# Estimation of P(X < Y) Stress–Strength Reliability Measures for a Class of Asymmetric Distributions: The Case of Three-Parameter p-Max Stable Laws

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## Abstract

**:**

## 1. Introduction

- to analytically derive R in terms of special functions, for each three-parameter p-max stable law with fewer parameter restrictions compared to previous results in the literature;
- to propose an estimator for R;
- to apply the results to the modeling of real datasets. In particular, two real scenarios are investigated, showing the versatility of stress–strength reliability (SSR) modeling approaches using p-max models. First, soccer pass completion proportions of two different championships (UEFA Champions League and 2022 FIFA World Cup) were compared, allowing scouting professionals to use the SSR results as a proxy for technical level comparison of teams that competed at those tournaments. Then, a second application involved the modeling and comparison of the strength of carbon fibers of different lengths when subjected to tension efforts. In both modeling scenarios, the best fitting p-max stable distribution (both qualitatively (by graphical methods) and quantitatively (by information criteria)) was taken as a starting point.

## 2. Preliminaries

#### 2.1. Special Functions

#### 2.2. Three-Parameter p-Max Stable Laws

- $${H}_{1}(x;\alpha )=\left\{\begin{array}{ccc}0,\hfill & if& x<1,\hfill \\ exp\{-{(logx)}^{-\alpha}\},\hfill & if& x\ge 1,\hfill \end{array}\right.$$
- $${H}_{2}(x;\alpha )=\left\{\begin{array}{ccc}0,\hfill & if& x<0,\hfill \\ exp\{-{(-logx)}^{\alpha}\},\hfill & if& 0\le x<1,\hfill \\ 1,\hfill & if& x\ge 1,\hfill \end{array}\right.$$
- $${H}_{3}(x;\alpha )=\left\{\begin{array}{ccc}0,\hfill & if& x<-1,\hfill \\ exp\{-{(-log(-x))}^{-\alpha}\},\hfill & if& -1\le x<0,\hfill \\ 1,\hfill & if& x\ge 0,\hfill \end{array}\right.$$
- $${H}_{4}(x;\alpha )=\left\{\begin{array}{ccc}exp\{-{(log(-x))}^{\alpha}\},\hfill & if& x<-1,\hfill \\ 1,\hfill & if& x\ge -1,\hfill \end{array}\right.$$
- $${H}_{5}\left(x\right)=\left\{\begin{array}{ccc}0,\hfill & if& x<0,\hfill \\ exp\{-{x}^{-1}\}\hfill & if& x\ge 0,\hfill \end{array}\right.$$
- $${H}_{6}\left(x\right)=\left\{\begin{array}{ccc}exp\left\{x\right\},\hfill & if& x<0,\hfill \\ 1\hfill & if& x\ge 0,\hfill \end{array}\right.$$

## 3. Reliability $\mathit{P}(\mathit{X}<\mathit{Y})$ for Three-Parameter $\mathit{p}$-Max Stable Laws

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

- (a)
- for $i=5$,$$\begin{array}{ccc}\hfill R=P(X<Y)& =& \frac{1}{{\gamma}_{1}}\mathbb{H}\left(\frac{1}{{\gamma}_{1}},\frac{1}{{\gamma}_{2}},\frac{{\beta}_{2}}{{\beta}_{1}},0,1,0\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=& \frac{{\gamma}_{2}^{{\beta}_{1}/{\beta}_{2}}{\beta}_{1}}{{\gamma}_{1}{\beta}_{2}}{\mathcal{H}}_{1,1}^{1,1}\left[\frac{{\gamma}_{2}^{{\beta}_{1}/{\beta}_{2}}}{{\gamma}_{1}}|\begin{array}{c}(\frac{{\beta}_{2}-{\beta}_{1}}{{\beta}_{2}},\frac{{\beta}_{1}}{{\beta}_{2}})\\ (0,1)\end{array}\right].\hfill \end{array}$$In particular, if $\beta ={\beta}_{1}={\beta}_{2}$, we have$$R=\frac{{\gamma}_{2}}{{\gamma}_{1}+{\gamma}_{2}};$$
- (b)
- for $i=6$,$$\begin{array}{ccc}\hfill R=P(X<Y)& =& {\gamma}_{1}\mathbb{H}\left({\gamma}_{1},{\gamma}_{2},\frac{{\beta}_{2}}{{\beta}_{1}},0,1,0\right)\hfill \\ & =& \frac{{\gamma}_{1}{\beta}_{1}}{{\gamma}_{2}^{{\beta}_{1}/{\beta}_{2}}{\beta}_{2}}{\mathcal{H}}_{1,1}^{1,1}\left[\frac{{\gamma}_{1}}{{\gamma}_{2}^{{\beta}_{1}/{\beta}_{2}}}|\begin{array}{c}(\frac{{\beta}_{2}-{\beta}_{1}}{{\beta}_{2}},\frac{{\beta}_{1}}{{\beta}_{2}})\\ (0,1)\end{array}\right].\hfill \end{array}$$In particular, if $\beta ={\beta}_{1}={\beta}_{2}$, we have$$R=\frac{{\gamma}_{1}}{{\gamma}_{1}+{\gamma}_{2}};$$

