# Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Motivation and Organization

## 2. The Lifetime Distribution and Statistical Methods

**Method I:**The MH-GS MCMC method.**Initial Step:**Give arbitrary initial vales of ${\beta}^{\left(0\right)}$, ${\delta}_{1}^{\left(0\right)}$, ${\delta}_{2}^{\left(0\right)}$, ${\theta}^{\left(0\right)}$, and ${p}_{1}^{\left(0\right)}$ from the respective parameter domains; propose a transition probability function, ${\omega}_{j}({\theta}_{j}^{*}\mid {\theta}_{j}^{\left(i\right)})$, to generate a possible ${\theta}_{j}^{*}$, given ${\theta}_{j}^{i}$ for $j=1,2,3,4,5$; and let $i=0$.

**Step 1:**- Implementing Step 2 N times, i.e., $1\le i\le N$, where N is a big positive number.
**Step 2:**- For $i=i+1$, use Step 2.1 to Step 2.5 to update ${\beta}^{\left(i\right)}$, ${\delta}_{1}^{\left(i\right)}$, ${\delta}_{2}^{\left(i\right)}$, ${\theta}^{\left(i\right)}$ and ${p}^{\left(i\right)}$. The acceptance criterion is based on the M-H algorithm, see in [35].
- Step 2.1:
- Generate ${\beta}^{*}\sim {\omega}_{1}\left({\beta}^{*}\mid {\beta}^{\left(i\right)}\right)$ and $u\sim U(0,1)$.If $u\le min\left\{1,\frac{\pi \left({\beta}^{*}\mid {\delta}_{1}^{\left(i\right)},{\delta}_{2}^{\left(i\right)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{1}\left({\beta}^{\left(i\right)}\mid {\beta}^{*}\right)}{\pi \left({\beta}^{\left(i\right)}\mid {\delta}_{1}^{\left(i\right)},{\delta}_{2}^{\left(i\right)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{1}\left({\beta}^{*}\mid {\beta}^{\left(i\right)}\right)}\right\}$, update ${\beta}^{(i+1)}$ by ${\beta}^{*}$;otherwise, ${\beta}^{(i+1)}={\beta}^{\left(i\right)}$.
- Step 2.2:
- Generate ${\delta}_{1}^{*}\sim {\omega}_{2}\left({\delta}_{1}^{*}\mid {\delta}_{1}^{\left(i\right)}\right)$ and $u\sim U(0,1)$.If $u\le min\left\{1,\frac{\pi \left({\delta}_{1}^{*}\mid {\beta}^{(i+1)},{\delta}_{2}^{\left(i\right)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{2}\left({\delta}_{1}^{\left(i\right)}\mid {\delta}_{1}^{*}\right)}{\pi \left({\delta}_{1}^{\left(i\right)}\mid {\beta}^{(i+1)},{\delta}_{2}^{\left(i\right)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{2}\left({\delta}_{1}^{*}\mid {\delta}_{1}^{\left(i\right)}\right)}\right\}$, update ${\delta}_{1}^{(i+1)}$ by ${\delta}_{1}^{*}$;otherwise, ${\delta}_{1}^{(i+1)}={\delta}_{1}^{\left(i\right)}$.
