1. INTRODUCTION
Duration analysis is an integral component of empirical and theoretical econometrics. Of particular concern is the treatment of unobserved heterogeneity. Authors such as Lancaster (1979), Heckman and Singer (1984), and Baker and Melino (2000) have shown that ignoring unobserved heterogeneity can lead to biased estimates of the hazard function and that different specifications for the distribution of the unobserved heterogeneity can lead to very different estimates of the hazard function. As a consequence, several authors have considered estimation of these models without specifying any parametric form for the unobserved heterogeneity.
The Heckman and Singer (1984) approach consistently estimates the parameters of the hazard function. To our knowledge, no one has worked out the asymptotic distribution of their estimator. Most applied researchers base their inferences conditional on estimates of the unobserved heterogeneity. This can lead to incorrect inferences.1
One might argue in defense of these approaches that the distributions can always be bootstrapped. However, this raises the question of whether the bootstrap is consistent in those cases.
This paper introduces a new estimator of hazard rates without imposing functional form restrictions on the distribution of the unobserved heterogeneity. To our knowledge, this is the first efficient semiparametric estimator for the class of models considered. Unlike Heckman and Singer (1984), we derive the asymptotic distribution of our estimator and show that it obtains the semiparametric efficiency bounds. One drawback is with respect to the usual Weibull model, for which Hahn (1994) has shown that there exists no
-consistent estimator. However, the asymptotic results that we develop do apply for a wide class of duration models, including non-Weibull and certain Weibull-like models.
Horowitz (1996) has proposed an
-consistent estimator for a certain proportional hazards model with unobserved heterogeneity. This estimator is more general than ours in that it does not specify a functional form for the baseline hazard. On the other hand, because it leaves the baseline hazard unspecified, it will generally not be efficient. The Nielsen, Linton, and Bickel (1998) approach parameterizes the baseline hazard rate but leaves the dependency on observed covariates unspecified.
Our approach is quite straightforward. The likelihood function for continuous duration models with unobserved multiplicative heterogeneity is written as a function of the hazard rate, whose parametric form is known, and the density function of the integrated hazard rate. We propose using nonparametric (kernel) estimates of this density function in a two-step procedure to obtain an asymptotically efficient estimator.
The following section discusses the estimation procedure in greater detail. Section 3 presents the asymptotic properties of the estimator. The estimator and consistent standard errors can be calculated using the optimization procedures of most intermediate-level computer packages. Section 4 discusses additional estimation issues and the semiparametric efficiency bounds. Section 5 concludes. Proofs are collected in three Appendixes.2
Monte Carlo results demonstrating the viability of the technique can be obtained from the authors.
2. THE ESTIMATOR
We focus on estimating mixed hazard rate models of the form λi(β)νi = λ(ti,xi;β)νi where ti is the duration variable, xi is a vector of observable covariates,
is the parameter of interest, and νi is an unobservable nonnegative error with unit mean and is independent of xi.
Consider first a benchmark estimator of β, supposing that we know the distribution of the unobserved heterogeneity. The survival function of ti, conditional on xi, is written
where
is the integrated hazard, Fν is the distribution function of νi, and
is the corresponding Laplace transform. With Fν (or
) known, the conditional density function of ti can be written as
where
is the derivative of
and the single observation log-likelihood function is
An efficient, albeit infeasible, estimator is obtained by maximizing the log-likelihood function based on (2.3). Alternatively, if a preliminary
-consistent estimator is available, an asymptotically efficient two-step estimator can be obtained using the score function and Hessian based on the log-likelihood in (2.3).
To get an intuitive understanding of our approach, let S and f denote the survivor and density functions of Λ. Following Lancaster (1990),
or
. To see this, simply transform in the usual way from the conditional density of t to obtain the conditional density of Λ:
The last expression follows because
is a function only of Λ. Alternatively, note that, conditional on νi, the integrated hazard has a unit exponential distribution with location parameter νi.
3This remains true in the presence of time-dependent covariates, extending the applicability of our results.
as in equation (2.1). The density function of Λi is thus given by
.
Given this, a maximum likelihood approach could proceed by substituting nonparametric estimates of f (Λi) into equations (2.3) and (2.4). This neglects several technical difficulties. The issues involved are similar to the Klein and Spady (1993) treatment of the binary choice model. Let Fε denote the error term from a standard latent variable model. If Fε were known, the single observation log-likelihood function would be written as di log Fε(xi′ β) + (1 − di) ×log(1 − Fε(xi′ β)) where di is an indicator function and xi is a vector of covariates. The term Fε cannot be estimated directly, but if
one can proceed by replacing Fε(xi′ β) in the likelihood function by nonparametric estimates of
. In our case, although νi is unobservable, we can exploit the fact that
and use nonparametric estimates of f (Λ) in the likelihood function.
