Chapter 14.1: Logistic Confidence Bounds
In this section, we present the methods used in the application to estimate the different types of confidence bounds for logistically distributed data. The complete derivations were presented in detail (for a general function) in Confidence Bounds.
Bounds on the Parameters
The lower and upper bounds on the location parameter are estimated from
The lower and upper bounds on the scale parameter are estimated from:
where is defined by:
If is the confidence level, then for the two-sided bounds, and for the one-sided bounds.
The variances and covariances of and are estimated from the Fisher matrix, as follows:
is the log-likelihood function of the normal distribution, described in Parameter Estimation and Appendix D.
Bounds on Reliability
The reliability of the logistic distribution is:
Here , , . Therefore, also is changing from to . Then the bounds on are estimated from:
The upper and lower bounds on reliability are:
Bounds on Time
The bounds around time for a given logistic percentile (unreliability) are estimated by first solving the reliability equation with respect to time as follows:
The upper and lower bounds are then found by: