Reliability HotWire

Reliability HotWire

Issue 123, May 2011

Reliability Basics

The MLE Solution for the Location Parameter of the 2-Parameter Exponential Distribution

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

In Weibull++, when using the 2-parameter exponential distribution, the software always sets the location parameter, gamma, equal to the first time-to-failure. Is there any theoretical explanation for it? In this article we will explain this special case of the MLE solution for the 2-parameter exponential distribution.

Maximum Likelihood Estimation Method

The general likelihood function is given by:

Equation

where:

  • f(x12,...,θk), is the pdf of a continuous random variable x.
  • θ12,...,θk are k unknown parameters that need to be estimated.
  • x1,x2,...,xR are R independent observations, which correspond to failure times in life data analysis.

The logarithmic likelihood function is given by:

Equation

The maximum likelihood estimators of θ12,...,θk are obtained by maximizing f(x) = ln Λ.

For the 2-parameter exponential distribution, the log-likelihood function is given as:

Equation

To find the pair solution , the equations and have to be solved.

Equation

and

Equation

Now let us first examine Eqn. (5). To get the MLE solution for γ, Eqn. (5) has to be set to zero. We can see that Eqn. (5) is satisfied if and only if:

Equation

or:

Equation

However, Eqn. (6) cannot be the solution, because we want to maximize Eqn. (3); and Eqn. (5) could not be achieved because the summation of failure times will not be zero for the common cases. This indicates there is no non-trivial MLE solution that satisfies both and .

To solve this dilemma, let us look at the effect of γ on the exponential distribution. The location parameter γ, if positive, shifts the beginning of the distribution by a distance of γ to the right of the origin, signifying that failures start to occur only after γ hours of operation and could not occur before, which means γ T1 (the first failure time). As an example, Figure 1 displays the effect of γ on the exponential distribution with parameters (λ = 0.001, γ = 500) and (λ = 0.001, γ = 0). The physical meaning of γ has shed the light on solving this 2-parameter exponential distribution using the MLE method. With the failure data, the partial derivative Eqn. (5) will be greater than zero. This implies that the likelihood function in Eqn. (2) increases when γ is changed from 0 to T1. (For a continuous function g(x), if its first derivative is always greater than 0 during the interval [a, b], then g(x) will increase in the interval [a, b].) To get the maximum likelihood estimators, we need to maximize Eqn. (2) so γ should be set to T1, the first time-to-failure. And then find λ such that .

Reliability vs. Time plot
Figure 1: The effect of the location parameter on the exponential distribution

Example

A reliability engineer conducted a reliability test on 14 units and obtained the following data set. From the previous testing experience, the engineer knew that the data were supposed to follow a 2-parameter exponential distribution.

Table 1 - Life Test Data

Data point index Time-to-Failure (days)
1 8
2 10
3 15
4 22
5 25
6 30
7 35
8 40
9 50
10 60
11 68
12 73
13 82
14 90

To get the MLE solution for this data analysis, the reliability engineer used Weibull++ and entered the data in a standard folio. To set the analysis to MLE, the engineer clicked the blue link in the Analysis Settings area of the control panel, as shown next.

Analysis Method

The engineer then clicked the Calculate icon. As shown in the following picture, the results shows that λ = 0.0282, and γ = 8.0, indicating that γ is set equal to the first failure time.

Analysis Method

As seen in the following Reliability vs. Time plot, the failures started only when the time equaled 8 hours.

Reliability vs. Time plot
Figure 4: Reliability vs. Time plot for life test data

Conclusion

In the article, we discussed the MLE solutions for the 2-parameter exponential distribution. While there is no regular MLE solution for this distribution, the physical effect of the location parameter, γ, leads us towards the solution. By setting γ equal to the first time-to-failure and obtaining λ such that , a maximum can be achieved along the λ axis, and a local maximum along the γ axis at γ = T1, constrained by the fact that γ T1.

References

[1] ReliaSoft Corporation, Life Data Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2008.