Rearranging (2.2) and (2.4), we can write
An interesting by-product of this result is that we can express the hazard function as
Without unobserved heterogeneity, the hazard function is defined as the ratio of the density of the duration variable to its survival function. The presence of f (Λ) in the denominator may seem odd until one notes, again, that in the absence of heterogeneity, the integrated hazard has an exponential distribution, in which case the survivor function and the density function are identical.
In essence, our estimator consists of replacing
in (2.3) by kernel estimates of the density function of Λi(β). However, there are several technical problems when one does this. First, the density of Λi is nonzero at the left edge of its support
, and the usual kernel estimates will be biased.
4A number of corrections for this “boundary” problem have been proposed in the statistics literature. Most of these (e.g., Gasser and Muller, 1979) are based on kernels with finite support and do not have the degree of smoothness or level of bias correction we require.
and multiply it by a polynomial chosen to reduce bias. Thus, we estimate the density of Λi using
where ιsT = (1 0 0 ··· 0), N is the sample size, γ = γ(N) ↓ 0 is the window width, and K* is the “inner” kernel or smoothing function and a superscript T indicates matrix transposition.
The term
can be nonpositive, leading to analytical and numerical difficulties. Like Klein and Spady (1993), we circumvent this problem by using
rather than
. Under the usual regularity conditions,
is, pointwise,
-consistent. Although many large-sample results exist in the literature, we are unable to find some that are usefully applied to our problem given the dependence of
and its derivatives on β. As a result, Appendix C derives a number of results that are applicable in our context.
5In the case of censoring it is necessary to estimate f by some other technique, for example, using some sort of smoothed Kaplan–Meier estimator. Although it is intuitive that the resulting estimator, with the corresponding modification to the likelihood function, will be consistent, we have found that this introduces many technical difficulties and is beyond the scope of the current paper.
A further issue is that f can get arbitrarily small so that the corresponding likelihood contribution need to be downweighted. Also, some adjustment must be made to the nonparametric estimates themselves to keep the score function bounded. We handle these problems using two smoothed trimming functions as in Klein and Spady (1993). The first downweights observations with small
; the second adds a small trimming factor to the denominator of the score function in this situation. See Klein and Spady (1993) for a more detailed explanation. A serious technical difficulty (which Klein and Spady and several authors with similar estimation problems do not have) is that our estimand can get arbitrarily close to zero. (In all cases f → 0 as t → ∞.) Klein and Spady and others suppose that the support of their functions is such that their estimands are strictly positive. The strict positivity assumption is not plausible in our case.
We presume that a
-consistent estimate
is used for trimming. Many duration models lend themselves to
-consistent estimation. For example, a mixed proportional hazard (MPH) model with Λi(α,δ) = Φ(ti;α)exp(−xiδ) can be written as log Φ(ti;α) = xiδ + ui where Φ is the integrated baseline hazard and ui is a residual independent of xi (Kiefer, 1988). Method of moments can consistently estimate (α,δ) using functions of the xi's as instruments. Similarly, models of the type in (2.1) can be written as log Λ(ti,xi;β) = ui where ui is a residual independent of xi. As before, we can specify a vector of instruments using functions of the xi's. Finally, a
-consistent estimator of the vector of parameters β can be obtained based on the conditional moment restriction that
with
.
6Clearly, certain models may require additional identification assumptions.
Given this, the proposed estimator is written in two-step form:
The parameters ε and ε′ are such that 0 < ε < ε′ and satisfy some technical conditions spelled out in the following section. The term
is defined to keep the denominator of
positive and bounded away from zero. In fact (see Remark A3 in Appendix A)
.
Matrix
could be replaced by any other weighting matrix that is consistent for (minus one times) the information matrix. Matrix
is a natural weighting matrix because we examine the properties of
in the demonstration of normality.
7The previous version of this paper defined
as the minimizer of a semiparametric likelihood function, showing consistency and asymptotic normality. In that version we also used an
-consistent estimator to define the trimming functions. As the co-editor and a referee pointed out, given this reliance on a preliminary
-consistent estimator, we might as well just examine a two-step version of the approach.
