# Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint

## Abstract

**:**

## 1. Introduction

## 2. The Class of SMLNFT Models

**Definition**

**1.**

- (i)
- LNFT model:$$R(y;\mu ,{\sigma}^{2})=1-\mathrm{\Phi}\left(B\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}h(y;\mu ,{\sigma}^{2})=\frac{\varphi \left(B\right)}{\sigma \phantom{\rule{0.277778em}{0ex}}y\phantom{\rule{0.277778em}{0ex}}R(y;\mu ,{\sigma}^{2})},$$
- (ii)
- L${t}_{\nu}$FT model:$$R(y;\mu ,{\sigma}^{2})=1-{F}_{\nu}\left(B\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}h(y;\mu ,{\sigma}^{2})=\frac{{f}_{\nu}\left(B\right)}{\sigma \phantom{\rule{0.277778em}{0ex}}y\phantom{\rule{0.277778em}{0ex}}R(y;\mu ,{\sigma}^{2})},$$
- (iii)
- LCFT model:$$R(y;\mu ,{\sigma}^{2})=\frac{1}{2}-\frac{arctan\left(B\right)}{\pi}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}h(y;\mu ,{\sigma}^{2})=\frac{1}{\sigma \phantom{\rule{0.277778em}{0ex}}y\pi \left(\right)open="["\; close="]">1+{B}^{2}}$$
- (iv)
- LLFT model:$$R(y;\mu ,{\sigma}^{2})=\frac{1}{1+{e}^{B}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}h(y;\mu ,{\sigma}^{2})=\frac{{e}^{B}\phantom{\rule{0.277778em}{0ex}}R(y;\mu ,{\sigma}^{2})}{\sigma \phantom{\rule{0.277778em}{0ex}}y}.$$
- (v)
- LSFT model:$$\begin{array}{ccc}\hfill R(y;\mu ,{\sigma}^{2})& =& \left(\right)open="\{"\; close>\begin{array}{c}1-\mathrm{\Phi}\left(B\right)+\left(\right)open="["\; close="]">\varphi \left(0\right)-\varphi \left(B\right)/B\hfill \\ \mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{0.277778em}{0ex}}B\ne 0,\hfill \end{array}1/2\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{0.277778em}{0ex}}B=0,\hfill \hfill \end{array}\hfill h(y;\mu ,{\sigma}^{2})& =& \left(\right)open="\{"\; close>\begin{array}{c}\left(\right)open="["\; close="]">\varphi \left(0\right)-\varphi \left(B\right)/\left(\right)open="\{"\; close="\}">{B}^{2}\sigma \phantom{\rule{0.277778em}{0ex}}y\phantom{\rule{0.277778em}{0ex}}R(y;\mu ,{\sigma}^{2})\hfill \end{array}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{0.277778em}{0ex}}B\ne 0,\hfill \hfill & 2\left(\right)open="["\; close="]">\varphi \left(0\right)-\varphi \left(B\right)/\left(\right)open="\{"\; close="\}">{B}^{2}\sigma \phantom{\rule{0.277778em}{0ex}}y\hfill \\ \mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{0.277778em}{0ex}}B=0.\hfill $$

## 3. Two-Stage MaxEnt Prior

#### 3.1. Stochastic Constraint on Reliability Measures

#### 3.2. Two-Stage MaxEnt Prior

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Bayesian Hierarchical SMLNFT Model

