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This derivative is an odd function and is always positive. It means that the growth
of the distance with respect to ε is the same on both the left and right of ζ = 0 . 5.
In particular, for ζ = 0 . 5 (TLS),
TLS = 2 σ
d ε
d
2
i
(6.77)
6.5.2 Analysis of the Maximum
α 1
The analysis of the maximum
α
1 is the same as in the case of saddles. However,
some crossovers miss.
For
ζ
0
.
5, the following strictest
inequalities derived from the previous
theory are true:
0
OLS
1 ζ
b T b
2 ( 1 ζ )
l 3
2
1
2 ( 1 ζ )
l 2
2
1
2 ζ
u 1
ζ ζ 0 . 5
γ
σ
σ
=
α 1
(6.78)
σ 1
2
2
1
0 ζ ζ
σ
α i
ζ
l 4
ζ
u 1
2
ζ
0 (OLS) the lower bound l 4 and the upper bound u 1tendtoinfinity;
as seen before,
For
α 1 →∞
ζ
.
.For
0
5, there is no right crossover and the strictest
inequality is given by
2
1
2 ζ
l 3
2
1
2 ( 1 ζ )
u 2
σ
σ
α 1
(6.79)
However, from the inequality (6.40) it follows that
0
OLS
1 ζ
γ
α
(6.80)
1
l
ζ
For
1 (DLS), this lower bound tends to infinity, as seen previously. The
same is true for the upper bound u 2. Indeed, the maximum eigenvalue for DLS
is infinite.
6.5.3 Analysis of the Minimum
α n +1
Also in the analysis of the minimum α n + 1 , previous considerations for the max-
imum can be repeated. For ζ 0 . 5, there is no left crossover and the strictest
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