The name is absent

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

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

— 2hj (hj + hε) — 2hj (2hj + hε),

which is clearly negative. Since lim_hj→₀ dhj = ∞ and lim_hj→∞ 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,

From (17) it is easy to see that daj > 0 if and only if (m_k — mj)² < σ² (hj, h_k, h_ε). Since
lim_hk→0 σ² = ∞ and σ² (hj , h_k, h_ε) is strictly decreasing in h_k (see (4)), there exists a
unique threshold h_k such that dhj ≥ 0 if and only if h_k = h_k. Observe that h_k > hj. ■

(Lemma A1) limh_j→ da* = ∞

First note that both lim_hj→₀ σ = ∞ and lim_hj→₀ dh- = ∞∙ To determine
we first simplify the limit of (14) as follows:

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