#### 4.1. Bayesian Hierarchical Model

#### 4.2. The Gibbs Sampler

- (1)
- The full conditional distribution of $\mu $ is an univariate normal given by:$$\begin{array}{c}\hfill \left(\right)open="["\; close="]">\mu |{\mu}_{0},\tau ,\mathit{\eta},\mathit{x}\\ \sim & N\left(\right)open="("\; close=")">\frac{{\tau}^{-1}{\mu}_{0}+\Delta {\sum}_{i=1}^{n}\kappa {\left({\eta}_{i}\right)}^{-1}{x}_{i}}{{\tau}^{-1}+\Delta {\sum}_{i=1}^{n}\kappa {\left({\eta}_{i}\right)}^{-1}},\phantom{\rule{0.277778em}{0ex}}\frac{1}{{\Delta}^{-1}+\tau {\sum}_{i=1}^{n}\kappa {\left({\eta}_{i}\right)}^{-1}},\hfill \end{array}$$
- (2)
- The full conditional distribution of ${\mu}_{0}$ is a truncated normal given by:$$\begin{array}{c}\hfill \left(\right)open="["\; close="]">{\mu}_{0}|\mu ,\tau ,\mathit{\eta},\mathit{x}\\ \sim & TN\left(\right)open="("\; close=")">\frac{\Delta {\theta}_{0}+{\Delta}_{0}\mu}{\Delta +{\Delta}_{0}},\frac{1}{{\Delta}^{-1}+{\Delta}_{0}^{-1}}I({\mu}_{0}\in \mathcal{C}),\hfill \end{array}$$
- (3)
- The full conditional distribution of $\tau $ is a Gamma distribution:$$\begin{array}{c}\hfill \left(\right)open="["\; close="]">\tau |\mu ,{\mu}_{0},\mathit{\eta},\mathit{x}\\ \sim & Gamma\left(\right)open="("\; close=")">{\nu}_{1}+\frac{n}{2},{\nu}_{2}+\frac{{\sum}_{i=1}^{n}\kappa {\left({\eta}_{i}\right)}^{-1}{({x}_{i}-\mu )}^{2}}{2}.\hfill \end{array}$$
- (4)
- The full conditional distributions of ${\eta}_{i}$s are independent and their densities are$$\begin{array}{c}\hfill p\left({\eta}_{i}\right|\mu ,{\mu}_{0},\tau ,\mathit{x})\propto g\left({\eta}_{i}\right)\kappa {\left({\eta}_{i}\right)}^{-1/2}exp\left(\right)open="\{"\; close="\}">-\frac{\tau \kappa {\left({\eta}_{i}\right)}^{-1}{({x}_{i}-\mu )}^{2}}{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,n,\end{array}$$

#### 4.3. Markov Chain Monte Carlo Method

- note 1:
- With given initial values of $(\mu ,{\mu}_{0},\tau ,\mathit{\eta})$, implementation of the Gibbs (or Metropolis-within-Gibbs) sampling algorithm consists of drawing repeatedly from distributions Equation (15) through Equation (18). The R package
`tmvtnorm`and the R package`mvtnorm`can be used to sample from the conditionals and to calculate $\delta $ for a given $\alpha $ from Equation (11). - note 2:
- In cases using the LNFT model, ${\eta}_{i}$ degenerates at $\kappa \left({\eta}_{i}\right)=1.$ This means that the conditional distribution Equation (18) can be eliminated from the Gibbs sampler by setting $\kappa \left({\eta}_{i}\right)=1$ for the conditionals of $\mu $ and $\tau .$
- note 3:
- When the L${t}_{\nu}$FT model is used for reliability analysis, the last stage of the Bayesian hierarchical model in Equation (13) becomes ${\eta}_{i}\sim Gamma(\nu /2,\nu /2)$ with $\kappa \left({\eta}_{i}\right)={\eta}_{i}^{-1}.$ Thus, the conditional distribution in Equation (18) yields$$\left(\right)open="["\; close="]">{\eta}_{i}|{\mathit{\eta}}_{\backslash {\eta}_{i}},\mu ,{\mu}_{0},\tau ,$$
- note 4:
- The conditional distribution Equation (18) for LSFT model is$$\left(\right)open="["\; close="]">{\eta}_{i}|{\mathit{\eta}}_{\backslash {\eta}_{i}},\mu ,{\mu}_{0},\tau I({\eta}_{i}\in (0,1)),$$
- note 5:
- It is easily seen that the approximate conditional distribution ${\eta}_{i}^{2}$ for LLFT model is$$\left(\right)open="["\; close="]">{\eta}_{i}^{2}|{\mathit{\eta}}_{\backslash {\eta}_{i}},\mu ,{\mu}_{0},\tau .$$
- note 6:
- The convergence of an MCMC algorithm is an important issue for the correct estimation of the posterior distribution of interest. See [30] for an example of multiple convergence diagnosis and output analysis. When the Markov chain is converged, Rao–Blackwellization yields good estimates of $\mu $ and $\tau .$
- note 7:
- As measures of model comparison among SMLNFT models, a deviance information criterion (DIC) can be used. This measure can be calculated based on extensions of the MCMC method. See [31] and references therein for a review and comparisons of such extensions.

