Characteristics of Optimalf 205
And we see that the function peaks at .25, where the slope of the tangent
is zero, exactly at the optimal f, and no other local extremum can exist
because of the restriction caused by the Pythagorean Theorem.
Lastly, we will see that optimalfis indifferent toT. We can take the
first derivative of the estimated TWR, Equation (5.10a) with respect toTas:
dTWR
dT=
(
A^2 −S^2
)T/ 2
* ln(
A^2 −S^2
)
(5.12)
Since ln(1)=0, then ifA^2 −S^2 =1, that is,A^2 − 1 =S^2 (or variance),
the function peaks out and the single optimal maximum TWR is found with
respect tof.Notice, though, that bothA, the arithmetic average HPR, and
S, the standard deviation in those HPRs, are not functions ofT.Instead,
they are indifferent toT; thus, (5.10a) is indifferent toTat the optimalf.
Thefthat is optimal in the sense of maximizing the estimated TWR will
always be the same value regardless ofT.