1. INTRODUCTION
Empirical likelihood (EL) is introduced by Owen (1988) as a way of extending parametric methods of
inference to certain nonparametric situations. In the simplest
situation, one is interested in obtaining a confidence region for the
mean of the unknown distribution of a sample of independent and
identically distributed (i.i.d.) observations. Without specifying a
parametric form for the unknown distribution, Owen (1988, 1990) shows that
under very weak conditions the EL ratio function can be used to
construct confidence regions for the mean using an asymptotic
χ2 calibration. EL confidence regions have a
data-determined shape (i.e., they tend to be more concentrated in
places where the density of the parameters estimator is greatest), and
they are not necessarily ellipsoidal, as in the case of confidence
regions based on the normal approximation. Furthermore, they are
Bartlett-correctable (DiCiccio, Hall, and Romano,
1991), so that the actual coverage error differs from the
nominal by a term of order O(n−2).
These attractive properties have motivated various researchers to
extend the EL method to more general statistical models. See the
monograph of Owen (2001) for an excellent
account of recent developments and applications of EL.
In this paper we consider EL inference for the class of statistical
models where the parameters are defined implicitly through a set of
estimating equations. This class of models is particularly relevant in
econometrics, because most maximum and quasi-maximum likelihood,
nonlinear least squares, and M estimators are defined through
estimating equations. We shall refer to this set of estimating
equations as generalized scores (GS).
The main objective of the paper is to develop some higher order
asymptotic theory for the EL ratio (ELR) test of a simple hypothesis
about the parameters defined by GS. The approach we follow exploits an
important connection relating EL to dual likelihood (DL) (Mykland, 1995): in the case of independent
observations, the ELR statistic coincides with the DL ratio (DLR)
statistic. The importance of this connection stems from the fact that
the DL concept is introduced by Mykland (1995) as a device for using likelihood methods in
the context of martingale inference. Thus, in the special case of
independent observations, the connection between DL and EL implies the
possibility of using likelihood methods in the context of EL inference.
This is very convenient because we can then rely on techniques
developed for ordinary parametric likelihood models and can use the
classical methods of Bhattacharya and Ghosh (1978), Chandra and Ghosh (1979, 1980), and Hill and
Davies (1968) to derive rigorously results on
the higher order properties of the ELR statistic.
The results obtained in the paper extend and/or complement
results of Owen (1988, 1991), DiCiccio and Romano (1989), DiCiccio et al. (1991), Chen (1993, 1994), Mykland (1995),
and others. The new results are the following: first, we prove the
validity of the Edgeworth expansion for the maximum DL estimator and
for the ELR statistic under a sequence of local alternatives. The
latter extends to the multiparameter case and to the third order
Mykland's asymptotic analysis (1995, Sect.
7) of the alternative hypothesis induced by the DL. Incidentally, we note
here that, with the exception of Chen (1994) and
Lazar and Mykland (1998), much of the research on
the higher order properties of the ELR statistic has been focused on its
accuracy rather than on its power. The Edgeworth expansion for the ELR
under a sequence of local alternatives derived in the paper fills this
gap, because it can be used to approximate the power of the ELR test
for particular values of the local alternatives.
Second, we show that, under the null hypothesis, the ELR statistic for
the parameters defined by GS admits always a Bartlett correction. This
result in itself is not new in the sense that the Bartlett correction factor
obtained in the paper coincides in practice1
Our
calculations support Keith Baggerly's conjecture, as reported in the
Errata section of Owen's empirical likelihood Web page:
http://www-stat.stanford.edu/∼owen/empirical, that there
is a mistake in the formula of DiCiccio et al. (1991, p. 1056) of the Bartlett correction. Specifically
the term t2 should be 0, whereas the term
t1 should be 1/3. With this correction, the formula
of DiCiccio (1991) coincides with the one obtained
in this paper (cf. (15)) in the case of GS.
with the one obtained
originally by DiCiccio et al. (
1991) in the case of
a multivariate mean (see also, in the case of linear regressions,
Chen, 1993; in the case of one parameter
M
functionals, see
Zhang, 1996). What is new instead
is the way in which the Bartlett correction is obtained. Our method relies
on certain Bartlett-type identities (
Mykland,
1994) (see (5), which follows) and produces an alternative proof
of the Bartlett-correctability of the ELR that complements the results
of DiCiccio et al. (
1991) and, we believe,
provides a key to understanding the Bartlett phenomenon in the context
of EL inference for GS; see the discussion following Corollary 3 in
Section 3. We also propose a simple consistent estimator for the
Bartlett correction.
Third, we provide asymptotic justification for a bootstrap version of
the ELR statistic originally proposed by Owen (1988) as a “hybrid” method in which EL
confidence regions for the parameters of interest are based on the
bootstrap distribution of the ELR rather than the asymptotic
χ2 calibration. We show that the bootstrap approximates
the distribution of the ELR to an order
o(n−1) except if the observed sample
is contained in a set of probability
o(n−1), and we also show that the
ELR with bootstrap calibration accomplishes the Bartlett correction
automatically; i.e., the ELR with bootstrap critical values has the
same level of accuracy as the Bartlett-corrected one. The same
phenomenon was noted by Beran (1988, p. 694)
in the case of parametric bootstrap likelihood ratio tests.
Finally, we prove in the Appendix a general theorem on the density of
generalized noncentral quadratic forms, which is of its own independent
interest, because it can be used to calculate formal Edgeworth
expansions of asymptotic χ2 test statistics admitting a
stochastic expansion under a sequence of local alternatives.
The results in the paper are useful to obtain ELR tests that have a
desirable higher order accuracy property. The bootstrap version of the
ELR is particularly useful because the bootstrap calibration is likely
to be more accurate than the Bartlett-corrected χ2
calibration. Monte Carlo simulations (see, e.g., Corcoran,
Davison, and Spady, 1995; Baggerly, 1998;
and Tables 1 and 2 in
Section 4 of this paper) seem to suggest in fact that the χ2
calibration itself is not very reasonable, implying that the practical
usefulness of the Bartlett correction is limited.
Nonlinear regression models
The results of the paper can be applied to test simple hypotheses for
parameters defined in practice by any GS: e.g., they can be applied to
the ELR test for generalized linear models considered by Kolaczyk
(1994) and to the ELR test for generalized
projection pursuit models considered by Owen (1992). Other examples include ELR tests for
“moment” models similar to those considered (albeit in the
more general context of weakly dependent observations) by Burnside and
Eichenbaum (1996), and for nonlinear and
robust regression models; the latter are considered in Section 4 of
this paper.
