Proof of Theorem 9 To prove
nlim→∞Eθ
Mc−Eθ[Mc]
√ Vθ[Mc]
− M†−Eθ[M†]
√Vθ[M†]
2
= 0,
it is sufficient to show
nlim→∞
√ 1
Vθ[Mc]Vθ[M†] (
Eθ[M Mc †]−Eθ[Mc]Eθ[M†] )
= 1.
We assume thatN is an odd number; then it is easy to see that for {(i, j) :Xi <Z, Xˇ j <
Z, iˇ ̸=j}, the conditional probability density κθ|zˇ of (Xi, Xj,Z) given by ˇˇ Z = ˇz is κθ|zˇ(x, y) = f(x)
F(ˇz)φ(ˇz−x)× f(y)
F(ˇz)φ(ˇz−y).
Using Bayes’ theorem, the joint probability density for {(i, j) :Xi <Z, Xˇ j <Zˇ} is obtained as follows:
κθ(x, y, z) = f(x)f(y)
{F(z)}2 φ(z−x)φ(z−y)
×
min{(N∑−1)/2,m} u=1
hθ(u, z)Pθ[U =u],
wherehθ(u, z) is the joint probability density of (U,Z) given in Section 5.3. In the same way,ˇ we defineπθ(x, z) as the joint probability density of (Xk,Z) forˇ {k :Xk<Zˇ}. By considering the conditional expectation, if f(1)(·) exists and is a continuous function on R, we have
Eθ[Mc|U] = U
∫∫
K∗
(z−x h
)
πθ(x, z)dxdz
= U
∫ (
πθ[1](z, z)−hA∗1,1πθ(z, z) +· · ·) dz, where
πθ[1](x, z) =
∫ x
−∞
πθ(s, z)ds.
Suppose that both Eθ[f( ˇZ)/F( ˇZ)] and Eθ[f(1)( ˇZ)/F( ˇZ)] are bounded. Then, since
∫ πθ[1](z, z)dz = 1, we have
Eθ[Mc] =Eθ[U] {
1−hA∗1,1
∫
πθ(z, z)dz+O(h2) }
.
In a similar way, we get Eθ[Mc2] = Eθ[U2−U]
∫
· · ·
∫ K∗
(z−x h
) K∗
(z−y h
)
κθ(x, y, z)dxdydz +Eθ[U]
∫∫ { K∗
(z−x h
)}2
πθ(x, z)dxdz
= Eθ[U2−U]
∫ (
κ[1,2]θ (z, z, z)−hA∗1,1{κ[1]θ (z, z, z) +κ[2]θ (z, z, z)}+· · ·) dz +Eθ[U]
∫
πθ[1](z, z)dz+o(N) and
Eθ[M Mc †] = Eθ[U2−U]
∫
· · ·
∫ K∗
(z−x h
)
φ∗(z−y)κθ(x, y, z)dxdydz +Eθ[U]
∫∫
K∗
(z−x h
)
φ∗(z−x)πθ(x, z)dxdz
= Eθ[U2−U]
∫ (
κ[1,2]θ (z, z, z)−hA∗1,1κ[2]θ (z, z, z) +· · ·) dz +Eθ[U]
∫
πθ[1](z, z)dz+o(N), where
κ[1,2]θ (x, y, z) =
∫ x
−∞
∫ y
−∞
κθ(s, t, z)dsdt, κ[1]θ (x, y, z) =
∫ x
−∞
κθ(s, y, z)ds and
κ[2]θ (x, y, z) =
∫ y
−∞
κθ(x, t, z)dt.
Note that ∫
κ[1]θ (z, z, z)dz =
∫
κ[2]θ (z, z, z)dz =
∫
πθ[1](z, z)dz and the term of order (N2h) of Eθ[Mc2]− {Eθ[Mc]}2 is given by
−2hA∗1,1Eθ[U2]
∫
πθ(z, z)dz−(−2hA∗1,1){Eθ[U]}2
∫
πθ(z, z)dz
= −2hA∗1,1Vθ[U]
∫
πθ(z, z)dz =O(N h).