**Proof.**

**Corollary**

**1.**

## 4. Estimation

**Remark**

**1.**

#### 4.1. A Random Optimization Method for Approximating the MLE

**Algorithm**

**1.**

#### 4.2. Bootstrap

- Generate independent bootstrap samples $\mathbf{X}$ and $\mathbf{Y}$ of sizes ${n}_{x}$ and ${n}_{y}$, respectively.
- Compute the parameter estimation based on $\mathbf{X}$ and $\mathbf{Y}$.
- Obtain $\widehat{R}$.
- Repeat steps $1\u20133$ $M=1000$ times.
- The approximate $100(1-\alpha )\%$ confidence interval of R is given by $[{\widehat{R}}_{M}(\alpha /2),{\widehat{R}}_{M}(1-\alpha /2)]$, where ${\widehat{R}}_{M}\left(\alpha \right)\approx {\widehat{G}}^{-1}\left(\alpha \right)$ and $\widehat{G}$ are the cumulative distribution function of $\widehat{R}$.

**Algorithm**

**2.**

- Step 1
- Generate independent bootstrap samples $\mathbf{X}$ and $\mathbf{Y}$.
- Step 2
- Compute the parameter estimation based on $\mathbf{X}$ and $\mathbf{Y}$.
- Step 3
- Obtain $\widehat{R}$.
- Step 4
- Repeat steps $\mathbf{1}$ to $\mathbf{3}$ M times.
- Step 5
- The approximate $100(1-\nu )\%$ confidence interval of R is given by $[{\widehat{R}}_{M}(\nu /2),{\widehat{R}}_{M}(1-\nu /2)]$, where ${\widehat{R}}_{M}\left(\nu \right)\approx {\widehat{G}}^{-1}\left(\nu \right)$ and $\widehat{G}$ is the cumulative distribution function of $\widehat{R}$.

## 5. Applications

#### 5.1. Simulation Results

**Remark**

**2.**

- 1000 random samples of $X\sim {H}_{2}({\alpha}_{2},{\beta}_{2},{\gamma}_{2})$ and $Y\sim {H}_{2}({\alpha}_{1},{\beta}_{1},{\gamma}_{1})$ are simulated;
- for each simulation, the parameter $R=P(X<Y)$ is estimated, according to the likelihood function (34) and Algorithm 1;
- the mean of the 1000 corresponding $\widehat{R}$ (denote ${\widehat{R}}_{MC}$) is obtained;
- the Bias and the Root Mean Squared Error (RMSE) are computed.

- in general, the estimation of R had good results, indicated by the small value of the bias;
- the bias values were within the fixed range $\epsilon =0.1$;
- RMSE did not increase as we increased the search space ${\Theta}_{0}$.

#### 5.2. Real Dataset Applications

#### 5.2.1. Medium Pass Completion Proportion

#### 5.2.2. Carbon Fibers

- According to the AIC, BIC, and EDC criteria, PDF ${h}_{2}$ is the one that best fits data X and Y. This was expected since the same data were already modeled via Weibull distribution (see [11,28]) and having positive right endpoint, Theorem 3.1 in [20] establishes that ${H}_{2}$ would be the corresponding p-max stable distribution;
- The p-values of the Kolmogorov–Smirnov test are 0.9404 and 0.8390, respectively, which indicate that we cannot reject the null hypotheses that the X and Y CDFs are ${H}_{2}$.