On the other hand, the results of this paper do not cover the case of
composite hypothesis. When nuisance parameters are present (even in the
form of overidentifying restrictions), it is not clear from DL
arguments what the higher order behavior of the ELR statistic should
be, because the Bartlett-type identities do not hold for the nuisance
parameters. This is the principal way in which dual (and hence
empirical) likelihood shows different behavior from an ordinary
parametric likelihood, and this implies that the profile ELR test
obtained by maximizing out the nuisance parameters is in general not
Bartlett-correctable (Lazar and Mykland,
1999). Whether it is possible to obtain higher order refinements
to the distribution of profile ELRs is an open question that is
relevant to econometrics, because the latter is typically characterized
by (possibly overidentified) models with many nuisance parameters. One
possibility is to adjust the critical values rather than the test
statistics themselves using generalized Cornish–Fisher expansions
(Hill and Davies, 1968). The calculations
involved in such expansions are however notoriously difficult:
computerized algebra or possibly a suitable bootstrap procedure might
be very useful in this respect; see the discussion at the end of
Corollary 7 in Section 3.
The remainder of the paper is organized as follows. The next section
reviews briefly EL inference for GS, emphasizing its DL interpretation.
Section 3 contains the main results of the paper. Section 4 considers
two examples that illustrate the theory and reports the results of some
Monte Carlo simulations used to assess the effectiveness of the
Bartlett correction and bootstrap calibration in finite samples.
Section 5 contains some concluding remarks and indications for future
research. All the proofs are contained in the Appendix.
Throughout the rest of the paper we follow tensor notation and
indicate arrays by their elements. Thus, for any index 1 ≤
rj ≤ q (j =
1,2,…,k), ar is an
-valued
vector, ars is an
-valued
matrix, etc. We also follow the summation convention; i.e., for any two
repeated indices, their sum is understood. Finally the sum [sum ] and
product [prod ] operators are intended, unless otherwise stated, as
.
2. THE RELATIONSHIP BETWEEN EMPIRICAL AND
DUAL LIKELIHOOD
Suppose that the observations
(zi)i=1n
are i.i.d.
-valued
random vectors from an unknown distribution F. Let
be an unknown parameter vector associated with F and E
denote the expectation operator with respect to F. We assume that
the information about F and θ is available in the form
for some specified value θ0 of θ, with the GS
-valued
vector of known functionally independent functions, satisfying the following
standard regularity conditions.
GS1. (i) E
[fr(z,θ)] = 0 for
a unique θ0 ∈ int{Θ}, (ii)
fr(z,θ) is continuous at
θ0 with probability 1, (iii) E
[∂fr(z,θ0)/∂θs]
is of full column rank q, (iv) E
[fr(z,θ0)
fs(z,θ0)] is
positive definite.
The EL approach to inference for the parameter θ defined by the
constraint (1) is based on the profile ELR function
where pi = Pr(z =
zi), which shows that EL inference may be
interpreted as parametric likelihood inference using a data-determined
multinomial distribution supported on the observations. Assume that
where ch{·} denote convex hull of the points
fr(zi,θ),
and suppose that we are interested in testing the hypothesis
H0 : θ = θ0. A Lagrange
multiplier argument shows (Owen, 1990) that
the unique solution of (2) evaluated at θ = θ0 is
given by pi = [n(1 +
λrfr(zi,θ0))]−1,
where λr =
λr(θ0) is an
-valued
vector of Lagrange multipliers determined by
[sum ]fr(zi,θ0)/(1
+
λrfr(zi,θ0))
= 0. Simple algebra shows that the log ELR test
for the null hypothesis H0 : θ =
θ0 is given by
and Owen (1990) proves that, under
GS1(iv) and EL,
—similarly,
confidence regions for θ may be obtained as the set of points θ
such that W(θ) ≤ cα, where
cα = Pr(χq2
≤ cα) = 1 − α.
Suppose now that, for a fixed θ0, λ is regarded
as a free-varying vector of unknown parameters and consider
the following statistic:
where the subscript λ is used to emphasize that (4) now depends
on the unknown parameter λ. In the case of independent
observations, Wλ(θ0) coincides
with the logarithm of the DL Mykland (1995);
the latter corresponds to the product integral of
λrfr(zi,θ0)
and can be viewed as a likelihood function for the parameter λ
because, as long as
fr(zi,θ0)
is bounded and λ is contained in a neighborhood of 0,
for small λ. A standard optimization argument shows that the ELR
test of θ = θ0 defined in (3) is then equivalent
to a DL ratio (DLR) test on the dual hypothesis H0
: λ = 0, i.e.,
where W0(θ0) =
Wλ(θ0)|λ=0 and
is the maximum DL estimator solution of the first-order conditions
∂Wλ(θ0)/∂λr
= 0; we shall call the maximum DL estimator DLE.
As mentioned in the introduction, the equivalence between the ELR
test of θ = θ0 and the DLR test of λ = 0 is
crucial in the paper because it implies the possibility of analyzing
the higher order properties of the ELR statistic using likelihood
methods on the dual parameter λ instead. To elaborate further, let
us introduce some additional notation: for k =
(k1,…,kl)
∈ (Z+)l write
;
let ∂kWλ(·) =
∂kWλ(·)/∂λr1…∂λrk
and Γτ = Γ(0,τ) be an open sphere centered
around 0 with radius τ > 0. Assume that for all λ ∈
Γτ
where the inequality should be interpreted componentwise. For
|k| = 0, DL1 is a sufficient condition for
showing the existence and consistency of the DLE
solution of ∂Wλ(θ0) = 0. To see
this, notice that the
-valued
matrix E
[∂2Wλ(θ0)]
is negative definite as long as 1 +
λrfr(zi,θ0)
≥ 1/n (which is implied by the fact that under EL the
set Sλ = {λr : 1 +
λrfr(zi,θ0)
≥ 1/n} is closed convex and bounded). Therefore, by the
implicit function theorem there exists a neighborhood
Γτ where
∂Wλ(θ0) = 0 have a unique
solution. Moreover, E
[Wλ(θ0)] is uniquely
maximized at λ = 0, and
for all λ ∈ Γτ. Thus, given the strict concavity
of Wλ(θ0) in
Γτ, it follows by standard consistency results for
concave objective functions (e.g., Theorem 2.7 of Newey and McFadden, 1994) that
.
For |k| ≥ 1, DL1 is a sufficient condition
for interchanging the differential and integral operators in the
equation ∂kE
{exp[Wλ(θ0)] −
1} = 0; the latter can then be used to produce a new set of regularity
conditions known in ordinary parametric likelihood theory as Bartlett
identities (Bartlett, 1953). The resulting
Bartlett-type identities relate linear combinations of expectations of
DL derivatives and can be summarized concisely as
where
∂kjW0(·)
=
∂kjWλ(·)|λ=0(j
= 1,…,l) and the sum is over all partitions
k1|k2|…|kl
of the set of indices Rk. In practice (5)
can be used as in parametric likelihood inference to simplify some of
the expressions arising in the calculations of the higher order
cumulants of the DLR statistic (see (A.20) in the Appendix), leading to
the general result (13) about the Bartlett-correctability property of
the DLR (whence of the ELR).