Suppose that both Eθ[{f( ˇZ)/F( ˇZ)}2] and Eθ[f(1)( ˇZ)/F( ˇZ)] are bounded. Then, if h=o(N−1/2) holds, we have
Vθ[Mc] = Eθ[U2−U]
∫
κ[1,2]θ (z, z, z)dz +Eθ[U]− {Eθ[U]}2+o(N).
From the above results, it is easy to see that Vθ[M†] = Eθ[U2−U]
∫
κ[1,2]θ (z, z, z)dz+Eθ[U]− {Eθ[U]}2 and
Eθ[M Mc †]−Eθ[Mc]Eθ[M†] = Eθ[U2−U]
∫
κ[1,2]θ (z, z, z)dz +Eθ[U]− {Eθ[U]}2+o(N).
Since Vθ[M†] =O(N), it follows that
(Vθ[Mc])−1/2 = (Vθ[M†])−1/2+o(N−1).
Consequently, we have
nlim→∞
√ 1
Vθ[M]Vc θ[M†] (
Eθ[M Mc †]−Eθ[Mc]Eθ[M†] )
= 1
and the desired result. We can conclude that asymptotic convergence still holds when N is even.
Proof of Corollary 1 Under the null hypothesis H0, we utilize the following Bahadur representation (Bahadur (1966)) of the combined median ˇZ,
Zˇ =z0+ 1 f(z0)
(1
2 −FN(z0) )
+RN,
where z0 is the median of F(·), FN(·) stands for the empirical distribution function of {Z1,· · · , ZN}, and RN satisfies
E0[RN] =− f(1)(z0)
4N f3(z0) +O(N−2) and
E0[RN2] =o(N−1)
(see Reiss (1976)). We assume that h=o(N−1/2) or thatA∗1,1 =A∗2,1 = 0 and h=o(N−1/4).
Then, by the Bahadur representation and direct computation, we have V0[Mc] = mn
4(m+n) +o(N) if in neighborhoods of z0, f(·) is positive and f(1)(·) is continuous.
Next, we utilize the Bahadur representation for the quantiles of two samples under H1 (Liu and Yin (1994)). Using the asymptotic expansion and the representation, we find that
Mc =
∑m i=1
[ K∗
(zθ,N −Xi h
)
+ 1
gθ,N(zθ,N)hk∗
(zθ,N −Xi h
) {1
2−Gθ,N(zθ,N) }]
+oP(N1/2), where
Gθ,N(x) =λNFX,m(x) + (1−λN)FY,n(x)
and FX,m(·) and FY,n(·) are the empirical distribution functions of {X1,· · · , Xm} and {Y1,· · · , Yn}, respectively. Therefore, we can show the following expectation underH1:
Eθ[Mc] = m
∫ ∞
−∞
k∗(v)F(zθ,N −hv)dy+o(N1/2)
= mF(zθ,N) +o(N1/2).
Combining the above results, we can prove that the Pitman efficacy is the same as the discrete one.
Proof of Theorem 10 Using the result of Lahiri and Chatterjee (2007), we have the fol-lowing Berry-Esseen bound:
−∞sup<x<∞
P0
[
v−31/2(M†−µ3)< x
]−Φ(x)=O ( 1
√N )
where µ3 =E0[M†] andv3 =V0[M†]. From the proof of Corollary 1, we have E0[D2] =O(h),
where
D=v−11/2(Mc−E0[Mc])−v−31/2(M†−µ3).
Therefore, D=OP(√
h) and P0[|D|> N−ϵh1/4] =Nϵh1/4 hold for any positive ϵ. Since sup
|t|<N−ϵh1/4,−∞<x<∞|Φ(x+t)−Φ(x)|=O(N−ϵh1/4), we can show that
−∞sup<x<∞
P0 [
v1−1/2(Mc−µ1)< x
]−Φ(x)
= sup
−∞<x<∞
P0
[{v3−1/2(M†−µ3) +D+o(1)}< x
]−Φ(x)
= o(1).
