# A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields

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## Abstract

**:**

## 1. Introduction

## 2. Closure Properties

- 1.
- Property of convolution: The NBRUL class is preserved under convolution, where$${\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{F}(x+y)dydx\le {\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{F}\left(x\right)\overline{F}\left(y\right)dydx.$$
**Example****1.**The convolution of the exponential distribution $F\left(x\right)=1-{e}^{-x}$with itself yields the gamma distribution of order 2: $G\left(x\right)=1-(1+x){e}^{-x}$, with strictly increasing failure rate. Thus, $G\left(x\right)$ is not NWRUL. - 2.
- Property of mixture: The NWRUL class is preserved under mixture, where$${\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{F}(x+y)dydx\ge {\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{F}\left(x\right)\overline{F}\left(y\right)dydx.$$
**Example****2.**Let ${\overline{F}}_{\alpha}\left(u\right)={e}^{-\alpha u},\alpha >0$ “scale parameter” and $\overline{G}\left(u\right)={\int}_{0}^{\infty}{\overline{F}}_{\alpha}\left(u\right){e}^{-\alpha}d\alpha ={(u+1)}^{-1}$. Then the failure rate function is ${r}_{g}\left(u\right)={(u+1)}^{-1},$ which is strictly decreasing; thus, $\overline{G}\left(u\right)$ is not NBRUL. - 3.
- The shock model under a homogeneous Poisson process: Suppose the device is subjected to a series of shocks that occur at random time intervals using the Poisson process with intensity $\lambda $. Further suppose that the device has a probability ${\overline{P}}_{k}$. From surviving the first shock k, where $1={\overline{P}}_{0}\ge {\overline{P}}_{1}\ge .......$ and $\phantom{\rule{4pt}{0ex}}{P}_{j}={\overline{P}}_{j-1}-{\overline{P}}_{j},\phantom{\rule{4pt}{0ex}}j\ge 1$. Then, the survival function of the device is given by$$\overline{H}\left(t\right)=\sum _{k=0}^{\infty}{\overline{P}}_{k}\frac{{\left(\lambda t\right)}^{k}}{k!}{e}^{-\lambda t},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\ge 0.$$$${\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{H}(x+y)dydx\le {\int}_{0}^{\infty}{\int}_{t}^{\infty}{e}^{-sx}\overline{H}\left(x\right)\overline{H}\left(y\right)dydx.$$

## 3. NBRUL Comparative Testing Alternatives

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**(i)**As n$\to \infty $, $\sqrt{n}(\widehat{\Delta}\left(s\right)-\Delta \left(s\right))$ is asymptotically normal with zero mean and variance ${\sigma}^{2}\left(s\right),$ where

**(ii)**Under ${H}_{\circ},$ the variance ${\sigma}_{\circ}^{2}\left(s\right)$ can be expressed as

**Proof.**

## 4. The Pitman Asymptotic Efficiency (PAE) of $\widehat{\Delta}\left(s\right)$

## 5. Critical Points for Monte Carlo Distribution

#### Estimations of Test Power

## 6. Censoring Data Testing

#### ${\widehat{\Delta}}_{c}\left(s\right)$ Test Power Estimates

## 7. Applications: Uncensored and Censored Observations

#### 7.1. Uncensored Data

#### 7.1.1. Data Set I: COVID-19-Italy

$4.571$ | $7.201$ | $3.606$ | $8.479$ | $11.410$ | $8.961$ | $10.919$ | $10.908$ | $6.503$ |

$18.474$ | $11.010$ | $17.337$ | $16.561$ | $13.226$ | $15.137$ | $8.697$ | $15.787$ | $13.333$ |

$11.822$ | $14.242$ | $11.273$ | $14.330$ | $16.046$ | $11.950$ | $10.282$ | $11.775$ | $10.138$ |

$9.037$ | $12.396$ | $10.644$ | $8.646$ | $8.905$ | $8.906$ | $7.407$ | $7.445$ | $7.214$ |

$6.194$ | $4.640$ | $5.452$ | $5.073$ | $4.416$ | $4.859$ | $4.408$ | $4.639$ | $3.148$ |

$4.040$ | $4.253$ | $4.011$ | $3.564$ | $3.827$ | $3.134$ | $2.780$ | $2.881$ | $3.341$ |

$2.686$ | $2.814$ | $2.508$ | $2.450$ | $1.518$ |

#### 7.1.2. Data Set II: COVID-19-Netherlands

$14.918$ | $10.656$ | $12.274$ | $10.289$ | $10.832$ | $7.099$ | $5.928$ | $13.211$ |