3. HIGHER ORDER ASYMPTOTICS AND THE
BOOTSTRAP FOR THE ELR STATISTIC
This section contains the main results of the paper. Section 3.1
derives valid Edgeworth expansions for the DLE and the ELR under both a
sequence of local alternatives and the null hypothesis. Section 3.2
discusses how to estimate in practice the Bartlett correction, and
Section 3.3 shows how the bootstrap and EL can be combined to produce
highly accurate inference for the parameters defined by GS.
3.1. A Valid Edgeworth Expansion for the DLE and for
the ELR Statistic
Let us introduce some additional notation and assumptions. Let
∥·∥ denote the Euclidean norm; assume that for all λ
∈ Γτ and some α,β ∈
Z+
where, for
denote the vector containing all the different DL derivatives up to
lth order evaluated at λr = 0.
Conditions DL2 and DL3 are sufficient for proving the validity of
Edgeworth expansions; notice that DL2 is satisfied if E∥
f (z,θ0)∥αβ
< ∞.
Let
denote the formal Edgeworth expansion (Bhattacharya and Ghosh, 1978, eq. (1.14)) of the distribution of
,
where φq,κrs(·)
denote the q-dimensional normal density with mean 0 and
covariance κrs = E
[fr(z,θ0)
fs(z,θ0)] .
The following theorem shows that the DLE admits a valid Edgeworth
expansion in the sense of Bhattacharya and Ghosh (1978).
THEOREM 1. Assume that GS1, EL, and DL2 with α = 4, β = 3
hold. Then, for some constant C > 0,
Furthermore, assume that the Cramér condition DL3 holds
with l = 3. Then,
where
is the class of Borel subsets in
satisfying
Before obtaining an Edgeworth expansion for the ELR under a sequence
of local alternatives, we first discuss what type of local alternatives
is consistent with the DL approach to EL inference. A simple
modification of the argument of Mykland (1995, pp. 410–411) shows that the sequence of
local alternatives induced by the DL is of the form
for some finite nonrandom vector
.
This type of local alternatives assumes implicitly that the DL under the
alternative hypothesis belongs to the same parametric subfamily specified
under the null hypothesis (i.e., the product integral of
λrfr(zi,θ0)),
ruling out effectively the possibility of considering local
alternatives where both λ and θ are allowed to depend
on a sequence of local alternatives. This type of local analysis is
introduced by Chesher and Smith (1997) as a
generalization of the standard likelihood based local analysis (see,
e.g., Peers, 1971) and requires the
specification of an augmented local alternative of the form
Hna :
φn =
[θn λn],
where θn = θ0 +
θ/n1/2 and
λn =
λ(θ/n1/2) =
λ/n1/2 for some finite
nonrandom vector θ. This general framework, however,
might break down the DL construction for it requires the consideration
of alternatives that might not necessarily lie in the (log)DL subfamily
defined in (4). Bearing this in mind, it should be clear why the only
sequence of local alternatives for the ELR consistent with the DL
construction takes the form
.
For simplicity we reparameterize Hn as
where κrs1/2 is the square root
of the symmetric positive definite matrix κrs,
and we define the scaled arrays (moments)
so that κrs =
δrs—the Kronecker delta, i.e.,
δrr = 1, δrs = 0 for
r ≠ s. Let
Gq,τ(·) denote the distribution
function of a noncentral chi square random variable with q
degrees of freedom and noncentrality parameter τ. The following
theorem shows that the ELR test under a sequence of local alternatives
Hn admits a valid Edgeworth expansion in
the sense of Chandra and Ghosh (1980).
THEOREM 2. Suppose that the assumptions of Theorem 1 hold. Then,
uniformly in c ∈ [c0,∞) for
some c0 ≥ 0 (c0 = 0 if
r > 1),
uniformly over compact subsets of
, where
and
where
and, e.g.,
is the sum over the three different ways to partition a set of four
indices into two subsets of two indices each.
The preceding expansion is useful for two reasons: first, it can be
used to compute the approximate local power function of the test
H0 : λ = 0 versus alternatives of the form
(i.e., Pr(W(θ0) >
cα|Hn)
where cα =
Pr(χq2 ≥
cα) = α). Second, it can be used to obtain
a valid Edgeworth expansion of the ELR statistic under
H0 with remainder of order
o(n−1). To improve the latter to the
order O(n−2) we need to strengthen
DL2 and DL3 as in the following corollary; let
gq(·) denote the density of a chi
square random variable with q degrees of freedom and let
B = Brr.
COROLLARY 3. Assume that GS1, EL, and DL2 with α = 6 and
β = 5 and DL3 with l = 4 hold. Then, for some
c0 ≥ 0
Let WB(θ0) =
W(θ0)/(1 + B/(nq))
and cα =
Pr(χq2 ≥
cα) = α. Then
Expansion (12) shows clearly that the Bartlett factor B is
the main error term for the ELR statistic; moreover because B
enters linearly in the expansion it is clear that scaling
W(θ0) by the same factor improves the coverage
error to the order O(n−2), as in
(13). This remarkable property of the ELR statistic is a direct
consequence of the connection relating dual and empirical likelihood in
the case of independent observations: because the ELR can be
interpreted as a DLR for λ, and a DLR inherits properties of an
ordinary parametric likelihood ratio statistic via the Bartlett-type
identities (5), it is perhaps not surprising that the ELR shares the
Bartlett-correctability property of an ordinary likelihood ratio
statistic. In particular, as shown in the proof of Theorem 2 in the
Appendix, the Bartlett-type identities (5) imply that, for DL2 with
α = 5 and β = 4 and DL3 with l = 3, the signed square
root Wr(θ0) of the ELR
(i.e., an
-valued
random vector such that
Wr(θ0)Wr(θ0)
= W(θ0) +
Op(n−3/2))
is asymptotically
where cr and
cr,s are constants defined as
kr2 and
kr,s3 (with
)
in (A.20) in the Appendix. This result was originally proved by DiCiccio
and Romano (1989), using a different technique
involving some lengthy algebra, and shows that
Wr(θ0) can be mean and
variance corrected so that the resulting adjusted statistic is
N(0,δrs) +
O(n−3/2), as is typically the
case for the signed square root of ordinary parametric likelihood
ratios. The existence of a Bartlett correction for the ELR then follows
from this result combined with an Edgeworth expansion argument. The
latter shows in fact that the density of
Wr(θ0)Wr(θ0)
+ Op(n−3/2)
is proportional to
exp(−x2/2)(x2)q/2−1[1
+ ψ(x2)/n] +
O(n−2) where
ψ(x2) is a linear function in
x2, so that scaling W(θ0)
by a factor 1 + B/(nq) with B =
cr cr +
crr eliminates the coefficient of
n−1 in the expansion of
WB(θ0) yielding (13).
