# Non-Parametric Hypothesis Testing for Unknown Aged Class of Life Distribution Using Real Medical Data

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Exponentiality Departure Measure

#### 2.1. Testing Exponentiality versus UBAC2L Class of Complete Data

**Lemma 1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 2.2. Monte Carlo Null Distribution Critical Points

#### 2.3. Pittman Asymptotic Relative Efficiency

#### 2.4. Power Estimates for Different Alternatives

#### 2.5. Applications for Complete Data

## 3. Testing Exponentiality for Censored Data

#### 3.1. Test for UBAC2L in Case of Right-Censored Data

#### 3.2. Applications

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

IFR | Increasing failure rate. |

IFRA | Increasing failure rate average. |

UBA | Used better than age. |

UBAC | Used better than age in convex order. |

UBAC2 | Used better than age in concave order. |

UBAC2L | Laplace transform for used better than age in concave order. |

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n | 90% | 95% | 99% |
---|---|---|---|

5 | 0.868731 | 1.2161 | 1.57217 |

10 | 0.488618 | 0.836443 | 1.23035 |

15 | 0.410873 | 0.611245 | 0.796065 |

20 | 0.361404 | 0.491752 | 0.696427 |

25 | 0.36791 | 0.485149 | 0.554637 |

30 | 0.298835 | 0.408053 | 0.433742 |

35 | 0.286803 | 0.404571 | 0.479514 |

40 | 0.235469 | 0.345774 | 0.396459 |

45 | 0.231345 | 0.322356 | 0.372959 |

50 | 0.252539 | 0.321946 | 0.377091 |

55 | 0.231475 | 0.330027 | 0.394961 |

60 | 0.250355 | 0.314627 | 0.360007 |

65 | 0.201125 | 0.283754 | 0.314362 |

70 | 0.186574 | 0.232399 | 0.27165 |

75 | 0.204348 | 0.266098 | 0.319055 |

80 | 0.187613 | 0.27441 | 0.332885 |

85 | 0.169602 | 0.229449 | 0.298501 |

90 | 0.18389 | 0.240486 | 0.298516 |

95 | 0.166333 | 0.213038 | 0.280515 |

100 | 0.161628 | 0.2132 | 0.24905 |

**Table 2.**PAE of ${\mathsf{\delta}}_{\mathrm{n}}^{\left(1\right)},{\delta}_{{U}_{2L}}$ and $\widehat{\delta}\left(s\right)$.

Distribution | ${\mathit{\delta}}_{\mathit{n}}^{\left(5\right)}$ | ${\mathit{\delta}}_{{\mathit{U}}_{2\mathit{L}}}$ | ${\widehat{\mathit{\delta}}}_{\mathit{n}}\left(\mathit{s}\right)$ | |||
---|---|---|---|---|---|---|

$\mathit{s}=0.09$ | $\mathit{s}=0.1$ | $\mathit{s}=0.3$ | $\mathit{s}=0.5$ | |||

LFR | 1.1456 | 1.3 | 1.385 | 1.379 | 1.262 | 1.225 |

Makeham | 0.5455 | 0.58 | 0.564 | 0.565 | 0.562 | 0.573 |

Weibull | ----- | ------ | 1.018 | 1.017 | 0.759 | 0.932 |

Distribution | n | $\mathit{\theta}=1$ | $\mathit{\theta}=2$ | $\mathit{\theta}=3$ |
---|---|---|---|---|

LFR | 10 | 1 | 1 | 1 |

20 | 1 | 1 | 1 | |

30 | 1 | 1 | 1 | |

Gamma | 10 | 0.0662 | 0.5736 | 0.9716 |

20 | 0.0673 | 0.7831 | 0.9997 | |

30 | 0.068 | 0.8973 | 1 | |

Weibull | 10 | 0.7024 | 1 | 1 |

20 | 0. 9398 | 1 | 1 | |

30 | 0.9453 | 1 | 1 |

Data # 1 | ${\underset{\mathsf{\delta}}{^}}_{n}\left(\mathrm{s}\right)=-149.074$ |

Data # 2 | ${\underset{\mathsf{\delta}}{^}}_{n}\left(\mathrm{s}\right)=0.06575$ |

n | 95% | 98% | 99% |
---|---|---|---|

5 | 9.06044 | 11.3263 | 11.3263 |

10 | 3.76773 | 4.24107 | 4.45821 |

15 | 2.16296 | 2.53999 | 2.78284 |

20 | 1.39771 | 1.64111 | 1.73429 |

25 | 0.946225 | 1.09012 | 1.23866 |

30 | 0.713409 | 0.852922 | 0.914456 |

35 | 0.599153 | 0.684245 | 0.774385 |

40 | 0.480519 | 0.543262 | 0.576404 |

45 | 0.424666 | 0.487289 | 0.534101 |

50 | 0.340545 | 0.394071 | 0.427245 |

55 | 0.300065 | 0.347867 | 0.374279 |

60 | 0.263235 | 0.29784 | 0.354673 |

65 | 0.241464 | 0.279319 | 0.306573 |

70 | 0.219645 | 0.253037 | 0.269572 |

75 | 0.194965 | 0.228086 | 0.255792 |

80 | 0.174686 | 0.209678 | 0.230627 |

85 | 0.158488 | 0.185419 | 0.205321 |

90 | 0.147107 | 0.16895 | 0.184741 |

95 | 0.136914 | 0.156613 | 0.166537 |

100 | 0.124793 | 0.142524 | 0.158562 |

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**MDPI and ACS Style**

Bakr, M.E.; A. Al-Babtain, A.
Non-Parametric Hypothesis Testing for Unknown Aged Class of Life Distribution Using Real Medical Data. *Axioms* **2023**, *12*, 369.
https://doi.org/10.3390/axioms12040369

**AMA Style**

Bakr ME, A. Al-Babtain A.
Non-Parametric Hypothesis Testing for Unknown Aged Class of Life Distribution Using Real Medical Data. *Axioms*. 2023; 12(4):369.
https://doi.org/10.3390/axioms12040369

**Chicago/Turabian Style**

Bakr, Mahmoud. E., and Abdulhakim A. Al-Babtain.
2023. "Non-Parametric Hypothesis Testing for Unknown Aged Class of Life Distribution Using Real Medical Data" *Axioms* 12, no. 4: 369.
https://doi.org/10.3390/axioms12040369