$7.968$ | $7.584$ | $5.555$ | $6.027$ | $4.097$ | $3.611$ | $4.960$ | $7.498$ |

$6.940$ | $5.307$ | $5.048$ | $2.857$ | $2.254$ | $5.431$ | $4.462$ | $3.883$ |

$3.461$ | $3.647$ | $1.974$ | $1.273$ | $1.416$ | $4.235$ |

#### 7.1.3. Data Set III: Aircraft Air Conditioning

$3.750$ | $0.417$ | $2.500$ | $7.750$ | $2.542$ | $2.042$ | $0.583$ |

$1.000$ | $2.333$ | $0.833$ | $3.292$ | $3.500$ | $1.833$ | $2.458$ |

$1.208$ | $4.917$ | $1.042$ | $6.500$ | $12.917$ | $3.167$ | $1.083$ |

$1.833$ | $0.958$ | $2.583$ | $5.417$ | $8.667$ | $2.917$ | $4.208$ |

$8.667$ |

#### 7.1.4. Data Set IV: Leukemia

$0.019$ | $0.129$ | $0.159$ | $0.203$ | $0.485$ | $0.636$ | $0.748$ |

$0.781$ | $0.869$ | $1.175$ | $1.206$ | $1.219$ | $1.219$ | $1.282$ |

$1.356$ | $1.362$ | $1.458$ | $1.564$ | $1.586$ | $1.592$ | $1.781$ |

$1.923$ | $1.959$ | $2.134$ | $2.413$ | $2.466$ | $2.548$ | $2.652$ |

$2.951$ | $3.038$ | $3.6$ | $3.655$ | $3.745$ | $4.203$ | $4.690$ |

$4.888$ | $5.143$ | $5.167$ | $5.603$ | $5.633$ | $6.192$ | $6.655$ |

$6.874$ |

#### 7.2. Censored Data

#### 7.2.1. Data Set V: Melanoma Patients

13 | 14 | 19 | 19 | 20 | 21 | 23 | 23 | 25 | 26 | 26 | 27 |

27 | 31 | 32 | 34 | 34 | 37 | 38 | 38 | 40 | 46 | 50 | 53 |

54 | 57 | 58 | 59 | 60 | 65 | 65 | 66 | 70 | 85 | 90 | 98 |

102 | 103 | 110 | 118 | 124 | 130 | 136 | 138 | 141 | 234 |

16 | 21 | 44 | 50 | 55 | 67 | 73 | 76 | 80 | 81 | 86 | 93 |

100 | 108 | 114 | 120 | 124 | 125 | 129 | 130 | 132 | 134 | 140 | 147 |

148 | 151 | 152 | 152 | 158 | 181 | 190 | 193 | 194 | 213 | 215 |

#### 7.2.2. Data Set VI: Blood Cancer

$0.030$ | $0.493$ | $0.855$ | $1.184$ | $1.283$ | $1.480$ | $1.776$ | $2.138$ |

$2.500$ | $2.763$ | $2.993$ | $3.224$ | $3.421$ | $4.178$ | $4.441+$ | $5.691$ |

$5.855+$ | $6.941+$ | $6.941$ | $7.993+$ | $8.882$ | $8.882$ | $9.145+$ | $11.480$ |

$11.513$ | $12.105+$ | $12.796$ | $12.993+$ | $13.849+$ | $16.612+$ | $17.138+$ | $20.066$ |

$20.329+$ | $22.368+$ | $26.776+$ | $28.717+$ | $28.717+$ | $32.928+$ | $33.783+$ | $34.221+$ |

$34.770+$ | $39.539+$ | $41.118+$ | $45.033+$ | $46.053+$ | $46.941+$ | $48.289+$ | $57.401+$ |

$58.322+$ | $60.625+$ |

$0.658$ | $0.822$ | $1.414$ | $2.500$ | $3.322$ | $3.816$ | $4.737$ | $4.836+$ |

$4.934$ | $5.033$ | $5.757$ | $5.855$ | $5.987$ | $6.151$ | $6.217$ | $6.447+$ |

$8.651$ | $8.717$ | $9.441+$ | $10.329$ | $11.480$ | $12.007$ | $12.007+$ | $12.237$ |

$12.401+$ | $13.059+$ | $14.474+$ | $15.000+$ | $15.461$ | $15.757$ | $16.480$ | $16.711$ |

$17.204+$ | $17.237$ | $17.303+$ | $17.664+$ | $18.092$ | $18.092+$ | $18.750+$ | $20.625+$ |