3.2. Estimation of the Bartlett Correction
The Bartlett correction for GS
depends on the third and fourth multivariate moments of
fr(z,θ0). The
computation of the moments involved in the threefold summation in (15)
takes O(nq3) computing time, so unless
q is very large the computational cost of the Bartlett
correction is not very high. Recall that B is based on the
scaled moments
κr1…rk
= E [(κr1
s1)1/2fs1(z,θ0)…(κrk
sk)1/2fsk(z,θ0)]
and suppose that
;
then a simple estimator of (15) can be based on
where
.
The following theorem shows that
is consistent for (15) and that replacing B with
in Corollary 3 does not alter the order of magnitude of the approximation
error (13). Let
denote a neighborhood of θ0 and make the following
assumption.
THEOREM 4. Assume that GS2 holds. Then,
.
Furthermore, suppose that the assumptions of Corollary 3 hold and that
.
Then, for some c0 ≥ 0
Remark. The assumption of i.i.d. sampling can be relaxed by considering
(zin)i≤n,n≥1
as a triangular array of
-valued
random vectors as in Owen (1991); i.e., for each
n,
fr(z1n,θ),…,fr(znn,θ)
are independent but not identically distributed
valued random vectors. The results presented in Sections 3.1 and 3.2 are still
valid by replacing some of the previous regularity conditions with the
following uniform (in n) version.
GS1. (i) E
[fr(zin,θ0)]
= 0 for a unique θ0 ∈ Θ for all n,
(ii) lim infn→∞
ζn /n > 0, where
ζn is the smallest eigenvalue of [sum ] E
[fr(zin,θ0)
fs(zin,θ0)],
GS2.
EL. Pr(0 ∈
ch{fr(z1n,θ0),…,fr(znn,θ0)})
= 1 as n → ∞,
DL2.
DL3. lim supn→∞
sup∥t∥≥δ|E
[exp(ıtrZnr
0l)]| < 1 for all δ > 0
and some l ∈ Z+,
where Wnλ(·) and
Znr
0l are, respectively, the log DL defined in
(4) and the
-valued
vector of DL derivatives evaluated at λ = 0 for the triangular array
fr(zin,θ0)
n ≥ 1. In particular GS1, GS2(i), EL, and DL2(iii) with
β = 3 and Znr
0l = Znr 0 for
l = 1 imply Theorem 2 of Owen (1991), whereas DL2(i)–(ii) and DL3 imply that
we can use Theorem 20.6 of Bhattacharya and Rao (1976) together with Theorem 3.2 and Remarks 3.3 and
3.4 in Skovgaard (1981) to justify the
validity of the Edgeworth expansions in Theorems 1 and 2. The results
shown in Section 3.3, which follows, are also valid by replacing BDL
with uniform versions similar to those just shown.
3.3. A Valid Edgeworth Expansion for the Bootstrap DLE
and the Bootstrap ELR Statistic
In this section we consider the higher order properties of bootstrap
dual/EL inference. Let χn* =
(z1*,…,zn*)
denote a bootstrap sample, i.e., a resample drawn independently and
uniformly from the observed sample χn with the
property that Pr(zi* =
zj|χn) =
1/n for 1 ≤ i,j ≤ n, and
let
fr(zi*,θ0)
denote the corresponding bootstrap GS.
We first consider bootstrapping the DLE
.
Let
denote the bootstrap (log) DL where, in analogy to Section 2, the
dual parameter λ*r is free-varying. The bootstrap DLE
(BDLE)
solves 0 =
∂Wλ**(θ0)/∂λr.
Assume that for some α,β ∈ Z+ and
γ > 0
BDL.
and let Pr* denote the bootstrap probability conditional on
χn. The following theorem establishes the
higher order equivalence between the original and BDLE.
THEOREM 5. Assume that GS1, EL, and BDL with α = 6, β =
3, and some γ > 0 hold. Then, for some constant C > 0
except if χn is contained in a set
with probability o(n−1). Furthermore,
assume that the Cramér condition DL3 holds with l = 3.
Then, for every class
of Borel subsets in
satisfying (8)
except if χn is contained in a set of
probability o(n−1).
Notice that another way to express (18) (and analogously (19) and
also (21), which follows) is
We now consider bootstrapping the ELR. It is important to note that
the validity of the bootstrap in the context of EL inference depends
crucially on the validity of the bootstrap moment condition
E*[fr(zi*,θ)]
= 0 at θ = θ0, where E* is the bootstrap
expectation under Pr*. A straightforward method to achieve this is to
consider the centered bootstrap GS
fr*(zi,θ)
=
fr(zi*,θ)
−
E*[fr(zi*,θ0)]
, which mimics the original GS as defined in (1) because by
construction the bootstrap moment condition
E*[fr*(zi,θ0)]
= 0. Alternatively, we can consider the so-called biased bootstrap GS
fr(zi[dagger],θ),
where the zi[dagger]'s are
resamples drawn independently from the original sample
χn with the property that
Pr(zi[dagger] =
zj|χn) =
pj for 1 ≤ i,j ≤
n, and the pjs are the estimated
EL probabilities. This resampling procedure was proposed, originally in
1992, by Brown and Newey (2001) in the
context of bootstrapping for generalized method of moments (GMM); the
term biased used here is borrowed from Hall and Presnell
(1999). Notice that
fr(zi[dagger],θ)
does not need to be centered, because by definition
E[dagger][fr(zi[dagger],θ0)]
= 0 where E[dagger] is the expectation under
Pr[dagger]. Thus both methods lead to unbiased (conditional
on χn) bootstrap GS at θ =
θ0. The latter fact is the key to the validity of
bootstrap ELR because it implies that the distribution of the bootstrap
ELR (BELR) resembles the null distribution of the original ELR
regardless of whether the null hypothesis is true or not. More
important, at least in the context of this paper, the results of Brown
and Newey (2001) (see also Horowitz, 2001) imply that biased bootstrap leads
to the same theoretical improvements to the finite-sample distribution
of the ELR statistic as those obtained using the standard (uniform)
bootstrap (cf. Corollary 7). Bearing this in mind, in the remaining
part of this section we consider bootstrap EL inference based on the
centered bootstrap GS
fr*(zi,θ).
