# Aging Intensity for Step-Stress Accelerated Life Testing Experiments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries of SSALT

## 3. AI Function for SSALT

- $\left(i\right)$
- $Exp\left({\theta}_{1}\right)$, $Exp\left({\theta}_{2}\right)$: ${g}_{1}\left(t\right)={g}_{2}\left(t\right)=t$$$A{I}_{1}^{Exp}\left(t\right)=1\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}A{I}_{2}^{Exp}\left(t\right)=\frac{t}{t-\left(1-\frac{{\theta}_{2}}{{\theta}_{1}}\right)\tau}$$
- $\left(ii\right)$
- $W({\theta}_{1},\delta )$, $W({\theta}_{2},\delta )$: ${g}_{1}\left(t\right)={g}_{2}\left(t\right)={t}^{\delta}$$$A{I}_{1}^{TF{R}_{\delta}}\left(t\right)=\delta \phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}A{I}_{2}^{TF{R}_{\delta}}\left(t\right)=\frac{\delta {t}^{\delta}}{{t}^{\delta}-\left(1-\frac{{\theta}_{2}}{{\theta}_{1}}\right){\tau}^{\delta}}$$
- $\left(iii\right)$
- $Gompertz({\theta}_{1},d)$, $Gompertz({\theta}_{2},d)$: ${g}_{1}\left(t\right)={g}_{2}\left(t\right)={\mathrm{e}}^{dt}-1$$$A{I}_{1}^{Gomp}\left(t\right)=t\xb7\frac{d{\mathrm{e}}^{dt}}{{\mathrm{e}}^{dt}-1}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}A{I}_{2}^{Gomp}\left(t\right)=\frac{dt}{1+\frac{{\theta}_{2}}{{\theta}_{1}}\left({\mathrm{e}}^{d(\tau -t)}-{\mathrm{e}}^{-dt}\right)-{\mathrm{e}}^{d(\tau -t)}}$$

**Remark**

**1.**

**Remark**

**2.**

## 4. AI-Based Estimation

**Example**

**1.**

**Remark**

**3.**

**Example**

**2.**

**Example**

**3.**

## 5. Goodness-of-Fit Testing for SSALT

## 6. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AI | Aging Intensity; |

ALT | Accelerated Life Testing; |

CDF | Cumulative Density Function; |

HR | Hazard Rate; |

Probability Density Function; | |

SSALT | Step-Stress ALT; |

SV | Survival Function; |

KL | Kullback–Leibler. |

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**Figure 1.**Aging intensity function on the second stress level for the Weibull and exponential distributions for the TFR and CE SSALT models, for $\delta =2$ (

**upper**), $\delta =0.5$ (

**lower**), ${\theta}_{2}/{\theta}_{1}=0.25$ (

**left**), and ${\theta}_{2}/{\theta}_{1}=0.5$ (

**right**). The dashed lines represent the AI of a CSALT model at level ${x}_{1}$ (truncated at $\tau $, it is also the AI on the first level of a SSALT model, i.e., the constant 1 for the exponential distribution and $\delta $ for the Weibull distribution.

**Figure 2.**True and estimated CDF, PDF, and AI function in red and blue, respectively, for two simulated samples of the considered SSALT exponential model.

**Figure 3.**True and estimated PDF and AI function in red and blue, respectively, based on the kernel method with data reflection, for the two simulated samples of Example 1.

**Figure 4.**Histograms with the estimates of ${\theta}_{1}$ and ${\theta}_{2}$ based on AI and MLE (${\stackrel{\u02c7}{\theta}}_{i}={\theta}_{i}^{\left(AI\right)}$, ${\widehat{\theta}}_{i}={\theta}_{i}^{\left(MLE\right)}$, $i=1,2$).

**Figure 5.**True and estimated AI function in red and blue, respectively, for two simulated samples of the considered SSALT exponential model with three stress levels.

**Figure 6.**True CDF (red), estimated CDFs based on MLE (black), and AI (blue) with points corresponding to empirical CDF (red dots) for two simulated samples from Example 3.

**Figure 7.**Histogram of the differences $KL(g,\widehat{g})-KL(g,\stackrel{\u02c7}{g})$ based on the simulation study described in Example 2.

**Figure 8.**True CDF (red), estimated CDFs based on MLE (black), and AI (blue) with points corresponding to empirical CDF (red dots) for sample with outliers (

**left**) and random selected sample (

**right**) from Example 2.

**Table 1.**Mean and standard deviation of the fitted (MLE) normal distributions for the estimators of ${\theta}_{1}=33$ and ${\theta}_{2}=7$ of the SSALT model (2) based on AI and MLE in the simulation study of Example 2 with $10.000$ replications.

Mean | Standard Deviation | |
---|---|---|

${\stackrel{\u02c7}{\theta}}_{1}$ | 25.6008 | 4.9854 |

${\widehat{\theta}}_{1}$ | 24.8684 | 4.3003 |

${\stackrel{\u02c7}{\theta}}_{2}$ | 6.9302 | 1.3083 |

${\widehat{\theta}}_{2}$ | 6.9849 | 1.1683 |

**Table 2.**Mean and standard deviation of the fitted (MLE) normal distributions for the estimators of ${\theta}_{1}=33$, ${\theta}_{2}=7$, and ${\theta}_{3}=3$ based on AI and MLE in the simulation study of Example 3 with $10.000$ replications.

Mean | Standard Deviation | |
---|---|---|

${\stackrel{\u02c7}{\theta}}_{1}$ | 29.2562 | 9.2000 |

${\widehat{\theta}}_{1}$ | 24.9046 | 4.3508 |

${\stackrel{\u02c7}{\theta}}_{2}$ | 6.4401 | 2.0313 |

${\widehat{\theta}}_{2}$ | 7.2282 | 1.6811 |

${\stackrel{\u02c7}{\theta}}_{3}$ | 2.7728 | 0.9291 |

${\widehat{\theta}}_{3}$ | 2.9939 | 0.7604 |

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**MDPI and ACS Style**

Buono, F.; Kateri, M.
Aging Intensity for Step-Stress Accelerated Life Testing Experiments. *Entropy* **2024**, *26*, 417.
https://doi.org/10.3390/e26050417

**AMA Style**

Buono F, Kateri M.
Aging Intensity for Step-Stress Accelerated Life Testing Experiments. *Entropy*. 2024; 26(5):417.
https://doi.org/10.3390/e26050417

**Chicago/Turabian Style**

Buono, Francesco, and Maria Kateri.
2024. "Aging Intensity for Step-Stress Accelerated Life Testing Experiments" *Entropy* 26, no. 5: 417.
https://doi.org/10.3390/e26050417