We let
denote an open ball centered on the true value of β, β0, with radius η for some η > 0. Regularity assumptions about functions of β are made to hold uniformly on
. Note that, for large enough N,
with probability approaching one. To avoid clutter, we leave this as understood in the derivations to follow.
3. ASYMPTOTIC RESULTS
In this section we establish the asymptotic normality of
.
8We denote the kth-order derivative of a function of Λ by a superscript (k). For a function q(β), [dtri ]q(β) = ∂q(β)/∂β, and [dtri ]2q(β) = ∂2q(β)/∂β∂βT. For a matrix A, we define ∥A∥ = (Vec[A]′Vec[A])1/2.
As discussed subsequently,
is the “efficient score” for the class of models in this paper; Σ(β0) is the semiparametric efficiency bound; 0 ≤ C < ∞ denotes a generic constant; Bi ≥ 0 refers to a generic dominating function. A random vector, qi(β), is Bi-dominated if, for all
, ∥qi(β)∥ ≤ Bi.
The following assumptions are used in this paper.
Assumption P.
(1) The observations, {zi}i=1N, zi = (ti,xi), are independent and identically distributed (i.i.d.); λi(β) and Λi(β) are, respectively, twice and four times continuously differentiable with respect to β, at all points
; Σ(β) is continuous in
and nonsingular at β0; log λi and log fi are Lipchitz-continuous with respect to β.
(2) For j = 0,1,2, the following functions (evaluated at β = β0) vanish at 0 and ∞:
,
,
.
(3) The following functions (evaluated at β = β0) have bounded derivatives with respect to Λi on
to order s + 2: fi,
,
,
.
(4) The following are Bi-dominated: ∥[dtri ](j)Λi(β)∥, j = 0,1,2,3,4, |log λi|,
, fi, |log fi|, and ∥[dtri ](j)Λi /Λi∥, j = 1,2,3.
(5)
.
Assumption P(1) provides standard regularity conditions. A couple of remarks should be made with respect to identification. First, the identification of duration models with unobserved heterogeneity has received a large amount of attention. Elbers and Ridder (1982) and others have shown the identification of the MPH model under various regularity conditions such as
. Ridder (1990) provides more general results on identification of generalized accelerated failure-time models. Neither of these papers uses any parametric restrictions. Because these results apply to a larger class of models than our own and we assume that an
-consistent estimator exists we leave identification as implicit. Note that the density function of Λi is also the first derivative of the moment generating function of the unobserved heterogeneity. Because
, this can be imposed by normalizing the estimates of f (0) to one. Incorporating the equivalent constraint into the estimator becomes tedious analytically. An alternative is to impose a normalization on the integrated hazard. For example, for the proportional hazard case with Λ(t,x;β) = Φ(t;α)exp(x′δ), it is often possible and sufficient to set Φ(1;α) = 1. With our identification condition we assume implicitly that this is done.
Assumption P(2) provides smoothness restrictions for bias reduction. Assumptions P(3) and (4) provide regularity conditions to simplify the asymptotic results. Assumption P(3) can be verified for most models using a Taylor expansion. It is used in the consistency proofs of Appendix C to put bounds on derivatives of kernel functions. Assumption P(5) has been discussed in Section 2.
Assumption K.
(1)
and ∫K*(w) dw = 1, and K* is continuously differentiable to order 2.
(2) K*(w)|w|q < C, |K*(1)(w)||w|q < C, |K*(2)(w)||w|q < C, for all q ≤ 2s.
(3) For all w ≥ 0: ∥μ(w)∥ ≤ C, ∥wμ(w)∥ ≤ C, ∥wjμ(j)(w)∥ ≤ C, j = 1,2,3, and ∥μ(1)(w)∥ ≤ C.
(4) The window width satisfies (a)
and (b) 1/(Nγ3+ε′) → 0.
(5) ∫|w|qK*(w) dw < C, for all q ≤ 2s, ∫|PK*(w)|2 dw < C, ∫|PK*(w)(1)|2 dw < C.
Assumption K(1) implies that K*, the “inner kernel,” is a probability density on the real line. It allows one to easily verify that its partial moments exist. The restrictions in Assumption K(2) bound K* and polynomial multiples of it. The restrictions on the tails of K* are easily satisfied by normal kernels or any other density functions that decay at a suitably high geometric rate. Assumption K(3) is simple to verify when K* is a probability density and will follow if the sth-order partial moment matrices of K* are nonsingular. This will follow for the probability densities typically used in kernel estimation. The conditions in Assumption K(4) are standard for semiparametric estimation. The
term in Assumption K(4)(a) is the order of bias for the score estimate. The term (Nγ3)−1 is the order of the variance of density derivative estimators. As is shown in Remark A3 (Appendix A), γε′/5 represents a lower bound for the density estimator and hence the presence of the γ−ε′ term in Assumptions K(4)(a) and (b). The term ε′ can be made arbitrarily small. Choosing γ = CN−1/q′, the conditions will be satisfied if q′ is slightly greater than 3 and s, the order of the kernel, is greater than 3/2 + 1, so that a third- or higher order kernel should be used.
