# Scale Mixture of Rayleigh Distribution

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Definition and Properties

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**2.**

#### Moments

**Lemma**

**1.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

**Remark**

**3.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 3. Lifetime Analysis

**Proposition**

**7.**

- 1.
- The survival function is $S(t;\sigma ,q)={\left(\right)}^{\frac{{t}^{2}}{2\sigma}}-\frac{q}{2}$, $t>0$.
- 2.
- The hazard function is$$\begin{array}{ccc}\hfill h(t;\sigma ,q)& =& {\displaystyle \frac{q\phantom{\rule{0.277778em}{0ex}}t}{{t}^{2}+2\sigma}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}t>0.\hfill \end{array}$$

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

#### Mean Residual Life

**Definition**

**3.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

#### Order Statistics

**Proposition**

**10.**

**Proof.**

**Remark**

**6.**

## 4. Entropy

**Proposition**

**11.**

**Proof.**

## 5. Inference

#### 5.1. Moment Method Estimators

**Proposition**

**12.**

**Proof.**

#### 5.2. ML Estimation

#### 5.3. ML Estimation Using EM-Algorithm

**Lemma**

**2.**

- 1.
- ${X}^{m}\sim GG\left(\right)open="("\; close=")">{\beta}^{m},\frac{k}{m},\frac{1}{m}$, $m>0$, with pdf given in (2).
- 2.
- $\mathbb{E}[log(X\left)\right]\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\Psi \left(k\right)-log\left(\beta \right)$ where $\Psi (\xb7)$ denotes the digamma function.

**E-step**: For $i=1,\dots ,n$ compute$$\begin{array}{cc}\hfill {\widehat{{y}_{i}^{2}}}^{(k+1)}={\displaystyle \frac{{\sigma}^{\left(k\right)}({q}^{\left(k\right)}+2)}{2{\sigma}^{\left(k\right)}+{t}_{i}^{2}}},& \phantom{\rule{1.em}{0ex}}{\widehat{log\left({y}_{i}\right)}}^{(k+1)}=\frac{1}{2}\left(\right)open="["\; close="]">\Psi \left(\right)open="("\; close=")">\frac{{q}^{\left(k\right)}+2}{2}-log\left(\right)open="("\; close=")">1+\frac{{t}_{i}^{2}}{2{\sigma}^{\left(k\right)}}\end{array}$$**M-step**: Update the vector of parameters $\theta =(\sigma ,q)$$$\begin{array}{cc}\hfill {\displaystyle {\widehat{\sigma}}^{(k+1)}={\displaystyle \frac{1}{2}}\phantom{\rule{4pt}{0ex}}\overline{{t}^{2}\phantom{\rule{4pt}{0ex}}{\widehat{{y}^{2}}}^{(k+1)}},}& \phantom{\rule{1.em}{0ex}}{\widehat{q}}^{(k+1)}=2\phantom{\rule{4pt}{0ex}}{\Psi}^{-1}\left(\right)open="("\; close=")">2\phantom{\rule{4pt}{0ex}}\overline{{\widehat{log\left(y\right)}}^{(k+1)}}\phantom{\rule{0.277778em}{0ex}}.\end{array}$$- E and M steps are repeated until a suitable convergence is reached.

## 6. Simulation Study

## 7. Real Data Illustration

#### 7.1. Application to Patients with Bladder Cancer

#### 7.2. Application to Number of Failures of an Air Conditioning System

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =1$, $q=1$ with $n=30,\dots ,200$.

**Figure 5.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =1$, $q=$1.5 with $n=30,\dots ,200$.

**Figure 6.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =1$, $q=2$ with $n=30,\dots ,200$.

**Figure 7.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =1$, $q=3$ with $n=30,\dots ,200$.

**Figure 8.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =10$, $q=1$ with $n=30,\dots ,200$.

**Figure 9.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =10$, $q=$1.5 with $n=30,\dots ,200$.

**Figure 10.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =10$, $q=2$ with $n=30,\dots ,200$.

**Figure 11.**Graphics of (

**a**) bias, (

**b**) RMSE and (

**c**) coverage of simulation for $\sigma =10$, $q=3$ with $n=30,\dots ,200$.

**Figure 12.**Density adjusted for the remission times of patients with bladder cancer in the R, SR and SMR distributions.

**Figure 13.**QQ-plot for distributions (

**a**) R, (

**b**) SR, (

**c**) SMR for the remission times of patients with bladder cancer.

**Figure 14.**Density adjusted for the number of failures of an air conditioning system in the R, SR and SMR distributions.

**Figure 15.**QQ-plot for distributions (

**a**) R, (

**b**) SR, (

**c**) SMR for the number of failures of an air conditioning system.

n | $\overline{\mathit{T}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ | $min\left(\mathit{T}\right)$ | $max\left(\mathit{T}\right)$ |
---|---|---|---|---|---|---|

128 | 9.366 | 10.508 | 3.287 | 18.483 | 0.08 | 79.05 |

**Table 2.**Estimates, standard errors (SE) in parenthesis, log-likelihood, AIC and BIC values for the remission times of patients with bladder cancer.

Estimaciones | R (SE) | SR (SE) | SMR (SE) |
---|---|---|---|

$\widehat{\sigma}$ | 98.639 (8.718) | 8.647 (2.051) | 15.369 (5.108) |

$\widehat{q}$ | - | 1.424 (0.224) | 1.772 (0.318) |

log-likelihood | −491.266 | −415.815 | −413.339 |

AIC | 984.531 | 835.631 | 830.677 |

BIC | 987.383 | 841.335 | 836.381 |

n | $\overline{\mathit{T}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ | $min\left(\mathit{T}\right)$ | $max\left(\mathit{T}\right)$ |
---|---|---|---|---|---|---|

188 | 92.074 | 107.916 | 2.139 | 8.023 | 1 | 603 |

**Table 4.**Estimates, SE in parenthesis, log-likelihood, AIC and BIC values for the number of failures of an air conditioning system.

Estimaciones | R (SE) | SR (SE) | SMR (SE) |
---|---|---|---|

$\widehat{\sigma}$ | 10,030.83 (730.135) | 264.611 (68.021) | 382.761 (113.843) |

$\widehat{q}$ | - | 0.902 (0.107) | 1.069 (0.136) |

log-likelihood | −1191.275 | −1053.503 | −1046.549 |

AIC | 2384.550 | 2111.006 | 2097.097 |

BIC | 2387.787 | 2117.479 | 2103.570 |

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**MDPI and ACS Style**

Rivera, P.A.; Barranco-Chamorro, I.; Gallardo, D.I.; Gómez, H.W.
Scale Mixture of Rayleigh Distribution. *Mathematics* **2020**, *8*, 1842.
https://doi.org/10.3390/math8101842

**AMA Style**

Rivera PA, Barranco-Chamorro I, Gallardo DI, Gómez HW.
Scale Mixture of Rayleigh Distribution. *Mathematics*. 2020; 8(10):1842.
https://doi.org/10.3390/math8101842

**Chicago/Turabian Style**

Rivera, Pilar A., Inmaculada Barranco-Chamorro, Diego I. Gallardo, and Héctor W. Gómez.
2020. "Scale Mixture of Rayleigh Distribution" *Mathematics* 8, no. 10: 1842.
https://doi.org/10.3390/math8101842