- Step 2.3:
- Generate ${\delta}_{2}^{*}\sim {\omega}_{3}\left({\delta}_{2}^{*}\mid {\delta}_{2}^{\left(i\right)}\right)$ and $u\sim U(0,1)$.If $u\le min\left\{1,\frac{\pi \left({\delta}_{2}^{*}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{2}\left({\delta}_{2}^{\left(i\right)}\mid {\delta}_{2}^{*}\right)}{\pi \left({\delta}_{2}^{\left(i\right)}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\theta}^{\left(i\right)},{p}^{\left(i\right)},\mathit{y}\right){\omega}_{2}\left({\delta}_{2}^{*}\mid {\delta}_{2}^{\left(i\right)}\right)}\right\}$, update ${\delta}_{2}^{(i+1)}$ by ${\delta}_{2}^{*}$;otherwise, ${\delta}_{2}^{(i+1)}={\delta}_{2}^{\left(i\right)}$.
- Step 2.4:
- Generate ${\theta}^{*}\sim {\omega}_{4}\left({\theta}^{*}\mid {\theta}^{\left(i\right)}\right)$ and $u\sim U(0,1)$.If $u\le min\left\{1,\frac{\pi \left({\theta}^{*}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\delta}_{2}^{(i+1)},{p}^{\left(i\right)},\mathit{y}\right){q}_{3}\left({\theta}^{\left(i\right)}\mid {\theta}^{*}\right)}{\pi \left({\theta}^{\left(i\right)}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\delta}_{2}^{(i+1)},{p}^{\left(i\right)},\mathit{y}\right){q}_{3}\left({\theta}^{*}\mid {\theta}^{\left(i\right)}\right)}\right\}$, update ${\theta}^{(i+1)}$ by ${\theta}^{*}$;otherwise, ${\theta}^{(i+1)}={\theta}^{\left(i\right)}$.
- Step 2.5:
- Generate ${p}^{*}\sim {\omega}_{5}\left({p}^{*}\mid {p}^{\left(i\right)}\right)$ and $u\sim U(0,1)$.If $u\le min\left\{1,\frac{\pi \left({p}^{*}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\delta}_{2}^{(i+1)},{\theta}^{(i+1)},\mathit{y}\right){q}_{3}\left({p}^{\left(i\right)}\mid {p}^{*}\right)}{\pi \left({p}^{\left(i\right)}\mid {\beta}^{(i+1)},{\delta}_{1}^{(i+1)},{\delta}_{2}^{(i+1)},{\theta}^{(i+1)},\mathit{y}\right){q}_{3}\left({p}^{*}\mid {p}^{\left(i\right)}\right)}\right\}$, update ${p}^{(i+1)}$ by ${p}^{*}$;otherwise, ${p}^{(i+1)}={p}^{\left(i\right)}$.