An immediate consequence of centering is that the bootstrap
equivalent of assumption EL holds, i.e.,
The latter implies that the bootstrap analogue of the profile ELR
function defined in (2),
admits a unique solution at θ = θ0 and that such
a solution can be obtained by the same Lagrangian argument of Section
2. Let
denote the resulting log BELR. Solving (20) and calculating
for b = 1,…,B bootstrap samples
χn* yields an estimator of the distribution of the
ELR that can be used to form critical values for test statistics (and/or
confidence regions) in the usual way.
As in Section 2, the BELR
can be interpreted as a bootstrap DLR test for the dual parameter
λ*r, i.e.,
where
is the bootstrap DL based on the centered bootstrap GS
fr*(zi,θ0),
W0*(θ0) is the
“restricted” bootstrap DL evaluated at λ* = 0, and
is the BDLE that solves 0 =
∂Wλ**(θ0)/∂λr
using centered bootstrap GS
fr*(zi,θ0).
Note that because of the centering
converges (in bootstrap probability) to 0 and not to the DLE
as in Theorem 5. Because W0*(θ0) = 0, let
;
the following theorem shows that the BELR statistic approximates the
distribution of the ELR statistic under the null hypothesis up to the order
o(n−1).
THEOREM 6. Suppose that the assumptions of Theorem 2 with α =
6 hold. Then for some c0 ≥ 0
except if χn is contained in a set
with probability o(n−1).
Notice that the moment assumption of Theorem 6 (which is slightly
stronger than the one assumed in Theorem 2) is necessary to ensure that
the BELR test is accurate up to
o(n−1); i.e., the BELR test has
rejection probabilities that are correct up to the same order. To see
this note that (21) is equivalent to
thus using the same arguments as Andrews (2002, pp. 149–150), it follows that for
cα* = inf{c ∈
[c0,∞) :
Pr*(W*(θ0) ≥ c) ≥ α}
which implies that, for large n, |1 − α
− Pr(W(θ0) ≤
cα*)| ≤
n−1, or equivalently
Thus, with the bootstrap calibration the level of ELR test is correct
through the order O(n−1). In
contrast, as shown in Theorem 2, the ELR test with χ2
calibration is accurate up to
O(n−1). On the other hand, using a
slightly refined argument based on generalized Cornish–Fisher
expansions (Hill and Davies, 1968), the
approximation error in (22) can be improved to
O(n−2), as next corollary shows.
COROLLARY 7. Suppose that the assumptions of Corollary 3 with
α = 8 hold. Then,
Corollary 7 shows that bootstrapping the ELR delivers the same type
of higher order accurate inference implied by the existence of a
Bartlett correction. The same phenomenon was noted by Beran (1988) in the context of bootstrap likelihood ratio
inference. Unless dim[f (z,θ)] is very
large, it is clear that the Bartlett-corrected ELR is computationally
more convenient than the BELR. On the other hand, as already mentioned
in the Introduction and further illustrated in Section 4 by some
simulations, it seems that the BELR has better finite-sample accuracy
than the Bartlett-corrected ELR. Moreover, the higher order refinements
delivered by the bootstrap approach to ELR inference do not depend in
general on Bartlett-type identities. This should be important when
considering the ELR statistic with nuisance parameters. To see why, let
η be a finite-dimensional vector of nuisance parameters and
denote the resulting profile ELR (PELR) where
.
As shown by Lazar and Mykland (1999) the fact that
the Bartlett-type identities (5) do not hold for η implies that in
general
is not Bartlett-correctable. Suppose however that
admits a valid asymptotic expansion in the sense of Chandra and Ghosh (1979) and that
.
Then, at least theoretically, generalized Cornish–Fisher expansions can
be brought to bear to show that
with adjusted critical values is accurate to an order
O(n−2), independent of whether the
Bartlett-type identities hold or not. Furthermore, using arguments
similar to those used in the proofs of Theorem 6 and Corollary 7 in the
Appendix it might be possible to obtain higher order refinements to the
distribution of the PELR using critical values based on the bootstrap PELR
.
APPENDIX
A Theorem on the Density of Generalized Noncentral Quadratic
Forms and Its Application to Hermite Polynomials. The following
theorem derives a general formula useful to express the density of
generalized noncentral quadratic forms (i.e., of scalar random
variables obtained by contracting nonzero mean normal random vectors
over multidimensional arrays of constants) in terms of finite linear
combinations of noncentral chi square random variables. The formula can
be applied to obtain asymptotic expansions for test statistics under a
sequence of local alternatives starting from the Edgeworth expansion of
their corresponding signed square root, rather than from their
approximate moment generating function.
Let φq,γ(·) denote the
q-dimensional normal density with mean γ and identity
covariance matrix and let
gq,τ(·) denote the density of a
noncentral chi square random variate with q degrees of freedom
and noncentrality parameter τ < ∞.
THEOREM 8. Let wRk =
wr1…wrk
where each
wrj ∼
φq,γ(w) (j =
1,2,…,k),
bRk =
br1…rk
an
-valued
array of constants not depending on n,
tr
an
-valued
vector of auxiliary variables, and ∂k(·) =
∂k(·)/∂tr1…∂trk.
Consider the function
and make the following assumptions.
GQF1.
,
the space of k times continuously differentiable functions on an open
set
of tr = 0,
GQF2.
where the
-valued
vector vr is defined subsequently. Then, for any
arbitrary noncentral k form
wRkbRk,
the following holds:
where
,
the symbol #Rk denotes
the number of different indices in the set Rk,
and the sum is over ϒ =
{ν2,…,νk}—the
number of ways of partitioning a set of Rk
indices into νj subsets containing j
indices (j = 2,…,k) such that the resulting
homogeneous polynomial in
γRk1 is even or
odd according to the number of indices in the set
Rk.
Proof. We use the transformation from
where x =
wrwr,
vr =
wr/(wsws)1/2
and the following identity:
Using T, the density for x is obtained by
integrating out the vector vr ∈
V1,q (i.e., over the unit sphere
vrvr = 1 in
), i.e.,
where |JT| =
xq/2−1/2 is the Jacobian of
the transformation T and (dvr)
denotes the unnormalized Haar measure on the Stiefel manifold
V1,q. Upon normalizing the Haar measure on
V1,q by the constant
2πq/2/Γ(q/2) and
interchanging differentiation and integration in (A.2), which is
permissible under GQF2 for f
(·,vr;
tr) =
exp(x1/2vrtr),
we can then use Theorem 7.4.1 in Muirhead (1982) to obtain
with
,
and (c)j = Γ(c +
j)/Γ(c). Differentiating (A.3) and evaluating
the resulting derivatives at t = 0 yields a polynomial in
x of degree at most k with coefficients given by
The constant
C(b,γ,#Rk)
in (A.4) is itself an even or odd polynomial in
γRk1 with
coefficients obtained by contracting accordingly the components of
γRk1,
δRk2, and
bRk. Elementary
symmetry considerations show that the number of such contractions can
be found using standard combinatorial results on partitions reported,
e.g., in Abramowitz and Stegun (1970, Table
24.2, p. 831). Combining (A.3) and (A.4), because
gq,τ(x) =
exp[−(x +
τ)/2]xq/2−10
F1(;q/2;
xτ/4)/2q/2Γ(q/2),
(A.1) follows immediately. █
For generalized quadratic forms up to k = 6 (corresponding
to the terms appearing in third-order Edgeworth expansions), (A.1)
yields
from which the generalized quadratic forms
bRkhRk
(1 ≤ k ≤ 6) based on the multivariate Hermite polynomial
hRk =
hr1…rk
=
(−1)k∂kφq,γ(w)/∂wr1…∂wrk
have densities
where
∇kgq,τ(·)
is the kth-difference operator applied to the density
gq,τ(·), i.e.,
.
