Approximate Confidence Bounds on IPL- Lognormal

Bounds on the Parameters

Since the standard deviation, $images\otdash2.gif$ and $images\khat.gif$ are positive parameters, then ln ( $images\otdash2.gif$ ) and ln ( $images\khat.gif$ ) are treated as normally distributed, and the bounds are estimated from,

$images\8_31_1.gif$ (upper bound)

$images\8_311_0.gif$ (lower bound)

And,

$images\8_31_2.gif$ (upper bound)

$images\8_31_01.gif$ (lower bound)

The lower and upper bounds on n, are estimated from,

$images\8_31_3.gif$ (upper bound)

$images\8_31_03.gif$ (lower bound)

The variances and covariances of A, B and $images\ot2.gif$ are estimated from the local Fisher Matrix (evaluated at $images\ahat.gif$ , $images\bhat.gif$ , $images\otdash2.gif$ ), as follows,

$images\8_31_4.gif$

where,

$images\8_31_5.gif$

Bounds on Reliability

The reliability of the lognormal distribution is,

$images\8_32_1.gif$

Let $images\zhat2.gif$ (t, V; K, n, $images\ot.gif$ ) = $images\eqn__6.gif$ , then = $images\dzdt.gif$ .

For t = $images\tdash2.gif$ , $images\zhat2.gif$ = $images\eqn__7.gif$ and for t = $images\oo.gif$ , $images\zhat2.gif$ = $images\oo.gif$ . The above equation then becomes,

$images\8_32_2.gif$

The bounds on z are estimated from,

$images\8_32_3.gif$

where,

$images\8_32_4.gif$

or,

$images\8_32_5.gif$

The upper and lower bounds on reliability are,

$images\8_32_6.gif$ (upper bound)

$images\8_32_06.gif$ (lower bound)

Confidence Bounds on Time

The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows,

$images\8_33_1.gif$