This completes the proof of the theorem.
Proof of Theorem 11Puttingθ =N−1/2ξ, let us examine the approximation ofzθ,N. Since the following expansion holds,
N
2 = mF(zθ,N) +nF(zθ,N −θ)
= N F(zθ,N)− n
√Nf(zθ,N)ξ+O(N−1),
we have √
N f(z0)η = n
√Nf(z0)ξ+o(N1/2h).
Thus, we find that
zθ,N =z0+N−1/2(1−λ)ξ+o(N−1/2h).
Combining
V0[Mc] = mn
4(m+n) −2(mh)A∗1,1,1f(z0) +O( N h2) and
Eθ[Mc]−E0[M] =c mF(zθ,N)−m
2 +O(N h2)
= λ(1−λ)√
N f(z0)ξ+o(N1/2h),
we get the following local power of Mc: LP√ξ
N,α[Mc] =P√ξ
N
[
v∗1−1/2(Mc−µ1)> q1−α ]
= 1−Φ(q1−α−c1) +O(N−1/2+h2).
Proof of Theorems 12, 13 and 14 We can easily show that Vθ[cW2] =
∑m i=1
∑m s=1
∑n j=1
∑n t=1
Eθ [
K
(Yj −Xi h
) K
(Yt−Xs h
)]
− [
mn
∫ ∞
−∞
f(y)F(y−θ)dy ]2
+o(N3)
= m2nCovθ [
K
(Y1−X1
h )
, K
(Y1−X2
h
)]
+mn2Covθ [
K
(Y1 −X1
h )
, K
(Y2−X1
h )]
+o(N3).
By direct computation, we have Eθ
[ K
(Y1−X1 h
) K
(Y1−X2 h
)]
=
∫
· · ·
∫ K
(y−x h
) K
(y−u h
)
f(x)f(u)f(y−θ)dxdudy
=
∫ ∞
−∞
F2(y)f(y−θ)dy+O(h2).
and
Eθ
[ K
(Y1−X1 h
) K
(Y2−X1 h
)]
=
∫
· · ·
∫ K
(y−x h
) K
(w−x h
)
f(x)f(y−θ)f(w−θ)dxdydw
=
∫ ∞
−∞
f(x){1−F(x−θ)}2dx+O(h2) Then, we easily obtain
Vθ[Wc2] = m2n (∫ ∞
−∞
f(x){1−F(x−θ)}2dx− [∫ ∞
−∞
F(y)f(y−θ)dy ]2)
+mn2 (∫ ∞
−∞
F2(y)f(y−θ)dy− [∫ ∞
−∞
F(y)f(y−θ)dy ]2)
+o(N3).
Since
Eθ[Wc2W2†] = m(m−1)n(n−1)Eθ [
K
(Y1−X1 h
) φ
(Y2−X2 h
)]
+m2nEθ [
K
(Y1 −X1 h
) , φ
(Y1−X2 h
)]
+mn2Eθ [
K
(Y1 −X1 h
) , φ
(Y2−X1 h
)]
+o(N3)
= m(m−1)n(n−1)
([∫ ∞
−∞
F(y)f(y−θ)dy ]2
+O(h2) )
+m2n (∫ ∞
−∞
F2(y)f(y−θ)dy+O(h2) )
+mn2 (∫ ∞
−∞
f(x){1−F(x−θ)}2dx+O(h2) )
and Eθ[Wc2] =Eθ[W2†] +O(h2), it is easy to see that
√ 1
Vθ[cW2]Vθ[W2†] (
Eθ[Wc2W2†]−Eθ[Wc2]Eθ[W2†] )
=o(1) and that Theorem 12 holds.