**Step 3:**- Go to Step 2 until $i=N$.
**Step 4:**- Denote the obtained Markov chains by $\left\{{\beta}^{\left(i\right)},i=1,2,\dots ,N\right\}$, $\left\{{\delta}_{1}^{\left(i\right)},i=1,2,\dots ,N\right\}$, $\left\{{\delta}_{2}^{\left(i\right)},i=1,2,\dots ,N\right\}$, $\left\{{\theta}^{\left(i\right)},i=1,2,\dots ,N\right\}$, and $\left\{{p}^{\left(i\right)},i=1,2,\dots ,N\right\}$, respectively. Based on the square error loss function, the Bayes estimator can be the sample mean of the obtained Markov chain after burn-in, ${N}_{0}$, and is denoted by$${\widehat{\sigma}}_{B}=\frac{1}{N-{N}_{0}}\sum _{t={N}_{0}+1}^{N}{\sigma}^{\left(t\right)},\phantom{\rule{3.33333pt}{0ex}}\sigma =\beta ,{\delta}_{1},{\delta}_{2},\theta ,p.$$

## 3. Discussions

## 4. Monte Carlo Simulations

- The estimation results in Table 3, Table 4 and Table 5 indicate that the estimation quality of the Bayes estimate for the mixture proportion is worse than the estimation quality of the Bayes estimates for other parameters. Moreover, the censoring rate significantly influences the estimation quality of Bayes estimates.
- The maximum likelihood estimation method is less reliable for estimating the mixture proportion and shape parameter. Moreover, the acceptance rate is low for the Markov chains of the parameters of $\theta $ and p if non-informative prior is used to implement the Bayesian estimation methods. The rBias and rsqMSE of the MLEs of $\beta $ and p in Table 6, Table 7 and Table 8 are large even when the sample size increases to 100 or the censoring rate is high. The findings mean that it could fail to use the maximum likelihood estimation method to obtain reliable estimates of the model parameters if the component lifetimes in CDS follow a baseline mixture distribution.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Kamps, U. A concept of generalized order statistics. J. Stat. Plan. Inference
**1995**, 48, 1–23. [Google Scholar] [CrossRef] - Cramer, E.; Kamps, U. Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Ann. Inst. Stat. Math.
**1996**, 48, 535–549. [Google Scholar] [CrossRef] - Cramer, E.; Kamps, U. Estimation with sequential order statistics from exponential distributions. Ann. Inst. Stat. Math.
**2001**, 53, 307–324. [Google Scholar] [CrossRef] - Cramer, E.; Kamps, U. Marginal distributions of sequential and generalized order statistics. Metrika
**2003**, 58, 293–310. [Google Scholar] [CrossRef] - Revathy, S.A.; Chandrasekar, B. Equivariant estimation of parameters based on sequential order statistics from (1,3) and (2,3) Systems. Commun. Stat. Theory Methods
**2007**, 36, 541–548. [Google Scholar] [CrossRef] - Zhuang, W.; Hu, T. Multivariate stochastic comparisons of sequential order statistics. Probab. Eng. Inform. Sci.
**2007**, 21, 47–66. [Google Scholar] [CrossRef] - Balakrishnan, N.; Beutner, E.; Kamps, U. Modeling parameters of a load-sharing system through link functions in sequential order statistics models and associated inference. IEEE Trans. Reliab.
**2011**, 60, 605–611. [Google Scholar] [CrossRef] - Burkschat, M. Systems with failure-dependent lifetimes of components. J. Appl. Probab.
**2009**, 46, 1052–1072. [Google Scholar] [CrossRef] [Green Version] - Beutner, E.; Kamps, U. Order restricted statistical inference for scale parameters based on sequential order statistics. J. Stat. Plan. Inference
**2009**, 139, 2963–2969. [Google Scholar] [CrossRef] - Deshpande, J.V.; Dewan, I.; Naik-Nimbalkar, U.V. A family of distributions to model load sharing systems. J. Stat. Plan. Inference
**2010**, 140, 1441–1451. [Google Scholar] [CrossRef] - Beutner, E. Nonparametric model checking for k-out-of-n systems. J. Stat. Plan. Inference
**2010**, 140, 626–639. [Google Scholar] [CrossRef] - Bedbur, S. UMPU tests based on sequential order statistics. J. Stat. Plan. Inference
**2010**, 140, 2520–2530. [Google Scholar] [CrossRef] - Schenk, N.; Burkschat, M.; Cramer, E.; Kamps, U. Bayesian estimation and prediction with multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions. J. Stat. Plan. Inference
**2011**, 141, 1575–1587. [Google Scholar] [CrossRef] - Navarro, J.; Burkschat, M. Coherent systems based on sequential order statistics. Nav. Res. Log.
