Approximate Confidence Bounds for the Arrhenius-Lognormal

Bounds on the Parameters

The lower and upper bounds on B are estimated from,

$images\6_31_1.gif$

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

$images\6_31_2.gif$

and,

$images\6_31_3.gif$

The variances and covariances of B, C and $images\otdash.gif$ are estimated from the local Fisher Matrix (evaluated at $images\bhat.gif$ , $images\chat.gif$ , $images\otdash2.gif$ , as follows,

$images\6_31_5.gif$

$images\6_31_4.gif$

Bounds on Reliability

The reliability of the lognormal distribution is,

$images\6_32_1.gif$

Let $images\zhat.gif$ (t, V; B, C, $images\ot.gif$ ) = $images\eqn1.gif$ , then $images\eqn__2.gif$ .

For t = $images\tdash2.gif$ , $images\zhat.gif$ = $images\eqn__3.gif$ and for t = $images\oo.gif$ , $images\zhat.gif$ = $images\oo.gif$ . The above equation then becomes,

$images\6_32_2.gif$

The bounds on z are estimated from,

$images\6_32_3.gif$

where,

$images\6_32_4.gif$

or,

$images\6_32_5.gif$

The upper and lower bounds on reliability are,

$images\6_32_6.gif$

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,

$images\6_33_1.gif$