Next, using the result of Maesono (1985), we will show briefly the validity of the Edge-worth expansion:
−∞sup<x<∞
P0 [
v2−1/2(Wc2−µ2)< x
]−Qm,n(x)=o(N−1/2)
where
Qm,n(x) = Φ(x)−ϕ(x)x2−1 6τm,n3
[ 1
m2E0[g31,0(X1)] + 1
n2E0[g0,13 (Y1)]
+ 6
mnE0[g1,0(X1)g0,1(Y1)g1,1(X1, Y1)]
] , g1,0(X1) =
∫
k(v)F(X1+hv)dv−1
2 +o(N−1/2), g0,1(Y1) =
∫
k(v)F(Y1−hv)dv−1
2 +o(N−1/2), g1,1(X1, Y1) = K
(Y1 −X1 h
)
−g1,0(X1)−g0,1(Y1)− 1
2+o(N−1/2), τm,n = m+n
12mn +o(N−1/2), E0[g31,0(X1)] = o(N−1/2),
E0[g30,1(Y1)] = o(N−1/2), E0[g1,0(X1)g0,1(Y1)g1,1(X1, Y1)] =o(N−1/2),
and ϕ(·) is the derivative of Φ(·) under the assumption. First, using Hoeffding’s decomposi-tion, we have
v2−1/2(Wc2−µ2) =: cW2∗+rN, where
Wc2∗ = 1 mn
( n
∑m i=1
g1,0(Xi) +m
∑n j=1
g0,1(Yj) +
∑m i=1
∑n j=1
g1,1(Xi, Yj) )
and rN =o(N−1/2) under the assumption of Theorem 3.2. Using Ess´een’s smoothing lemma, the supremum is
−∞sup<x<∞
P0
[cW2∗ < x
]−Qm,n(x)
≤ 1 π
∫ N1/2logN
−N1/2logN
Cm,n∗ (t)−Cm,n(t) t
dt+o(N−1/2),
where Cm,n∗ (·) and Cm,n(·) are characteristic functions of P0[cW2∗ < x] and Qm,n(x), respec-tively. Dividing the interval into two and using the fact that
∫
logN≤|t||t−1Cm,n(t)|dt =o(N−1/2), it is sufficient to prove that
I1 :=
∫
|t|≤N1/4/logN
|t−1(Cm,n∗ (t)−Cm,n(t))|dt=o(N−1/2) and
I2 :=
∫
N1/4/logN≤|t|≤√ NlogN
|t−1Cm,n∗ (t)|dt =o(N−1/2).
Using Taylor expansions of Cm,n∗ (·) and evaluating the moments of g0,1(X1), g1,0(Y1) and g1,1(X1, Y1), we can see that I1 holds. In order to prove equation I2, we need Cram´er’s condition,
|t|→∞lim |E0[eitg1,0(X1)]|<1 and lim
|t|→∞|E0[eitg0,1(Y1)]|<1.
Using the result of Garc´ıa-Soid´an et al. (1997), we can prove them. For all real a and a positive number η, we can prove that there exist c0 >0 such that
|E0[exp(itg1,0(X1))]|
=
∫ exp
( it
∫
k(v)F(x−hv)dv )
f(x)dx
≤
(∫ a−η
−∞
+
∫ ∞
a+η
)
f(x)dx +
∫ a+η a−η
exp (
it
∫
k(v)F(x−hv)dv )
f(x)dx
≤ 1−2ηc0f(a) +c0f(a) ∫ a+η
a−η
exp (
it
∫
k(v)F(x−hv)dv )
dx . Here, we consider
1 2η
∫ a+η a−η
exp (
it
∫
k(v)F(x−hv)dv )
dx as the characteristic function of λ(U) := ∫
k(v)F(U−hv)dv, where U is a uniform random variable on (a−η, a+η). There exista and η >0 such that ∫
k(v)F(x−hv)dv >0 for all x∈[a−η, a+η]. Hence,λ(U) satisfies Cram´er’s condition:
|tlim|→∞E0[
(2η)−1eitλ(U)] =ρ(a, η)<1.
Thus, we find
|t|→∞lim |E0[exp(itg1,0(X1))]| ≤1−2ηc0f(a) (
1−ρ(a, η) )
<1.
