A sequence of independent lifetimes

Maxwell distribution plays an important role in physics and other allied sciences. This paper introduces a discrete analogue of the Maxwell distribution, called discrete Maxwell distribution (dMax distribution). This distribution is suggested as a suitable reliability model to fit a range of discrete lifetime data.

In reliability theory many continuous lifetime models have been suggested and studied. However, it is sometimes impossible or inconvenient to measure the life length of a device, on a continuous scale. In practice, we come across situations, where lifetime of a device is considered to be a discrete random variable. For example, in an on/off switching device, the lifetime of the switch is a discrete random variable. Also, the number of voltage fluctuations which an electrical or electronic item can withstand before its failure is a discrete rv.

If the lifetimes of individuals in some population are grouped or when lifetime refers to an integral number of cycles of some sort, it may be desirable to treat it as a discrete rv. When a discrete model is used with lifetime data, it is usually a multinomial distribution, which arises because effectively continuous data have been grouped. Some situations may demand for another discrete distribution, usually over the non negative integers. Such situations are best treated individually, but generally one tries to adopt one of the standard discrete distributions.

In the last two decades, standard discrete distributions like geometric and negative binomial have been employed to model lifetime data. However, there is a need to find more plausible discrete lifetime distributions to fit to various types of lifetime data. For this purpose, popular continuous lifetime distributions can be helpful in the following manner.

The Maxwell distribution defines the speed of molecules in thermal equilibrium under some conditions as defined in statistical mechanics. For example, this distribution explains many fundamental gas properties in kinetic theory of gases; distribution of energies and moments, and so forth.

Tyagi and Bhattachary ([

Chaturvedi and Rani [

In this section we propose change point model on discrete Maxwell distribution. We also derived the Bayes estimates for the model. Let

Later

The ML method, as well as other classical approaches, is based only on the empirical information provided by the data. However, when there is some technical knowledge on the parameters of the distribution available, a Bayes procedure seems to be an attractive inferential method. The Bayes procedure is based on a posterior density, say

We also suppose that some information on

Following Calabria and Pulcini [

As in the study by Broemeling and Tsurumi [

And

Marginal posterior density of

The marginal posterior density of change point

Bayes estimator of

Other Bayes estimators of

A noninformative prior is a prior that adds no information to that contained in the empirical data. Thus, a Bayes inference based upon noninformative prior has generally a theoretical interest only, since, from an engineering view point, the Bayes approach is very attractive for it allows incorporating expert opinion or technical knowledge in the estimation procedure. Let the joint noninformative prior density of

The marginal posterior density of change point

Bayes estimator of

The loss function

In this section, we derive Bayes estimators of change point

The posterior expectation of the Linex loss function is

Minimizing expected loss function

Minimizing expected loss function

Minimizing expected loss function

Another loss function, called General Entropy (GE) loss function, proposed by Calabria and Pulcini [

Minimizing expectation

Minimizing expected loss function

Minimizing expected loss function

Putting

Note that, for

We have generated 30 random observations from dmax distribution involving change point discussed in Section

Generated samples from dMax distribution.

Sample no. | Sample | Actual reliability | ||||

1 | 30 | 15 | ||||

2 | 50 | 25 | 0.0460 | 0.0011 | ||

3 | 50 | 35 | ||||

4 | 30 | 15 | 0, | |||

5 | 50 | 25 | 0, | 0.8495 | 0.6594 | |

6 | 50 | 35 |

We also compute the Bayes estimators

We have generated 6 random samples from discrete Maxwell distribution involving change point discussed in Section

Bayes estimate of

Bayes estimates of | Bayes Estimates of | |||
---|---|---|---|---|

Sample no. | ||||

1 | 30 | 15 | 1.0 | 0.5 |

2 | 50 | 25 | 1.0 | 0.5 |

3 | 50 | 35 | 1.0 | 0.5 |

4 | 30 | 15 | 5.0 | 2.0 |

5 | 50 | 25 | 5.0 | 2.0 |

6 | 50 | 35 | 5.0 | 2.0 |

The Bayes estimates using Linex loss function. (

Informative prior | 0.09 | 15 | 1.0 | 0.53 |

0.10 | 15 | 1.0 | 0.52 | |

0.20 | 15 | 1.0 | 0.51 | |

1.2 | 14 | 0.97 | 0.48 | |

1.5 | 13 | 0.90 | 0.44 | |

−1.0 | 16 | 1.5 | 0.57 | |

−2.0 | 17 | 1.7 | 0.59 | |

Noninformative Prior | 0.09 | 14 | 0.93 | 0.54 |

0.10 | 14 | 0.92 | 0.51 | |

0.20 | 14 | 0.91 | 0.50 | |

1.2 | 13 | 0.85 | 0.46 | |

1.5 | 12 | 0.82 | 0.41 | |

−1.0 | 16 | 1.5 | 0.56 | |

−2.0 | 17 | 1.7 | 0.57 |

The Bayes estimates using General Entropy loss function. (

Informative Prior | 0.09 | 1.3 | 15 | 0.53 |

0.10 | 1.3 | 15 | 0.51 | |

0.20 | 1.2 | 15 | 0.50 | |

1.20 | 1.0 | 13 | 0.47 | |

1.50 | 0.93 | 12 | 0.45 | |

−1.0 | 1.5 | 16 | 0.55 | |

−2.0 | 1.7 | 17 | 0.58 | |

Noninformative Prior | 0.09 | 1.2 | 14 | 0.53 |

0.10 | 1.2 | 14 | 0.53 | |

0.20 | 1.0 | 14 | 0.52 | |

1.2 | 0.90 | 12 | 0.47 | |

1.5 | 0.84 | 11 | 0.44 | |

−1.0 | 1.4 | 17 | 0.58 | |

−2.0 | 1.8 | 19 | 0.61 |

Table

For

Table

It can be seen from Tables

In this section, we study the sensitivity of the Bayes estimator, obtained in Section

This can be seen from Tables

Bayes estimate of

1.0 | 0.3 | 15 |

1.0 | 0.5 | 15 |

1.0 | 0.7 | 15 |

0.7 | 0.5 | 15 |

0.9 | 0.5 | 15 |

1.4 | 0.5 | 15 |

Bayes estimate of

5.0 | 1.0 | 25 |

5.0 | 3.0 | 25 |

5.0 | 4.0 | 25 |

4.0 | 2.0 | 25 |

6.0 | 2.0 | 25 |

6.5 | 2.0 | 25 |

Bayes Estimate of

1.0 | 0.3 | 35 |

1.0 | 0.5 | 35 |

1.0 | 0.7 | 35 |

0.7 | 0.5 | 35 |

0.9 | 0.5 | 35 |

1.4 | 0.5 | 35 |

Table

In Section

Frequency distributions of the Bayes estimates of the change point.

Bayes estimate | % frequency for | ||

01–08 | 09–11 | 12–20 | |

Posterior mean | 12 | 78 | 10 |

| 20 | 65 | 15 |

| 22 | 66 | 12 |

The value of Bayes estimator of change point

Table

The authors would like to thank the editor and the referee for their valuable suggestions which improved the earlier version of the paper.