Engineered Software

Normal Distribution - Maximum Likelihood Estimation


Cumulative Distribution Function

Cumulative Hazard Function

Weibull Distribution

Normal Distribution

Lognormal Distribution

Exponential Distribution


Engineered Software Home Page

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:

  1. Enter the data using one of the data entry grids, or connect to a database.
  2. Select the "Parameter Estimation"
  3. Select "Normal"
  4. 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 maximum likelihood equations for the normal distribution are

where r is the number of failures,

k is the number of censored observations,

is the sample mean of the failures,

s is the sample standard deviation for the failures,

is the standard normal deviate

, and

is the hazard function evaluated at the ith point

where is the standard normal probability density function evaluated at the ith point, and

is the standard normal cumulative distribution function evaluated at the ith point.

Note that if no censored data are involved these expressions reduce to the sample mean and the sample standard deviation.

Iterative techniques are necessary to these equations. A standard method based on Taylor series expansions involves repeatedly estimating the parameters until a desired level of accuracy is reached. Estimates of m and s are given by the expressions

where h is a correction factor for the distribution mean, and

k is a correction factor for the distribution standard deviation.

For each iteration, the correction factors are estimated from the expressions



The estimated parameters are asymptotically normal. The variances of the estimates can be found by inverting the local information matrix.

After inversion, the variances are

Approximate (1-a )100% confidence intervals for the estimated parameters are

where is the inverse of the standard normal probability density function.

These confidence intervals are approximate, but approach exactness as the sample size increases. Confidence intervals for reliability can be found using the expressions

Confidence intervals for percentiles can be found using the expressions