Proofs. Let C denote a generic positive constant
not depending on n that may vary from one (in)equality to
another. For simplicity of notation let W(θ0)
= W and Wλ(θ0) =
Wλ.
Proof of Theorem 1. We first establish (6). Let
λRk =
λr1…λrk;
a Taylor expansion of the DL first-order condition 0 =
∂Wλ about 0 gives, for any
λr ∈ Γτ,
where Rn(λ) ≤
C∥λr∥3max|k|=4
sup∥λ−λ∥≤τ|∂kWλ|/n.
By DL2 for some Cε < ∞ and all
ε > 0
where the first inequality follows by a moderate deviations result of
Bhattacharya and Rao (1976, Corollary 17.12),
the second by Rosenthal and Markov inequality, and the third by
combining the second equality with the triangle inequality. Let
Sn denote the union of the sets in (A.7);
clearly Pr(Sn) =
o(n−1), so that on the set
Snc we can write (A.6)
as λr =
gn(λr) where
gn(·) is a continuous function from
satisfying ∥gn(u)∥ ≤
Cn−1/2(log n)1/2
for all ∥u∥ ≤
Cn−1/2(log n)1/2.
Application of Brower's fixed point theorem as in Theorem 2.1 of
Bhattacharya and Ghosh (1978) and the
concavity of Wλ show that there exists a unique
sequence λnr such that
Pr(∥λnr∥ <
Cn−1/2(log n)1/2)
≥ 1 − Cn−1(log
n)−2, whence (6). To derive the Edgeworth
expansion (7), we first derive a stochastic expansion for
.
For notational convenience let
;
define the following Op(1) random
arrays:
where
κr1…rk
= E
[fr1(z,θ0)…
frk(z,θ0)],
and let
denote the vth-order (mixed) cumulant of
ZRj =
Zr1…rj.
Combining (A.7) and (A.8) yields
for ∥λ∥ ≤ ∥λ∥, where
Pr(∥Rn(λ)∥ >
Cn−2(log n)5/2) =
o(n−1). Because
λr =
κrsZs +
ζ1nr, where
Pr(∥ζ1nr∥ >
Cn−1/2(log n)) =
o(n−1), and
by the weak law of large numbers, (A.10) can be written as
where Pr(∥ζ2nr∥
> Cn−1(log n)3/2)
= o(n−1). Upon substituting this
last stochastic expansion into (A.10), because
,
we obtain the stochastic expansion
where Zr =
κrsZs and
Pr(∥ζ3nr∥ >
Cn−3/2(log n)2) =
o(n−1). To derive the Edgeworth
expansion of λr on
(the class of all Borel subsets of
),
notice that λr is a function of the three
random arrays Zr,
Zrs, and Zrst
and recall that Zr3 is the
-valued
vector containing all the different DL derivatives under
H0 up to third order. Let
γZr3(B)
denote the (signed) measure corresponding to the characteristic
function of Zr3 for
(the class of all Borel subsets of
)
and consider the following continuous transformation:
Ur =
χ(Zr3),
where
and κr,st and
κr,stu are defined in (A.9). Let
r1,…,rk
denote indices each with
components,
E(UrUs)
= Σrs,
E(UrUs)
= Σrs = κrs,
E(UrUs)
= Σrs,
κr1…rk
=
CUM(Ur1,…,Urk),
the kth-order multivariate cumulant of
Ur, and
hr1…rk(u)
=
(−1)k(∂k/∂Ur1…∂Urk)φq,
Σrs(Ur).
Given the continuity of (A.12) and the independence of
Ur from Urs
and Urst we have that, for
Zr3 ∈
χ−1(B),
γUr(B)
=
γZr3
[χ−1(B)] is given by
Consider the following continuous transformation:
Xr =
ψ(Ur),
for
where Ur =
κrsUs,
κrst =
κrr′κss′κtt′κr′s′t′,
and κrstu =
κrr′κss′κtt′κuu′κr′s′t′u′.
By (A.7) there exists a C > 0 such that on the set
SnU = {U :
∥Ur∥ <
C(log n)1/2} the transformation ψ
of (A.14) is a C∞ diffeomorphism on
SnU onto its image. Set
;
clearly
where |Jx| is the Jacobian
of the transformation (A.14). Because
can be expressed as
for some polynomials qj(·), we
can integrate Xr out, obtaining
for some polynomials
qj′(·). Thus by (A.15) we
have that for λnr ∈
Bε with ε =
Cn−3/2(log n)2
It remains to show that the density
of the formal two-terms Edgeworth expansion of the DLE
λr obtained by the delta method coincides with
.
Because λr is a maximum likelihood type estimator,
its cumulants of order bigger than 4 are
o(n−1), whence
,
where
is the Fourier transform of
defined in (A.15). By Cauchy's estimates for derivatives of analytic
functions (Bhattacharya and Rao, 1976, Lemma 9.2),
we then have that all the derivatives of
differ by o(n−1) at η = 0, implying
that the two expansions are identical. █
Proof of Theorem 2. A Taylor expansion of W about the
DLE λr and (A.7) show that on the same set
Snc of Theorem 1
W =
λrλs(∂2Wλ)pn(Zr1…rk,λr)
+ ζ4n, where
pn(·) is a quartic polynomial in
λr and the random variable
ζ4n is such that
Pr(|ζ4n| >
Cn−3/2(log n)5/2)
= o(n−1). By a further Taylor
expansion of W about the normalized deviation
n1/2ηr =
ηr = (λr −
λnr) and the second
inequality in (A.7), we have that on
Snc with
Pr(Snc) = 1 −
o(n−1) W admits the
following stochastic expansion,
uniformly over compact subsets of
.