$23.158$ | $27.730+$ | $31.184+$ | $32.434+$ | $35.921+$ | $42.237+$ | $44.638+$ | $46.480+$ |

$47.467+$ | $48.322+$ | $56.086$ |

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Models | |||
---|---|---|---|

Test | Makeham | LFR | Weibull |

Mugdadi and Ahmad [23] | 0.039 | 0.408 | 0.170 |

Kango [24] | 0.144 | 0.433 | 0.132 |

Abdel-Aziz [25] | 0.184 | 0.535 | 0.223 |

Etman et al. [26] | 0.233 | 0.932 | 1.046 |

EL-Sagheer et al. [13] | 0.287 | 0.901 | 1.158 |

Proposed test
$\widehat{\Delta}\left(0.99\right)$ | 0.280 | 0.946 | 1.116 |

Sample Size | Confidence Levels | ||
---|---|---|---|

$\mathit{n}$ | $\mathbf{90}\%$ | $\mathbf{95}\%$ | $\mathbf{99}\%$ |

5 | $0.001238$ | $0.001473$ | $0.001939$ |

10 | $0.000818$ | $0.000955$ | $0.001259$ |

15 | $0.000657$ | $0.000791$ | $0.001005$ |

20 | $0.000576$ | $0.000689$ | $0.000884$ |

25 | $0.000512$ | $0.000612$ | $0.000793$ |

29 | $0.000476$ | $0.000565$ | $0.000736$ |

30 | $0.000470$ | $0.000564$ | $0.000734$ |

35 | $0.000434$ | $0.000518$ | $0.000675$ |

40 | $0.000409$ | $0.000488$ | $0.000633$ |

43 | $0.000393$ | $0.000474$ | $0.000619$ |

45 | $0.000392$ | $0.000478$ | $0.000605$ |

50 | $0.000361$ | $0.000432$ | $0.000566$ |

59 | $0.000337$ | $0.000408$ | $0.000530$ |

n | $\mathit{\theta}$ | Weibull | Gamma |
---|---|---|---|

10 | 2 3 4 | $0.9345$ $0.9997$ $1.0000$ | $0.6718$ $0.9371$ $0.9902$ |

20 | 2 3 4 | $0.9959$ $1.0000$ $1.0000$ | $0.8372$ $0.9962$ $0.9999$ |

30 | 2 3 4 | $0.9998$ $1.0000$ $1.0000$ | $0.9235$ $0.9996$ $1.0000$ |

Sample Size | Confidence Intervals | ||
---|---|---|---|

$\mathit{n}$ | $\mathbf{90}\%$ | $\mathbf{95}\%$ | $\mathbf{99}\%$ |

10 | $0.011499$ | $0.014484$ | $0.022093$ |

20 | $0.007095$ | $0.008642$ | $0.012312$ |

30 | $0.005581$ | $0.006751$ | $0.009710$ |

40 | $0.004488$ | $0.005425$ | $0.007276$ |

50 | $0.004075$ | $0.004885$ | $0.006603$ |

51 | $0.004014$ | $0.004857$ | $0.006554$ |

60 | $0.003660$ | $0.004448$ | $0.005918$ |

70 | $0.003254$ | $0.003928$ | $0.005155$ |

80 | $0.003016$ | $0.003589$ | $0.004904$ |

81 | $0.002998$ | $0.003615$ | $0.004873$ |

n | $\mathit{\theta}$ | Weibull | LFR | Gamma |
---|---|---|---|---|

10 | 2 3 4 | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ |

20 | 2 3 4 | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ |

30 | 2 3 4 | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ | $1.0000$ $1.0000$ $1.0000$ |

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## Share and Cite

**MDPI and ACS Style**

Etman, W.B.; El-Morshedy, M.; Eliwa, M.S.; Almohaimeed, A.; EL-Sagheer, R.M.
A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields. *Axioms* **2023**, *12*, 331.
https://doi.org/10.3390/axioms12040331

**AMA Style**

Etman WB, El-Morshedy M, Eliwa MS, Almohaimeed A, EL-Sagheer RM.
A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields. *Axioms*. 2023; 12(4):331.
https://doi.org/10.3390/axioms12040331

**Chicago/Turabian Style**

Etman, Walid B., Mahmoud El-Morshedy, Mohamed S. Eliwa, Amani Almohaimeed, and Rashad M. EL-Sagheer.
2023. "A New Reliability Class-Test Statistic for Life Distributions under Convolution, Mixture and Homogeneous Shock Model: Characterizations and Applications in Engineering and Medical Fields" *Axioms* 12, no. 4: 331.
https://doi.org/10.3390/axioms12040331