**2011**, 58, 123–135. [Google Scholar] [CrossRef] - Balakrishnan, N.; Kamps, U.; Kateri, M.A. Sequential order statistics approach to step-stress testing. Ann. Inst. Stat. Math.
**2012**, 64, 303–318. [Google Scholar] [CrossRef] - Burkschat, M.; Navarro, J. Dynamic signatures of coherent systems based on sequential order statistics. J. Appl. Probab.
**2013**, 50, 272–287. [Google Scholar] [CrossRef] - Burkschat, M.; Torrado, N. On the reversed hazard rate of sequential order statistics. Stat. Probabil. Lett.
**2014**, 85, 106–113. [Google Scholar] [CrossRef] - Park, C. Parameter estimation from load-sharing system data using the expectation-maximization algorithm. IIE Trans.
**2013**, 45, 147–163. [Google Scholar] [CrossRef] - Bedbur, S.; Burkschat, M.; Kamps, U. Inference in a model of successive failures with shape-adjusted hazard rates. Ann. Inst. Stat. Math.
**2016**, 68, 639–657. [Google Scholar] [CrossRef] - Sutar, S.S.; Naik-Nimbalkar, U.V. Accelerated failure time models for load sharing systems. IEEE Trans. Reliab.
**2014**, 63, 706–714. [Google Scholar] [CrossRef] - Esmailian, M.; Doostparast, M. Estimation based on sequential order statistics with random removals. Probab. Math. Stat.
**2014**, 34, 81–95. [Google Scholar] - Shafay, R.; Balakrishnan, N.; Sultan, K.S. Two-sample Bayesian prediction for sequential order statistics from exponential distribution based on multiply Type-II censored samples. J. Stat. Comput. Simul.
**2014**, 84, 526–544. [Google Scholar] [CrossRef] - Hashempour, M.; Doostparast, M. Estimation with non-homogeneous sequential k-out-of-n system lifetimes. Reliab. Theory Appl.
**2015**, 49, 49–52. [Google Scholar] - Bedbur, S.; Kamps, U.; Kateri, M. Meta-analysis of general step-stress experiments under repeated type-II censoring. Appl. Math. Model.
**2015**, 39, 2261–2275. [Google Scholar] [CrossRef] - Balakrishnan, N.; Jiang, N.; Tsai, T.R.; Lio, Y.L.; Chen, D.G. Reliability Inference on composite dynamic systems based on Burr type-XII distribution. IEEE Trans. Reliab.
**2015**, 64, 144–153. [Google Scholar] [CrossRef] - Hashempour, M.; Doostparast, M. Statistical evidences in sequential order statistics arising from a general family of lifetime distributions. Istatistik
**2016**, 9, 29–41. [Google Scholar] - Burkschat, M.; Cramer, E.; Górny, J. Type-I censored sequential k-out-of-n systems. Appl. Math. Model
**2016**, 40, 8156–8174. [Google Scholar] [CrossRef] - Hashempour, M.; Doostparast, M. Bayesian inference on multiply sequential order statistics from heterogeneous exponential populations with GLR test for homogeneity. Commun. Stat. Theory Methods
**2017**, 46, 8086–8100. [Google Scholar] [CrossRef] - Baratnia, M.; Doostparast, M. Modelling lifetime of sequential r-out-of-n systems with independent and heterogeneous components. Commun. Stat. Simul. Comput.
**2017**, 46, 7365–7375. [Google Scholar] [CrossRef] - Burkschat, M.; Navarro, J. Ageing properties of sequential order statistics. Probab. Eng. Inform. Sci.
**2011**, 25, 1–19. [Google Scholar] [CrossRef] - Barakat, H.M.; Nigm, E.M.; El-Adll, M.E.; Yusuf, M. Prediction of future generalized order statistics based on exponential distribution with random sample size. Stat. Pap.
**2018**, 59, 605–631. [Google Scholar] [CrossRef] - Katzur, A.; Kamps, U. Classification using sequential order statistics. Adv. Data Anal. Classif.
**2020**, 14, 201–230. [Google Scholar] [CrossRef] - Tsai, T.R.; Xin, H.; Kao, C.H. Bayesian estimation based on sequential order statistics for heterogeneous baseline Gompertz distributions. Mathematics
**2021**, 9, 145. [Google Scholar] [CrossRef] - Tsai, T.R.; Ng, H.K.; Pham, H.; Lio, Y.L.; Chiang, J.Y. Reliability inference for VGA adapter from dual suppliers based on contaminated type-I interval-censored data. Qual. Reliab. Eng. Int.
**2019**, 35, 2297–2313. [Google Scholar] [CrossRef] - Robert, C.P. The Metropolis-Hastings Algorithm. In Wiley StatsRef: Statistics Reference Online; Wiley: Hoboken, NJ, USA, 2016. [Google Scholar]