## 6. Conclusions

- General expressions were analytically derived for $R=P(X<Y)$ when X and Y follow three-parameter p-max stable laws with fewer parameter restrictions compared to previous results in the literature;
- A stochastic optimization procedure was proposed to build an estimator for R based on the novel expressions derived;
- The validity of the expressions and of the general methodological framework developed were demonstrated by Monte Carlo simulations;
- The suitability of the p-max distributions to model real datasets was attested by study cases.

- The stochastic optimization procedure relies on compact search spaces ${[0,N]}^{3}$ for fixed N, whose impact is exponential on the computational effort required;
- The amount of data used to illustrate the methodology and equations developed in the paper is limited; thus, the superiority of the p-max distributions over other possible models needs to be assessed in a case-by-case fashion.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Plots for X (

**left**) and Y (

**right**). On (

**top**), histogram and fitted PDF; on (

**bottom**), empirical CDF and fitted CDF.

**Figure 7.**Plots for X (

**left**) and Y (

**right**). On top, histogram and fitted PDF; on bottom, empirical CDF and fitted CDF.

**Table 1.**Mean, bias, and RMSE of $\widehat{R}$ for PDF ${h}_{2}$ ($\epsilon =0.1$, ${\delta}_{\epsilon}=0.1$, and $n=30$).

N | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\gamma}}_{1}$ | R | ${\widehat{\mathit{R}}}_{\mathbf{MC}}$ | Bias | RMSE |
---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5203 | −0.0036 | 0.1550 |

2 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.3716 | −0.0342 | 0.1487 |

2 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3524 | −0.0493 | 0.1480 |

2 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2873 | −0.0888 | 0.1537 |

2 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2915 | −0.0846 | 0.1575 |

2 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.2860 | −0.0910 | 0.1558 |

2 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6647 | 0.0007 | 0.1308 |

2 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6643 | −0.0011 | 0.1287 |

2 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6650 | −0.0009 | 0.1319 |

2 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6668 | 0.0002 | 0.1287 |

5 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5392 | 0.0152 | 0.1501 |

5 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.4051 | −0.0007 | 0.1379 |

5 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3606 | −0.0411 | 0.1435 |

5 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.3107 | −0.0654 | 0.1380 |

5 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.3110 | −0.0651 | 0.1412 |

5 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.3056 | −0.0713 | 0.1438 |

5 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6621 | −0.0020 | 0.1255 |

5 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6537 | −0.0117 | 0.1309 |

5 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6575 | −0.0085 | 0.1301 |

5 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6552 | −0.0115 | 0.1292 |

10 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5320 | 0.0080 | 0.1492 |

10 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.4078 | 0.0021 | 0.1412 |

10 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3765 | −0.0251 | 0.1419 |

10 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.3204 | −0.0558 | 0.1319 |

10 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.3248 | −0.0514 | 0.1325 |

10 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.3110 | −0.0660 | 0.1390 |

10 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6396 | −0.0244 | 0.1376 |

10 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6398 | −0.0256 | 0.1360 |

10 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6400 | −0.0259 | 0.1342 |

10 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6401 | −0.0266 | 0.1286 |

**Table 2.**Mean, bias, and RMSE of $\widehat{R}$ for PDF ${h}_{2}$ ($\epsilon =0.01$, ${\delta}_{\epsilon}=0.05$, and $n=200$).

N | ${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\gamma}}_{1}$ | R | ${\widehat{\mathit{R}}}_{\mathbf{MC}}$ | Bias | RMSE |
---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5331 | 0.0090 | 0.0940 |

2 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.3671 | −0.0390 | 0.0880 |

2 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3418 | −0.0600 | 0.0940 |

2 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2811 | −0.0950 | 0.1170 |

2 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2801 | −0.0960 | 0.1170 |

2 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.2785 | −0.0980 | 0.1180 |

2 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6628 | −0.0010 | 0.0720 |

2 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6621 | −0.0030 | 0.0690 |

2 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6602 | −0.0060 | 0.0680 |

2 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6649 | −0.0020 | 0.0660 |

5 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5299 | 0.0060 | 0.0920 |

5 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.3632 | −0.0430 | 0.0900 |

5 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3395 | −0.0620 | 0.0960 |

5 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2854 | −0.0910 | 0.1140 |

5 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2820 | −0.0940 | 0.1160 |

5 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.2767 | −0.1000 | 0.1200 |

5 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6606 | −0.0030 | 0.0720 |

5 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6601 | −0.0050 | 0.0710 |

5 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6626 | −0.0030 | 0.0680 |

5 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6663 | 0.0000 | 0.0670 |

10 | 1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.5311 | 0.0072 | 0.0898 |

