da*
Since c > 0, dhj has the sign of the sum between square brackets in (14). Using the
expression for σ in (4) and simplifying we find that
∂σ(hj ,hk,hε) /
dhj 1 1
σ (hj,hk,hε^) у
(mk
2
mj)
σ2 (hj, hk, hε)
ʌ__L_
hj + hε
hk (2hj + hε) (hk + hε) 1 —
σ(mmk-m⅛y) - 2hj [hj (hj + hε) + hk (hk + hε)]
2hj (hj + hε) [hj (hj + hε) + hk (hk + hε)]
(16)
da* 1 1 . 1 /- „ч m1 . η η „ ,
dhj has the same sign as the numerator in (16). The partial derivative of the numerator
in (16) with respect to hj is
— 2hk (hk + hε)
(mk
2
mj)
σ2 (hj, hk, hε)
(mk — mj)2 ∂σ2
+ hk (2hj + hε) (hk + hε) 2 z, T rʒ TTT-
σ2 (hj, hk, hε) ∂hj
<0
— 2hj (hj + hε) — 2hj (2hj + hε),
which is clearly negative. Since limhj→0 dhj = ∞ and limhj→∞ dhj = 0, it follows from
da*
this that given the other parameters there exists a unique hj > 0 such that dj < 0 if and
only if hj = hj.
Finally,
da*
dhk
exp
( (mk-mj )
∖ 2σ2(hj ,hk,,
)2
,hε)
√2πσ (hj,hk,he)
'----------------------v------
>0
Wjh + /
c'' ⅛ ) V
-------zC_
∂σ(hj ,hk ,hε)
∂hk
σ (hj, hk, hε)
SZ
>0
)(■
S
(mk
2
mj)
σ2 (hj, hk, hε)
. (17)
From (17) it is easy to see that daj > 0 if and only if (mk — mj)2 < σ2 (hj, hk, hε). Since
limhk→0 σ2 = ∞ and σ2 (hj , hk, hε) is strictly decreasing in hk (see (4)), there exists a
unique threshold hk such that dhj ≥ 0 if and only if hk = hk. Observe that hk > hj. ■
Lemma 1
(Lemma A1) limhj→ da* = ∞
Proof.
daj*
limh, →o -г;—,
hj→0 dhj
First note that both limhj→0 σ = ∞ and limhj→0 dh- = ∞∙ To determine
we first simplify the limit of (14) as follows:
daj
lim -ɪ
hj→0 dhj
limhj→0 exp -
(mk-mj )2 ∖ "I W hε
2σ2(hj ,hk,hε) J W Wj hj +hε
lim
hj→0
2π
∂σ(hj ,hk,hε)
dhj
c' (⅛')
Wj
σ2 (hj, hk, hε)
lim 1
hj →0
(mk
2
mj)
σ2 (hj, hk, hε)
ʌ 1- (
- lim
hj→0
(hj +hε)σ(hj,hk,hε
2∏C^c" (α*)
lim
hj →0
∂σ(hj ,hk,hε)
dhj
σ2 (hj, hk, hε)
41
More intriguing information
1. The Role of Trait Emotional Intelligence (El) in the Workplace.2. fMRI Investigation of Cortical and Subcortical Networks in the Learning of Abstract and Effector-Specific Representations of Motor Sequences
3. Individual tradable permit market and traffic congestion: An experimental study
4. Language discrimination by human newborns and by cotton-top tamarin monkeys
5. Automatic Dream Sentiment Analysis
6. The name is absent
7. Spatial patterns in intermunicipal Danish commuting
8. SOCIOECONOMIC TRENDS CHANGING RURAL AMERICA
9. Distortions in a multi-level co-financing system: the case of the agri-environmental programme of Saxony-Anhalt
10. The name is absent