**Figure 2.**The Markov chains of ${\delta}_{1}$. The horizontal line is the true value of ${\delta}_{1}$.

**Figure 3.**The Markov chains of ${\delta}_{2}$. The horizontal line is the true value of ${\delta}_{2}$.

BXIID, mBXIID | Burr type-XII distribution, mixture-BXIID |

CDF | cumulative density function |

CDS | composite dynamical system |

ECI | equal-tailed credible interval |

GOS | generalized order statistic |

GS | Gibbs sampling |

HCI | highest credible interval |

IS | important sampling |

LED | light emitting diode |

MCMC | Markov chain Monte Carlo |

M-H | Metropolis–Hastings |

MLE | maximum likelihood estimate |

MSE | mean square error |

probability density function | |

PTHR | power-trend hazard rate |

SOS | sequential order statistic |

VGA | video graphics array |

**Table 2.**The numerical sequential order statistic (SOS) sample from mixture-BXIID for $\beta =5$, ${\delta}_{1}=5$, ${\delta}_{2}=10$, $p=0.7$, and $\theta =1.02$ when $n=30$ and $r=15$.

0.0081 | 0.1869 | 0.1962 | 0.2422 | 0.2572 |

0.2649 | 0.3477 | 0.3752 | 0.4324 | 0.5402 |

0.6523 | 0.6579 | 0.7236 | 0.7536 | 0.8726 |

**Table 3.**The relative bias (rBias) and relative square root of the mean square error (rsqMSE) of Bayes estimates for $p=0.2$.

n | r | ${\widehat{\mathit{\beta}}}_{\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{1\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{2\mathit{B}}$ | ${\widehat{\mathit{\theta}}}_{\mathit{B}}$ | ${\widehat{\mathit{p}}}_{\mathit{B}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBiae | 0.1032 | −0.0279 | −0.0134 | 0.0039 | 0.6278 |

rsqMSE | 0.1034 | 0.0281 | 0.0135 | 0.0039 | 0.6282 | ||

50 | 25 | rBias | 0.0953 | 0.0001 | −0.0078 | −0.0055 | 0.6083 |

rsqMSE | 0.0955 | 0.0032 | 0.0080 | 0.0055 | 0.6086 | ||

80 | 30 | rBias | 0.0275 | 0.0506 | 0.0194 | −0.0060 | 0.6146 |

rsqMSE | 0.0280 | 0.0507 | 0.0195 | 0.0060 | 0.6149 | ||

100 | 30 | rBias | −0.1522 | 0.0073 | −0.0199 | −0.0070 | 0.5168 |

rsqMSE | 0.1522 | 0.0079 | 0.0200 | 0.0070 | 0.5172 |

n | r | ${\widehat{\mathit{\beta}}}_{\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{1\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{2\mathit{B}}$ | ${\widehat{\mathit{\theta}}}_{\mathit{B}}$ | ${\widehat{\mathit{p}}}_{\mathit{B}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBias | 0.0092 | −0.0029 | 0.0359 | 0.0053 | 0.2283 |

rsqMSE | 0.0112 | 0.0044 | 0.0359 | 0.0053 | 0.2288 | ||

50 | 25 | rBias | 0.2052 | −0.0036 | −0.0421 | −0.0053 | 0.2994 |

rsqMSE | 0.2054 | 0.0047 | 0.0422 | 0.0053 | 0.2997 | ||

80 | 30 | rBias | 0.3267 | 0.0425 | 0.0054 | −0.0099 | 0.2757 |

rsqMSE | 0.3268 | 0.0426 | 0.0057 | 0.0099 | 0.2761 | ||

100 | 30 | rBias | 0.0895 | −0.0443 | 0.0009 | −0.0086 | 0.2236 |

rsqMSE | 0.0897 | 0.0444 | 0.0019 | 0.0086 | 0.2241 |

n | r | ${\widehat{\mathit{\beta}}}_{\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{1\mathit{B}}$ | ${\widehat{\mathit{\delta}}}_{2\mathit{B}}$ | ${\widehat{\mathit{\theta}}}_{\mathit{B}}$ | ${\widehat{\mathit{p}}}_{\mathit{B}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBias | 0.0697 | −0.0113 | 0.0217 | 0.0041 | 0.0035 |

rsqMSE | 0.0701 | 0.0118 | 0.0217 | 0.0041 | 0.0099 | ||

50 | 25 | rBias | −0.0310 | 0.0114 | −0.0155 | −0.0044 | −0.0167 |

rsqMSE | 0.0315 | 0.0118 | 0.0156 | 0.0044 | 0.0192 | ||

80 | 30 | rBias | −0.2359 | <0.0001 | 0.0165 | −0.0075 | −0.0646 |

rsqMSE | 0.2359 | 0.0029 | 0.0166 | 0.0075 | 0.0653 | ||

100 | 30 | rBias | 0.1844 | −0.0119 | 0.0060 | −0.0066 | 0.0442 |

rsqMSE | 0.1845 | 0.0122 | 0.0062 | 0.0066 | 0.0452 |

n | r | $\widehat{\mathit{\beta}}$ | ${\widehat{\mathit{\delta}}}_{1}$ | ${\widehat{\mathit{\delta}}}_{2}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{p}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBias | 5.3671 | 0.0603 | 0.0549 | −0.0179 | −0.7863 |

