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We present Bayes estimators, highest posterior density (HPD) intervals, and maximum likelihood estimators (MLEs), for the Maxwell failure distribution based on Type II censored data, i.e. using the first r lifetimes from a group of n components under test. Reliability/Hazard function estimates, Bayes predictive distributions and highest posterior density prediction intervals for a future observation are also considered. Two data examples and a Monte Carlo simulation study are used to illustrate the results and to compare the performances of the different methods.

The prediction problems of lifetime models are very important and have been studied, among others by Aitchi- son & Dunsmore (1975) [

Tyagi and Bhattacharya (1989a, b) [

The purpose of this paper is to derive maximum likelihood estimators (or estimators that maximize the likelihood function), Bayes estimators in terms of the mean of the posterior distribution, and highest posterior density intervals for θ (which are intervals with a posterior probability of

The Maxwell probability density function (pdf) and cumulative distribution function (cdf), are respectively given by:

and

where

A result we will use in the developments of this paper is the following. If

From a Bayesian perspective, in this study, we consider an asymptotically locally invariant prior,

where

and

For the pdf (1), it can be shown that (Howlader and Hossain (1998) [

In addition, we consider Jeffreys prior given by

where

In Section 2, we describe the procedure for estimating the parameter θ, the reliability function

We assume a group of n components have lifetimes which follow a Maxwell distribution. The failure times are recorded as they occur until a fixed (known) number r of components have failed. As it is quite common in life testing situations, (e.g., destructive tests, high cost of testing an item, etc.) only the first r lifetimes in a sample of n units can be obtained. Let

Therefore, the likelihood function for θ in terms of y can be written as,

where

Furthermore,

and the log-likelihood is equal, except for a constant, to

The maximum likelihood estimator (MLE) of θ,

quantile of a

By the invariance property of MLE’s, we get that the MLE of the reliability function,

Also, the MLE for the failure rate or hazard function,

Combining the likelihood function and the prior

A squared error loss is appropriate when decisions become gradually more damaging for larger errors. The Bayes estimator of

In general, this expected value does not have a closed form solution. We rely on MCMC calculations to approximate

The Bayes estimates of

and

By graphing posterior realizations of

A credible interval for θ can be obtained by taking our sample

Analogously, for the reliability function

sample for

on these values provide a

It is not automatic to compute a highest posterior density (HPD) interval from a Monte Carlo sample specially if the posterior is far from symmetric or multimodal. However, we use the method in Liu, Gelman and Zheng (2013) [

Let z be a future observation which has already survived

and

1. Sample

2. Sample

3. Repeat steps (1) and (2) M times where M represent a fixed number of MCMC samples.

This procedure provides a sample,

A

sample quantiles of

The following data represent noise levels in cryogenic microwave receivers and were obtained from Darrell Hicks at the National Radio Astronomy Observatory, Socorro, NM. We arranged the observations in ascending order and dropped the last 11 data points to induce censoring, which leads to a situation with n = 86 and r = 75. After transforming the data with x^{2},

The MLE of θ obtained numerically with a Nelder-Mead, Quasi-Newton algorithm (in 100-th units) is

In

HPD intervals from our MCMC samples are as follows. The HPD interval for θ is (16.78278, 24.48906) which is not very different compared to an interval based on the 0.25 and 0.975 sample quantiles. The HPD for the predictive distribution of a future z observation is (9.36, 12.82) which also remains similar to the one reported with 0.25, 0.975 sample quantiles. In

We now use the Kazmi et al. (2012) [

determined by analyzing the pressure profiles in the spherical vessel and checked by direct observation of flame propagation. The data related to the burning velocity (cm/sec) of different chemical materials is given below: 68, 61, 64, 55, 51, 68, 44, 82, 60, 89, 61, 54, 166, 66, 50, 87, 48, 42, 58, 46, 67, 46, 46, 44, 48, 56, 47, 54, 47, 80, 38, 108, 46, 40, 44, 312, 41, 31, 40, 41, 40, 56, 45, 43, 46, 46, 46, 46, 52, 58, 82, 71, 48, 39, 41. The source of the above explained data related to the burning velocity of different chemical materials for the year 2005 is available from http://www.cheresources.com/mists.pdf. In this paper, and to compute our estimates, we consider the units of burning velocity as m/sec.

In

lengths of the HPD intervals for θ under Hartigan prior are slightly shorter compared to the intervals obtained with Jeffreys’ and Gamma priors for both censoring times considered.

In addition, we also computed the approximate 95% confidence interval for θ with the MLE asymptotic approximation to the Normal distribution as described in Section 2. For a censoring time of 64(0.64), the interval is (0.1706, 0.2790) and for a censoring time of 68 (0.68), the interval is (0.1681, 0.2697). Again, the expected Fisher’s information was approximated with the observed Fisher’s information at the MLE. In contrast to the NRAO data set example, the HPD intervals provide different results compared to a MLE approximation that relies in asymptotic normality.