Define the scaled arrays
where
κr1…rk
= E [(κr1
s1)1/2fs1(z,θ0)…(κrk
sk)1/2fsk(z,θ0)]
. Substituting the stochastic expansion λr
− λnr =
Zr +
Λnr +
op(n−1) (where
Λnr is as in (A.11)) in
(A.16), some algebra shows that the stochastic expansion for the ELR
statistic W under a sequence of local alternatives has signed
square root Wr (i.e.,
WrWr +
op(n−1) =
W) given by
To prove (10), we first show that Wr
admits a valid Edgeworth expansion in the sense of Bhattacharya and
Ghosh (1978). Because
Wr is a function of the first three DL
derivatives, let
γZr3(B)
denote the measure corresponding to the characteristic function of the
vector Zr3 (as in
the proof of Theorem 1) and consider the same continuous transformation
defined in (A.12), Ur =
χ(Zr3) where
and
.
Clearly (A.18) implies that
γUr(B) =
γZr3
[χ−1(B)] can be expressed
using the same Edgeworth measure as in (A.13), the only difference
being that the normal distribution appearing in the leading term has
now nonzero mean
.
Proceeding now as in the proof of Theorem 1, define
Xr =
ψ(Ur)
where
On the set SnU =
{U : ∥Ur∥
< C(log n)1/2}, the transformation
ψ defined in (A.19) is a C∞
diffeomorphism on SnU
onto its image. Then, as in the proof of Theorem 1, the measure
corresponding to the characteristic function of the inverse
transformation of (A.19) can be expressed as in (A.15) for some
polynomials qj′′(·)
whose degree is even or odd according to j. Integrating out
Xr, the joint expansion for
Ur is then reduced to that of
.
It remains to calculate the formal Edgeworth expansion of
.
Using the delta method, some algebra shows that the approximate cumulants for
Wr are
where kr =
E(Wr),
kr,s =
Cov(Wr,Ws),
etc., with
Applying now the Bartlett-type identities (5) to (A.20), it is
straightforward to verify that
κr,s,t −
[3]κr,st +
2κrst = 0 and
kr,s,t,u4
= 0. Thus, as in the conclusion of Theorem 1, for the class
of Borel subsets in
satisfying (8), the formal Edgeworth expansion of
,
is valid in the sense of Bhattacharya and Ghosh (1978). The validity of expansion (10) follows by
verifying that the conditions (a)–(c) of a theorem in Chandra and
Ghosh (1980, p. 173) that we will denote as
Theorem CG hold. By (A.17),
for 1 ≤ |k| ≤ 3 and
pn(·) is a polynomial; therefore
condition (a) of Theorem CG holds—in particular
As (iv) (see p. 172) holds on the same set
Snc of Theorem 1.
Condition (b) of Theorem CG is a Cramér condition as assumed in
DL3, whereas condition (c) of Theorem CG holds by the validity of
(A.21). Expansion (10) is therefore valid in the sense of Chandra and
Ghosh (1980). It remains to calculate the
polynomials pjk in (10). By (A.21) and the
linear transformation
,
we have that on Sc =
{yr : yr
yr ≤ c} for any c ∈
[c0,∞) with c0 ≥ 0
where ∂k =
∂k(·)/∂yr1…∂yrk
and the cumulants krj,
kr,sj
(j = 2,3), and
kr,s,t3 are
defined in (A.20). Integrating (A.22) and using (A.5) yields
where
with
from which (10) and (11) follow after some algebra, noting that
κr,stt = κrstt
and κr,st =
κrst. █
Proof of Corollary 3. By (essentially) the same arguments of
Theorem 2, it is possible to show that, except on a set
Sn with Pr(Sn) =
o(n−2),
Wr admits a stochastic expansion that
depends on the first four DL derivatives
Zr4 and that there
exists a transformation ψ′ similar to the one given in
(A.19) such that the measure corresponding to the characteristic
function of (ψ′)−1 can be expressed as the
product of a multivariate normal and some polynomials
qj′′′(·)/nj/2
for j = 1,…,4. Then, the same integration argument
giving (A.15) in the proof of Theorem 1 shows that
Wr has a valid Edgeworth expansion up to
o(n−2) whose coefficients cannot be
written down explicitly. Thus, using the moment condition and
Cauchy's estimates for derivatives of analytic functions
(Bhattacharya and Rao, 1976, Lemma 9.2) it is
possible to show that this expansion agrees with the formal four-terms
Edgeworth expansion
,
i.e., for the class
of Borel subsets in
satisfying (8), the following expression,
is a valid Edgeworth expansion in the sense of Bhattacharya and Ghosh
(1978). Thus to show (12) it suffices to
calculate
for j = 1,…,4, where Sc is a
sphere in
with radius c1/2 for some c ∈
[c0,∞). Integrating over the sphere
Sc, using (A.5) (with γ = 0) and the
standard argument based on the oddness/evenness property of the
Hermite polynomials appearing in the Edgeworth expansion of W
(Barndorff-Nielsen and Hall, 1988), gives the
following Edgeworth expansion,
whose validity in the sense of Chandra and Ghosh (1979) follows, using the same arguments of Theorem
2, by the validity of (A.24). Let
∇Gq(·) =
Gq+2(·) −
Gq(·); noting that
∇Gq(·) =
−2gq+2(·) and
gq+2(c)/gq(c)
= c/q, (12) follows immediately from (A.25). To
prove (13) notice that by a Taylor expansion
whence inserting (A.26) in (12) yields
from which (13) follows. █
Proof of Theorem 4. We first show that
for any
.
Let fRk(z,θ) =
fr1(z,θ)…
frk(z,θ)
and notice that by GS1(ii)
fRk(z,θ)
is continuous at θ0 with probability 1 (w.p.1) for
k = 2,3,4; the consistency of
implies that there exists a δn → 0 such that
.
Let Δf (θ) =
sup∥θ−θ0∥≤δn|
fRk(z,θ)
−
fRk(z,θ0)|;
continuity of
fRk(z,θ)
at θ0 shows that Δf (θ)
→ 0 with probability 1. The dominance condition GS2(i) implies that
whence by dominated convergence E [Δf (θ)] → 0. By Markov inequality
Pr(|[sum ]Δfi(θ)/n|
> ε) ≤ E [Δf (θ)]/ε → 0 for all ε > 0. By
the weak law of large numbers
;
moreover
Therefore, the triangle inequality implies that
.
The consistency of
follows then by the continuity of tensor multiplication and matrix inversion.
To establish (17) we use the following mean value expansion:
where
.
Because by the weak law of large numbers
,
it suffices to show that
.
This follows by the same argument used to establish
.
Because by GS2(ii) the derivative
∂fRk(z,θ)/∂θr
is continuous at θ0, it follows that
Δ∂f (θ) =
sup∥θ−θ0∥≤δn|∂fRk(z,θ)/∂θr
−
∂fRk(z,θ0)/∂θr|
→ 0 (w.p.1), so that by GS2(iii),
,
and the triangle inequality we have that
i.e., n−1 [sum ]
∂fRk(zi,θ)/∂θ
= Op(1). By the n1/2
consistency of
,
the stochastic expansion (A.27) shows that
,
whence by the delta method
.