10 | 1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.4057 | 0.3664 | −0.0394 | 0.0907 |

10 | 1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.4017 | 0.3365 | −0.0651 | 0.0984 |

10 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2839 | −0.0923 | 0.1155 |

10 | 1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3762 | 0.2827 | −0.0935 | 0.1167 |

10 | 1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.3769 | 0.2770 | −0.1000 | 0.1211 |

10 | 1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.6641 | 0.6634 | −0.0007 | 0.0703 |

10 | 1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.6654 | 0.6635 | −0.0019 | 0.0692 |

10 | 1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.6659 | 0.6620 | −0.0040 | 0.0688 |

10 | 1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.6667 | 0.6619 | −0.0047 | 0.0694 |

**Table 3.**Mean, bias, and RMSE of $\widehat{R}$ for PDF ${h}_{1}$ ($\epsilon =0.01$, ${\delta}_{\epsilon}=0.05$, $N=2$, and $n=30$).

${\mathit{\alpha}}_{2}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\gamma}}_{1}$ | R | ${\widehat{\mathit{R}}}_{\mathbf{MC}}$ | Bias | RMSE |
---|---|---|---|---|---|---|---|---|---|

1 | 0.50 | 0.30 | 0.70 | 0.50 | 0.30 | 0.5240 | 0.4161 | −0.1078 | 0.2443 |

1 | 0.50 | 0.30 | 0.70 | 0.75 | 0.30 | 0.3721 | 0.3728 | 0.0007 | 0.2103 |

1 | 0.50 | 0.30 | 1.00 | 0.75 | 0.30 | 0.3064 | 0.4284 | 0.1219 | 0.2032 |

1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3182 | 0.3056 | −0.0125 | 0.1478 |

1 | 0.30 | 0.55 | 0.70 | 0.55 | 0.90 | 0.3182 | 0.3097 | −0.0085 | 0.1562 |

1 | 0.30 | 0.55 | 1.00 | 0.55 | 0.90 | 0.2414 | 0.2493 | 0.0079 | 0.1214 |

1 | 1.00 | 1.00 | 0.90 | 0.50 | 0.50 | 0.7961 | 0.7569 | −0.0391 | 0.1347 |

1 | 1.00 | 1.00 | 0.95 | 0.50 | 0.50 | 0.7949 | 0.7661 | −0.0288 | 0.1215 |

1 | 1.00 | 1.00 | 0.97 | 0.50 | 0.50 | 0.7944 | 0.7668 | −0.0276 | 0.1151 |

1 | 1.00 | 1.00 | 1.00 | 0.50 | 0.50 | 0.7937 | 0.7657 | −0.0279 | 0.1236 |

**Table 4.**Mean, bias, and RMSE of $\widehat{R}$ for PDF ${h}_{5}$ ($\epsilon =0.01$, ${\delta}_{\epsilon}=0.05$, $N=10$, and $n=30$).

${\mathit{\beta}}_{2}$ | ${\mathit{\gamma}}_{2}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\gamma}}_{1}$ | R | ${\widehat{\mathit{R}}}_{\mathbf{MC}}$ | Bias | RMSE |
---|---|---|---|---|---|---|---|

0.30 | 0.50 | 0.20 | 0.30 | 0.7362 | 0.7811 | 0.0450 | 0.0783 |

0.30 | 0.50 | 0.60 | 0.50 | 0.3443 | 0.3309 | −0.0134 | 0.0743 |

0.30 | 0.50 | 0.90 | 1.00 | 0.2050 | 0.1337 | −0.0713 | 0.0848 |

0.50 | 0.70 | 0.20 | 0.30 | 0.8706 | 0.9080 | 0.0374 | 0.0545 |

0.50 | 0.70 | 0.60 | 0.50 | 0.5473 | 0.5249 | −0.0225 | 0.0796 |

0.50 | 0.70 | 0.90 | 1.00 | 0.3507 | 0.2960 | −0.0546 | 0.0853 |

1.00 | 1.00 | 0.20 | 0.30 | 0.9393 | 0.9666 | 0.0273 | 0.0344 |

1.00 | 1.00 | 0.60 | 0.50 | 0.7370 | 0.7050 | −0.0320 | 0.0755 |

1.00 | 1.00 | 0.90 | 1.00 | 0.5071 | 0.5247 | 0.0176 | 0.0778 |

1.00 | 1.00 | 1.00 | 1.00 | 0.5000 | 0.4997 | −0.0003 | 0.0755 |

Dataset | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | n |
---|---|---|---|---|---|---|---|

