for the data are given in Table

Step 3: let

Step 4: the neutrosophic Hotelling

Step 5: the critical region is using equation (

Step 6: as

The neutrosophic sweat data.

Individual | |||
---|---|---|---|

1 | [3.7, 3.7] | [48.5, 48.7] | [9.3, 9.3] |

2 | [5.7, 5.8] | [65.1, 65.1] | [8.0, 8.1] |

3 | [3.8, 3.8] | [47.2, 47.3] | [10.9, 10.9] |

4 | [3.2, 3.3] | [53.2, 53.3] | [12.0, 12.0] |

5 | [3.1, 3.1] | [55.5, 55.5] | [9.7, 9.8] |

6 | [4.6, 4.8] | [36.1, 36.2] | [7.9, 7.9] |

7 | [2.4, 2.4] | [24.8, 24.8] | [14.0, 14.0] |

8 | [7.2, 7.3] | [33.1, 33.2] | [7.6, 7.7] |

9 | [6.7, 6.7] | [47.4, 47.4] | [8.5, 8.6] |

10 | [5.4, 5.5] | [54.1, 54.2] | [11.3, 11.3] |

11 | [3.9, 3.9] | [36.9, 36.9] | [12.7, 12.7] |

12 | [4.5, 4.6] | [58.8, 58.9] | [12.3, 12.4] |

13 | [3.5, 3.5] | [27.8, 27.9] | [9.8, 9.8] |

14 | [4.5, 4.5] | [40.2, 40.2] | [8.4, 8.5] |

15 | [1.5, 1.7] | [13.5, 13.5] | [10.1, 10.2] |

16 | [8.5, 8.5] | [56.4, 56.5] | [7.1, 7.1] |

17 | [4.5, 4.7] | [71.6, 71.9] | [8.2, 8.2] |

18 | [6.5, 6.5] | [52.8, 52.8] | [10.9, 10.9] |

19 | [4.1, 4.2] | [44.1, 44.2] | [11.2, 11.3] |

20 | [5.5,5.5] | [40.9, 40.9] | [9.4, 9.5] |

In Section

In this paper, we introduced the Hotelling T-squared statistic under neutrosophic statistics (NS) which is the generalization of classical statistics and applied under uncertainty environment. We discussed the application and advantage of neutrosophic Hotelling T-squared statistic with the aid of data. The proposed neutrosophic Hotelling T-squared statistic is expressed in the indeterminacy interval and hence more flexible and information than the Hotelling T-squared statistic under classical statistics. Based on the comparison, we recommend using the proposed neutrosophic Hotelling T-squared statistic for the analysis of the data under uncertainty. Some more properties of the proposed neutrosophic Hotelling T-squared statistic can be studied as future research. The sensitivity of the proposed statistic to uncertainty and measurement errors can be studied in future work.

The data used to support the findings of this study are included in the paper.

The authors declare that they have no conflicts of interest regarding this paper.

This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge DSR technical and financial support with thanks.

^{2}statistic in monitoring multivariate quality characteristics

^{2}statistic to batch processes