rsqMSE | 5.3673 | 0.0604 | 0.055 | 0.0179 | 0.7863 | ||

50 | 25 | rBias | 5.5802 | 0.1087 | 0.0203 | −0.0185 | −0.8204 |

rsqMSE | 5.5804 | 0.1087 | 0.0204 | 0.0185 | 0.8205 | ||

50 | 30 | rBias | 3.3028 | 0.0643 | 0.0206 | −0.0187 | −0.8855 |

rsqMSE | 3.3029 | 0.0644 | 0.0207 | 0.0187 | 0.8855 | ||

100 | 30 | rBias | 4.2000 | 0.0955 | 0.0556 | −0.0185 | −0.8890 |

rsqMSE | 4.2001 | 0.0955 | 0.0557 | 0.0185 | 0.8891 |

n | r | $\widehat{\mathit{\beta}}$ | ${\widehat{\mathit{\delta}}}_{1}$ | ${\widehat{\mathit{\delta}}}_{2}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{p}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBias | 7.8958 | 0.0631 | 0.0374 | −0.0174 | −0.8314 |

rsqMSE | 7.8961 | 0.0632 | 0.0374 | 0.0174 | 0.8315 | ||

50 | 25 | rBias | 5.0357 | 0.0586 | 0.0811 | −0.0184 | −0.8866 |

rsqMSE | 5.0358 | 0.0587 | 0.0811 | 0.0184 | 0.8867 | ||

50 | 30 | rBias | 2.9665 | 0.0719 | 0.0419 | −0.0187 | −0.9196 |

rsqMSE | 2.9665 | 0.0720 | 0.0419 | 0.0187 | 0.9197 | ||

100 | 30 | rBias | 4.7010 | 0.0766 | 0.0574 | −0.0188 | −0.9245 |

rsqMSE | 4.7011 | 0.0766 | 0.0574 | 0.0188 | 0.9245 |

n | r | $\widehat{\mathit{\beta}}$ | ${\widehat{\mathit{\delta}}}_{1}$ | ${\widehat{\mathit{\delta}}}_{2}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{p}}$ | |
---|---|---|---|---|---|---|---|

30 | 15 | rBias | 6.5672 | 0.1020 | 0.0448 | −0.0175 | −0.8937 |

rsqMSE | 6.5674 | 0.1021 | 0.0448 | 0.0175 | 0.8937 | ||

50 | 25 | rBias | 4.5287 | 0.0565 | 0.0442 | −0.0182 | −0.9311 |

rsqMSE | 4.5288 | 0.0567 | 0.0442 | 0.0182 | 0.9312 | ||

50 | 30 | rBias | 5.6215 | 0.0777 | 0.0278 | −0.0186 | −0.9380 |

rsqMSE | 5.6216 | 0.0778 | 0.0279 | 0.0186 | 0.9381 | ||

100 | 30 | rBias | 4.4166 | 0.1011 | 0.0533 | −0.0188 | −0.9485 |

rsqMSE | 4.4167 | 0.1012 | 0.0534 | 0.0188 | 0.9485 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsai, T.-R.; Lio, Y.; Xin, H.; Pham, H.
Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution. *Mathematics* **2021**, *9*, 810.
https://doi.org/10.3390/math9080810

**AMA Style**

Tsai T-R, Lio Y, Xin H, Pham H.
Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution. *Mathematics*. 2021; 9(8):810.
https://doi.org/10.3390/math9080810

**Chicago/Turabian Style**

Tsai, Tzong-Ru, Yuhlong Lio, Hua Xin, and Hoang Pham.
2021. "Parameter Estimation for Composite Dynamical Systems Based on Sequential Order Statistics from Burr Type XII Mixture Distribution" *Mathematics* 9, no. 8: 810.
https://doi.org/10.3390/math9080810