Similarly, we can show that lim|t|→∞|E0[exp(itg0,1(Y1))]|<1. Using Cram´er’s condition, we can prove that equation I2 holds. This completes the proof of Theorem 13. We can easily prove Theorem 14 in a similar manner to that of Mc.
6 Conclusion and Future work
We have proposed new kernel smoothed statistics in nonparametric inference. Nonparametric statistics have good properties, and kernel type ones are smooth nonparametric statistics.
We confirmed that kernel type statistics have some problems, and one of the most serious problems is that most kernel statistics are inferior to parametric statistics in the sense of an optimal convergence rate. First, we considered to improve an asymptotic mean squared error (AM SE) of a naive kernel hazard ratio estimator given by H(x) =e fb(x)/(1−Fb(x)). By extending ´Cwik and Mielniczuk (1989)’s method, we obtained a ‘direct type’ kernel estimator H. When the bandwidth parameters are same, the proposed estimatorb Hb has uniformly smaller asymptotic variance than the naive kernel estimator. By choosing each optimal bandwidth respectively, the asymptotic variance of Hb is not always smaller. We showed, however, that the proposed method gives asymptotically precise estimation in some important cases.
As is pointed out by Faugeras (2009), naive kernel estimators of a ratio of functions may be numerically unstable when their estimated denominator terms are close to zero. However, direct estimators are not ratios. It is also possible to derive direct estimators of a ratio of some functions, which is based on modifications of ´Cwik and Mielniczuk (1989)’s method.
We also need to study numerical stability of the direct estimators.
As the second of this thesis, we discussed the boundary bias problem in naive kernel type estimation of an underlying density function. When the underlying density is non-zero at some boundary points, the naive kernel estimator loses even its consistency at the points because of the boundary bias. We showed that the naive density estimator possibly has a boundary bias when the support of the underlying density is unknown. We proposed a new method for nonparametric estimation of the probability density and obtained its asymptotic properties. The proposed method detects the unknown boundary and returns a modified density estimator fb† (and a distribution estimator Fb†) which is free from the unexpected boundary bias. Moreover, we discussed an extension to a simple multivariate case and pro-posed a new method for estimating joint probability density functions.
In kernel type estimation of the hazard ratio function, both the naiveHe and the proposed estimator Hb possibly have boundary biases when the support of the underlying density function does not cover the whole real line or is unknown. If we know the support exactly, modified ‘naive’ kernel estimators of the hazard ratio, which are free from the boundary bias, can be obtained by replacing kernel estimators fbandFb of He with estimators which are free from each boundary bias, respectively. We can also derive a modified (naive) estimator He† by replacing fband Fb with the proposed kernel estimators fb† and Fb† respectively. When the support of the underlying density is unknown, the obtained hazard estimator He†may be free from the unexpected boundary bias. By applying plug-in method, we can derive some (naive) estimators which may be free from unexpected boundary biases. However, we must put somewhat restrictive assumptions which ensure their bias reduction. To apply proposed
methods to other kernel type estimators and relax the assumptions is our future work.
Lastly, we considered application of the kernel smoothing method to some discrete distribution-free tests. Since p-values of the sign and Wilcoxon’s signed rank tests jump in response to a change in data values, the discrete tests have a problem with their p-values. We confirmed the problem and proposed new smoothed alternatives whose Pitman’s efficacies coincide with those of the discrete tests respectively. In addition, the smoothed tests are higher-order asymptotically distribution-free if the kernel functions and bandwidth parameters are suitably selected. We showed that the smoothed tests, which inherit such good properties, can solve the problem of the p-values. We also discussed smoothing the median and Wilcoxon’s rank sum tests and showed that the smoothed tests also have some good properties.
Like these discrete distribution-free tests, other discrete distribution-free tests may have the same problem with theirp-values. Smoothing most of discrete tests may be possible, but to derive their (asymptotic) properties is often quite difficult (see Remark 20). To devise smoothed tests, which are superior to the discrete tests in some sense, must be meaningful.
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