Censoring | 95% HPD Intervals | |
---|---|---|

0% | 6.583 | (1.959, 12.58) |

10% | 10.7 | (9.729, 13.12) |

13% | 10.33 | (9.328, 12.83) |

20% | 7.735 | (6.634, 10.34) |

30% | 5.807 | (4.637, 8.367) |

Priors | Censoring Time = 64 (0.64) | Censoring Time = 68 (0.68) |
---|---|---|

Hartigan | (0.2107, 0.3393) | (0.1947, 0.3110) |

Jeffreys | (0.2137, 0.3469) | (0.1970, 0.3171) |

Gamma (1,1) | (0.2163, 0.3485) | (0.1983, 0.3171) |

In order to assess the performance of the estimation and prediction approaches proposed in this paper, we perform a Monte Carlo simulation study based on 5000 simulated data samples of sizes n = 15, 25, 35, 50 and 100 with 10% and 30% censoring respectively. For each of our simulated data samples, we computed the Bayes estimates using a squared error loss function after a burn-in of 500 MCMC iterations. For different values of t and using the above mentioned sample sizes, we computed the average Bayes estimates,

Furthermore, for a second simulation based on 5000 samples and for various values of θ, in

The main contribution of this paper is to obtain the parameter estimates for the Maxwell failure distribution under Type II censoring via the Gamma distribution, using Bayesian estimation under different priors and compared it with Maximum Likelihood Estimation (MLE). MLE can be thought as the maximum of the posterior distribution under an unrestricted uniform prior. We observe from

Time | Estimates | n = 15 | n = 25 | n = 35 | n = 50 | n = 100 |
---|---|---|---|---|---|---|

t = 0.5 | 0.9992 | 0.9993 | 0.9993 | 0.9993 | 0.9994 | |

0.0049 | 0.0043 | 0.0041 | 0.0041 | 0.0037 | ||

t = 2.5 | 0.9132 | 0.9215 | 0.9251 | 0.9250 | 0.9315 | |

0.1027 | 0.0926 | 0.0883 | 0.0884 | 0.0807 | ||

t = 5.0 | 0.5609 | 0.5880 | 0.6001 | 0.5986 | 0.6236 | |

0.3073 | 0.2807 | 0.2694 | 0.2698 | 0.2488 | ||

t = 7.5 | 0.2145 | 0.2365 | 0.2464 | 0.2430 | 0.2684 | |

0.5308 | 0.4883 | 0.4701 | 0.4710 | 0.4368 |

Time | Estimates | n = 15 | n = 25 | n = 35 | n = 50 | n = 100 |
---|---|---|---|---|---|---|

t = 0.5 | 0.9995 | 0.9996 | 0.9996 | 0.9996 | 0.9996 | |

0.0031 | 0.0026 | 0.0025 | 0.0024 | 0.0022 | ||

t = 2.5 | 0.9434 | 0.9506 | 0.9535 | 0.9552 | 0.9572 | |

0.0667 | 0.0582 | 0.0547 | 0.0527 | 0.0503 | ||

t = 5.0 | 0.6789 | 0.7087 | 0.7216 | 0.7294 | 0.7388 | |

0.2087 | 0.1850 | 0.1752 | 0.1694 | 0.1627 | ||

t = 7.5 | 0.3432 | 0.3780 | 0.3945 | 0.4048 | 0.4181 | |

0.3703 | 0.3314 | 0.3153 | 0.3057 | 0.2944 |

sizes and for small values of t. When t increases the values of

For estimating the parameter θ of the Maxwell distribution under Type II censoring, it appears to be clear from all our numerical results that Bayes estimation is appropriate or in some cases, superior than MLE, but with the MLE method as a good competitor. In the second data example for which the sample size n = 55, an approximate 95% confidence interval for θ provides a result that is different to HPD intervals. A limitation of our approach is the use of a square data transformation for the Gamma distribution which can pose challenges under the Type II censoring for large numerical data values. The priors considered for comparisons are obtained under the non-censored case, therefore, a similar study can be attempted computing Jeffreys’ and Hartigan’s priors numerically and where the Type II censoring is incorporated into the likelihood function. An Openbugs model that implements the approach described in this manuscript with the Gamma and truncated Gamma distributions, is available by request from the second author.

We thank the referee and Editor for their comments and suggestions about our paper. This project was supported

by New Mexico Tech, Socorro, New Mexico, USA and research funds from the College of Arts and Sciences, The University of New Mexico,USA. G. Huerta performed all the computations and simulations applied in this paper. He also proposed the model from a Gamma distribution perspective. The Openbugs model for this paper is available from G. Huerta.

Anwar M.Hossain,GabrielHuerta, (2016) Bayesian Estimation and Prediction for the Maxwell Failure Distribution Based on Type II Censored Data. Open Journal of Statistics,06,49-60. doi: 10.4236/ojs.2016.61007