X | 0.02 | 0.28 | 0.46 | 0.45 | 0.60 | 0.91 | 37 |

Y | 0.77 | 0.84 | 0.86 | 0.86 | 0.89 | 0.93 | 32 |

Dataset | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | BIC | EDC | |
---|---|---|---|---|---|---|---|

X | ${h}_{1}$ | 0.7698 | 0.4113 | 5.4562 | 103.95 | 88.29 | 102.25 |

${\mathbf{h}}_{\mathbf{2}}$ | 1.8231 | 0.7822 | 0.8249 | −8.26 | −23.93 | −9.97 | |

${h}_{5}$ | – | 0.9778 | 3.5948 | 29.66 | 19.22 | 28.53 | |

Y | ${h}_{1}$ | 17.9328 | 1.1439 | 3.3069 | −113.48 | −128.27 | −114.94 |

${\mathbf{h}}_{\mathbf{2}}$ | 2.1254 | 9.7882 | 1.8085 | −127.37 | −142.17 | −128.83 | |

${h}_{5}$ | – | 17.8660 | 19.9746 | −112.29 | −122.15 | −113.26 |

Dataset | Min. | 1st Qu. | Median | Mean | 3rd Qu. | Max. | n |
---|---|---|---|---|---|---|---|

X | 1.31 | 2.10 | 2.48 | 2.45 | 2.77 | 3.58 | 69 |

Y | 1.90 | 2.55 | 3.00 | 3.06 | 3.42 | 5.02 | 63 |

Dataset | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | BIC | EDC | |
---|---|---|---|---|---|---|---|

X | ${h}_{1}$ | 26.1763 | 0.1583 | 2.4240 | 138.20 | 157.60 | 140.86 |

${\mathbf{h}}_{\mathbf{2}}$ | 2.1882 | 2.0072 | 0.0706 | 104.57 | 123.98 | 107.23 | |

${h}_{5}$ | – | 3.3015 | 0.1254 | 153.71 | 166.65 | 155.48 | |

WB | 3.8428 (shape) | – | 11.3142 (scale) | 101.74 | 114.68 | 114.68 | |

EWB | 0.4839 | 1.8690 | 1.84582 | 150.83 | 163.77 | 163.77 | |

Y | ${h}_{1}$ | 32.7124 | 0.1640 | 2.3043 | 113.27 | 94.41 | 110.74 |

${\mathbf{h}}_{\mathbf{2}}$ | 4.2267 | 1.1982 | 0.1073 | 106.26 | 87.40 | 103.73 | |

${h}_{5}$ | – | 2.4785 | 0.0639 | 162.05 | 149.48 | 160.37 | |

WB | 3.9090 (shape) | – | 38.5449 (scale) | 124.30 | 136.88 | 125.99 | |

EWB | 0.3860 | 1.7889 | 1.4415 | 183.50 | 196.07 | 185.18 |

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## Share and Cite

**MDPI and ACS Style**

Quintino, F.S.; Rathie, P.N.; Ozelim, L.C.d.S.M.; da Fonseca, T.A.
Estimation of *P*(*X* < *Y*) Stress–Strength Reliability Measures for a Class of Asymmetric Distributions: The Case of Three-Parameter *p*-Max Stable Laws. *Symmetry* **2024**, *16*, 837.
https://doi.org/10.3390/sym16070837

**AMA Style**

Quintino FS, Rathie PN, Ozelim LCdSM, da Fonseca TA.
Estimation of *P*(*X* < *Y*) Stress–Strength Reliability Measures for a Class of Asymmetric Distributions: The Case of Three-Parameter *p*-Max Stable Laws. *Symmetry*. 2024; 16(7):837.
https://doi.org/10.3390/sym16070837

**Chicago/Turabian Style**

Quintino, Felipe Sousa, Pushpa Narayan Rathie, Luan Carlos de Sena Monteiro Ozelim, and Tiago Alves da Fonseca.
2024. "Estimation of *P*(*X* < *Y*) Stress–Strength Reliability Measures for a Class of Asymmetric Distributions: The Case of Three-Parameter *p*-Max Stable Laws" *Symmetry* 16, no. 7: 837.
https://doi.org/10.3390/sym16070837