## 5. Numerical Illustrations

#### 5.1. Equipment Failure Data Example

#### 5.2. Artificial Data Example

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Densities of $Y\sim \mathcal{SMLN}(\mu ,{\sigma}^{2},\kappa ,G)$ for five different choices of the mixing variable distribution $G\left(\eta \right)$ and $\kappa \left(\eta \right)$: (

**a**) $(\mu ,\sigma )=(0,1)$ case and (

**b**) $(\mu ,\sigma )=(2,0.5)$ case.

**Figure 2.**Reliability and hazard functions for five $\mathcal{SMLN}(\mu ,{\sigma}^{2},\kappa ,G)$ models with $(\mu ,\sigma )=(2,0.5)$: (

**a**) reliability functions, $R(y;\mu ,{\sigma}^{2})$; (

**b**) hazard functions, $h(y;\mu ,{\sigma}^{2})$.

**Figure 3.**Estimated posterior predictive reliability measures for $\delta =0.9$: the label for the y-axis of (

**a**) is reliability and that of (

**b**) is hazard rate. The label for the x-axis is failure time and the value for tick marks in the y-axis of (

**b**) is multiplied by ${10}^{4}$.

Variable | Mean | S.D. | S–W (p-Value) | K–S (p-Value) |
---|---|---|---|---|

Y | 2201.488 | 2519.174 | 0.732 (<0.01) | 0.231 (<0.01) |

X | 7.143 | 1.088 | 0.982 (0.760) | 0.064 (>0.150) |

$\mathcal{C}$ | Parameter | $\mathit{\delta}=0$ | $\mathit{\delta}=0.5$ | $\mathit{\delta}=0.9$ | $\mathit{\delta}=1$ | S.D. | 2.5% | 97.5% | MC Error |
---|---|---|---|---|---|---|---|---|---|

$(7.5,\infty )$ | $\mu $ | 7.141 | 7.182 | 7.261 | 7.553 | 0.155 | 6.964 | 7.569 | 0.002 |

${\sigma}^{2}$ | 1.181 | 1.185 | 1.196 | 1.327 | 0.295 | 0.812 | 1.930 | 0.004 | |

$\alpha $ | 0.308 | 0.698 | 0.865 | 1.000 | - | - | - | - | |

$(8.0,\infty )$ | $\mu $ | 7.141 | 7.202 | 7.364 | 8.029 | 0.157 | 7.067 | 7.684 | 0.002 |

${\sigma}^{2}$ | 1.181 | 1.188 | 1.228 | 1.941 | 0.313 | 0.823 | 2.036 | 0.004 | |

$\alpha $ | 0.158 | 0.660 | 0.834 | 1.000 | - | - | - | - | |

$(9.0,\infty )$ | $\mu $ | 7.141 | 7.255 | 7.637 | 9.035 | 0.187 | 7.306 | 8.041 | 0.002 |

${\sigma}^{2}$ | 1.181 | 1.198 | 1.408 | 4.734 | 0.401 | 0.901 | 2.450 | 0.005 | |

$\alpha $ | 0.023 | 0.610 | 0.780 | 1.000 | - | - | - | - |

Model | Parameter | Mean | S.D. | MC Error | 2.5% | Median | 97.5% | DIC |
---|---|---|---|---|---|---|---|---|

LNFT | $\mu $ | 2.043 | 0.147 | <0.001 | 1.755 | 2.043 | 2.332 | 861.602 |

${\sigma}^{2}$ | 4.329 | 0.438 | 0.002 | 3.554 | 4.300 | 5.274 | - | |

LCFT | $\mu $ | 1.948 | 0.043 | <0.001 | 1.863 | 1.948 | 2.035 | 463.242 |

${\sigma}^{2}$ | 0.146 | 0.026 | <0.001 | 0.101 | 0.146 | 0.204 | - | |

L${t}_{5}$FT | $\mu $ | 1.948 | 0.043 | <0.001 | 1.862 | 1.948 | 2.033 | 394.339 |

${\sigma}^{2}$ | 0.289 | 0.036 | <0.001 | 0.225 | 0.286 | 0.365 | - | |

LSFT | $\mu $ | 1.992 | 0.118 | <0.001 | 1.761 | 1.992 | 2.227 | 867.325 |

${\sigma}^{2}$ | 0.557 | 0.098 | <0.001 | 0.389 | 0.548 | 0.776 | - | |

LLFT | $\mu $ | 2.025 | 0.132 | <0.001 | 1.767 | 2.025 | 2.282 | 842.847 |

${\sigma}^{2}$ | 1.363 | 0.176 | <0.001 | 1.056 | 1.350 | 1.745 | - |

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Kim, H.-J.
Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint. *Entropy* **2017**, *19*, 274.
https://doi.org/10.3390/e19060274

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Kim H-J.
Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint. *Entropy*. 2017; 19(6):274.
https://doi.org/10.3390/e19060274

**Chicago/Turabian Style**

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2017. "Bayesian Hierarchical Scale Mixtures of Log-Normal Models for Inference in Reliability with Stochastic Constraint" *Entropy* 19, no. 6: 274.
https://doi.org/10.3390/e19060274