The name is absent

5% level in most occasions, when W₁ is employed. It is obvious from our simulation
experiment that, when a small penalty term is used, the B(m) statistic tends to be
significantly larger B(m_o). In this instance the modified Bierens test may suffer from
overrejection of the null hypothesis. Notice however, that in Table (4) B(rn) equals
B(m_o) in most cases, even when a small penalty term is employed.

6 Appendix

Lemma A. Let q(s) : R → R be Borel measurable. The function Φ(s) : R → C is
bijective and continuously differentiable with C an open and bounded subset of R. If
q(.) is integrable, then

A [{s : q(s) = 0}] > 0 if and only if [ q(s')e^mφ^^s^ ds = 0

√R

for m E R in an arbitrarily small neighborhood of zero.

Proof of Lemma A: (i) The “if” part is trivial. We show the “only if” part. Consider
the Borel measurable functions

qι(s) = max {q(s), 0} and q₂(s) = max {-q(s) 0}

and notice that q = q₁ - q₂. Assume that

c₁ = ʃ q₁(s)ds > 0 and c₂ = ʃ q₂(s)ds > 0.

Define the probability measures⁶ F_i, i = {1,2} on the Borel field restricted on C (B_c)
as

Then

' q(s)e^mφV>ds = ∣ q√s)e^mφ¾ - ʃ q₂(s)e^mφ¾

Φ-¹(s) ds - q q₂(Φ~¹ (s))e^ms

Jc

c₂ f e^mrdF₂(r)

Jc

= C1η1(m) - c₂η₂(m),

where the second equality above is due to Billingsley (1979, Theorem 17.2). Notice
that η_i(m) is the moment generating function of F_i, i = {1, 2}. In view of this,

⁶Noticethat ʃɔ q_i(Φ ¹(s)) ∣Φ ¹(s)∣ ds = J_r q_i(s)ds = c⅛.

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