Exponential Distribution - Maximum Likelihood Estimation

	Exponential Distribution - Maximum Likelihood Estimation

The following section describes maximum likelihood estimation for the normal distribution using the Reliability & Maintenance Analyst. The manual method is located here.

The maximum likelihood estimation routine is considered the most accurate of the parameter estimation methods, but does not provide a visual goodness-of-fit test. It is recommended to verify goodness-of-fit using probability plotting or hazard plotting, and then, if the fit is acceptable, use maximum likelihood estimation to determine the parameters. Maximum likelihood estimation provides confidence limits for all parameters as well as for reliability and percentiles. To estimate the parameters of the normal distribution using maximum likelihood estimation, follow these steps:

Enter the data using one of the data entry grids, or connect to a database.
Select the "Parameter Estimation"
Select "Exponential"
Select "Maximum Likelihood (MLE)"

The estimated parameters are given along with 90% confidence limits; an example using the data set "Demo2.dat" is shown below.

The default confidence level is 90%. The confidence level can be changed using the spin buttons, or by typing over the existing value. Changing the confidence level erases the confidence limits for the parameters. To re-calculate the confidence limits, click the "Compute Confidence Limits" button.

Clicking the "Plot" button gives a plot of expected reliability with upper and lower confidence limits at the level specified. A plot of percentiles (time as a function of reliability) is produced by selecting the "Percentiles" option in the Plot Type frame before clicking the "Plot" button. The title of the graph can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button.

To predict reliability or time-to-fail using the estimated parameters use the Predicting Module.

Manual Maximum Likelihood Estimation

The exponential probability density function is

The maximum likelihood estimation for the parameter q is

where x_i is the ith data point; this may be a failure or a censoring point,
n is the total number of data points both censored and uncensored, and
r is the number of failures.

This estimate is unbiased and is the minimum variance estimator.

Example
The cycles to fail for seven springs are:

30,183 14,871 35,031 76,321

43,891 31,650 12,310

Assuming an exponential time to fail distribution, estimate the mean time to fail and the mean failure rate.

Solution
The mean time to fail is

cycles

The mean failure rate is the inverse of the mean time to fail

failures per cycle

Click here to download this example in an Excel spreadsheet.

Example:
Assume the data in the example above represents cycles to fail for seven springs, but an additional 10 springs were tested for 80,000 cycles without failure. Estimate the mean time to fail and the mean failure rate.

Solution: The mean time to fail is

cycles

The mean failure rate is

l = 1/149,179.6 = 0.0000067 failures per cycle

For a time truncated test a confidence interval for q is

Note that the degrees of freedom differ for the upper and lower limits.

Example
Fifteen items were tested for 1000 hours. Failures occurred at times of 120 hours, 190 hours, 560 hours and 812 hours. Construct a 90% confidence interval for the mean time to fail and the failure rate.

Solution
This is a time truncated test. The mean life estimate is

For a 90% confidence interval, a = 0.1; C²( 0.05,10) = 18.307, and C²( 0.95,10) = 2.733. The 90% confidence interval for q is

1,385.5<q<9280.6

The confidence interval for the failure rate is the inverse of the confidence interval for the mean time to fail.

0.0001077<l<0.0007217

Click here to download this example in an Excel spreadsheet.

For a failure truncated test and for multiple censored data, a confidence interval for q is

Note that the degrees of freedom are the same for the upper and lower limits.

Example
Twelve items were tested with failures occurring at times of 43 hours, 67 hours, 92 hours 94 hours, and 149 hours. At a time of 149 hours, the testing was stopped for the remaining seven items. Construct a 95% confidence interval for the mean time to fail.

Solution
This is a failure truncated test. The mean life estimate is

For a 95% confidence interval, a = 0.05; C²( 0.025,10) = 20.483, and C²( 0.975,10) = 3.247. The 95% confidence interval for q is

145.3<q<916.5

Click here to download this example in an Excel spreadsheet.

For failure free testing the one sided lower confidence limit simplifies to

where t is the testing time,
ln is the natural logarithm, and
a is the significance (a = 0.05 for a 95% limit).

Example 3.5
Twenty items are tested for 230 hours without failure. Determine a 90% lower confidence limit for q .

Solution

hours

A confidence interval for reliability is

where q_L is the lower confidence limit for the mean time to fail, and
q_U is the upper confidence limit for the mean time to fail.

A confidence interval for percentiles is

where P is the probability of failure prior to time = x.

Example
Twenty items are tested for 230 hours without failure. Determine a 90% lower confidence limit for reliability at time = 1000.

Solution
The lower 90% confidence limit for the mean is

hours

The lower 90% confidence limit for reliability at time = 1000 is

Click here to download this example in an Excel spreadsheet.