The conclusion of the theorem follows by the oddness/evenness property of
the Hermite polynomials as in Corollary 3. █
Let χn denote the observed sample. In what
follows we write Pr(χn) =
o(n−1) to denote the event
“the observed sample χn is contained in
a set of probability
o(n−1).”
Proof of Theorem 5. We first show (18). Recall that
∂kWλ** = [sum ]
fr1(zi*,θ0)…
frk(zi*,θ0)/(1
+
λ*rfr(zi*,θ0))k
and that the BDLE
solves 0 = ∂Wλ**(θ0). By (A.7)
and Markov inequality we have that for all ε > 0
except if Pr(χn) =
o(n−1). The triangle and Markov
inequalities show that
|E*(∂kW0*)|
≤ |E*(∂kW0*)
−
E(∂kW0)| +
|E(∂kW0)|
and
Pr(|E*(∂kW0*)
−
E(∂kW0)|
> ε) = o(n−1), implying
that for 2 ≤ |k| ≤ 3 except if
Pr(χn) =
o(n−1)
Furthermore the moderate deviations estimates of Bhattacharya and Rao
(1976, Corollary 17.12) show that for α = 6
except if Pr(χn) =
o(n−1). Let
Sn* denote the union of sets where
(A.28)–(A.30) hold, and consider a Taylor expansion of 0 =
∂Wλ**(θ0)/n
about the DLE
where
.
A further mean value expansion of each of the
in (A.31) about 0, the uniform moment condition BDL(i) and
(ii), and (A.28)–(A.30) show that outside the set
Sn* with bootstrap probability
Pr*(Sn*) =
o(n−1), we can rewrite (A.31) as
where the continuous function
satisfies ∥gn(u)∥ ≤
Cn−1/2(log n)1/2
for all ∥u∥ ≤
Cn−1/2(log n)1/2.
Thus as in the proof of Theorem 1, Brower's fixed point theorem
implies that there exists a sequence of random vectors
such that
except if Pr(χn) =
o(n−1). To prove (19) we follow the
same arguments of Theorem 1. First, by repeated application of
(A.28)–(A.30) it is possible to show that, except if
has the following stochastic expansion:
where
is an
-valued
function of the BDL derivatives
(and their expectations) evaluated at
whose norm is
Op*(n−1(log
n)3/2) except if
Pr(χn) =
o(n−1) (i.e.,
is the bootstrap analogue of Λnr
given in (A.11)). Next by Lemma 2 of Babu and Singh (1984) the characteristic function of
satisfies with probability 1
for some positive constants δ,γ,γ′, where
is the
-valued
vector containing all the different BDL derivatives (up to third order)
evaluated at
.
Thus, as in Theorem 1, it is possible to show that except if
Pr(χn) = o(n−1)
for every class
of Borel subsets in
satisfying
where
is the density of the empirical Edgeworth expansion obtained by
replacing the population moments with their bootstrap analogue. Thus
(19) follows if for all ε > 0
Because each bootstrap moment can be expressed as
,
and
where ZiRl is the
ith component of the
-valued
array of DL derivatives evaluated at an arbitrary λ ∈
Γτ, (A.33) follows by verifying that for 1 ≤
|k| ≤ 4
The first limit follows by a mean value expansion about 0. To see
this notice that by the second equality in (A.7), we have
so that by the triangle inequality
for some Cε < ∞, where
.
Moreover by (6) and Markov inequality
whence
The second limit in (A.34) follows by Rosenthal and Markov
inequality. █
Proof of Theorem 6. Let
Zr1…rk*
= [sum ]
fr1(zi*,θ0)…
fr1(zi*,θ0)
and
κr1…rk*
=
E*(Zr1…rk*/n)
and notice that ∂kW0* =
(−1)k−1(k −
1)(Zr1…rk*
−
κr1…rk*),
where W0* =
Wλ**|λ*=0 is the
“restricted” BDL evaluated at 0. As in Theorem 5 we have
that, except if Pr(χn) =
o(n−1), for 1 ≤
|k| ≤ 3
whereas by Markov inequality
|κr1…rk*
−
κr1…rk|
= op(n−1), so
that
except if Pr(χn) =
o(n−1). A Taylor expansion of
W* about 0 gives
where
.
As in Theorem 5 it is possible to show that
admits a stochastic expansion of the form
where κ*rs is the matrix inverse of
κrs* and
Λn*r is as in
(A.32). Repeated applications of (A.35) show that
except if Pr(χn) =
o(n−1), and thus by triangle
inequality it follows that
Substituting (A.37) into (A.36) shows that, except if
Pr(χn) =
o(n−1), the BELR admits the
following stochastic expansion:
where pn(·) is a polynomial in
Zr1…rk*
with coefficients depending on
κr1…rk*.
Let Zr*c =
Zr* − κr*;
as in Theorem 2, we can obtain the signed square root decomposition of
W* =
κ*rsWr*Ws*
+ op*(n−1)
with
which shows that, except if Pr(χn) =
o(n−1),
Wr* is a function of the first three BDL
derivatives outside a set Sn* with
bootstrap probability Pr*(Sn*) =
o(n−1). Thus, by the same arguments
of Theorem 5 it can be shown that Wr*
admits a valid Edgeworth expansion in the sense of Bhattacharya and
Ghosh (1978) except if
Pr(χn) =
o(n−1). Therefore following the same
steps of Theorem 2 it can be shown that, except if
Pr(χn) =
o(n−1), W* admits the
following valid (in the sense of Chandra and Ghosh,
1979) Edgeworth expansion
where B* =
κrstu*κ*rsκ*tu/2
−
κrtv*κsuw*κ*rsκ*tuκ*vw/3
is the bootstrap Bartlett correction, i.e., (16) at θ =
θ0. By (A.34) we have
and thus combining (A.38), (A.39), and (A.25) yields
except if Pr(χn) =
o(n−1). █
Proof of Corollary 7. As in the proof of Corollary 3 it is
possible to show that the BELR admits a valid Edgeworth expansion up to
o(n−2) except if
Pr(χn) =
o(n−2). Thus by the results of
Corollary 3, Theorem 6, and the generalized Cornish–Fisher
expansion formula of Hill and Davies (1968)
it follows that the asymptotic expansions for the α point of
W and W* are
respectively, where cα =
Pr(χq2 ≥
cα) = α. Therefore using the delta method,
a Taylor expansion, and the Edgeworth expansion of W, we have
which shows that effectively the BELR accomplishes the Bartlett
correction automatically. █