THEOREM 1. Let Assumptions P and K hold. Then,
and
in probability.
The proof of this theorem is provided in Appendix A.
From Lemma A3 in Appendix A (minus one times) the derivative of the score function will be consistent for Σ. An inspection of the proof of Lemma A3 will show that various other approximations can also be used such as
Under our assumptions this will be consistent for Σ(β0). The term
can be calculated using the estimated score functions. It follows that likelihood ratio (LR), Lagrange multiplier (LM), and Wald tests and a Hausman–Wu test for correct specification of Fν can be conducted using
(and/or restricted versions thereof).
4. EFFICIENCY ISSUES
The results of Section 3 rely implicitly on the usual identification condition that the expected log-likelihood function has a unique maximizer
9Elbers and Ridder (1982) and Heckman and Singer (1985) show that, apart from normalization restrictions, the presence of observable covariates is generally sufficient to identify mixed duration models.
See Horowitz (1996, pp. 148–150) for further discussion of this topic. Hahn (1994) also showed that the information matrix of the multispell MPH model is nonsingular even in the Weibull case.
-estimability of the Weibull model may be the rule, rather than the exception. It turns out that the issue is not that severe.
First, by fixing one of the elements of δ it is possible to estimate (α,δT)T at the usual
rate. This is discussed by Hahn and is done implicitly in the Monte Carlo experiments in Horowitz (1996). Furthermore, for choices of θ(x) other than exp(−xTδ), Σ(β0) often will be nonsingular. Finally, for the MPH family with θ(x) = e−xTδ Ridder and Woutersen (2003) show that Σ(β0) is singular only for those models with integrated baseline hazard of the form Λ(t) = h(t)α and for which h(0) is equal to zero or infinity. In the Weibull model the singularity of Σ(β0) can be avoided by considering h(t) = α(t + ε)(α−1) for ε > 0. Other examples of MPH models with a nonsingular Σ(β0) include the log-logistic specification (Trussel and Richards, 1985; Gritz, 1993), polynomial specifications (Kennan, 1985; Ham and Rea, 1987; Gurmu, Rilstone, and Stern, 1995), and the piecewise-constant hazard of Meyer (1990) and Han and Hausman (1990). In conclusion, there is a large class of non-Weibull, and nonstandard Weibull models that can be estimated at the
rate using our method.
11An earlier version of this paper included numerical results indicating that the members of the family of Box–Cox hazards (Heckman and Singer, 1985; Lancaster, 1990), including hazard specifications such as the Weibull, log-logistic, and polynomial, have nonsingular Σ(β0) except in the Weibull case.
The Hahn (1994) efficiency bound results are derived for separable MPH models with time-invariant covariates and coefficients. The use of such nonseparable models in numerous papers such as Meyer (1990), Follmann, Goldberg, and May (1990), Katz and Meyer (1990), Jurajda (2002), and Canals-Cerdá (2003) underscores their importance. As a consequence, we present an intuitive way to derive the bounds of models of the form (2.1) without the separability requirement used in Hahn (1994).
Following Newey (1990), conjecture the true model as belonging to a family of submodels described by a parameter of interest, β, and a nuisance parameter, η, that describes the distribution of Fν. The semiparametric efficiency bound is defined as the supremum of the Cramler–Rao lower bounds for all possible submodels. For a given submodel, the log-likelihood function is written
and the score function for this submodel as
Suppose that f (Λ) and
can be approximated using a basis series {ρj} such that
where {ρj} is normalized such that
For simplicity, assume that β is a scalar. As a consequence of these definitions the γj are such that
Consider a sequence of likelihood functions for these models for different M. Putting
we can write down the corresponding log-likelihood function as
The score function for this model is then
and the information matrix is
The Cramler–Rao lower bound for β in such a model is simply
This is increasing in M. In fact, from the properties of the γ's we have
Here
corresponds to the asymptotic variance of
, the estimator in this paper, and
is therefore first-order efficient. Furthermore, in the absence of covariates, Sβi is simply a function of the duration variable and, because Λi is a monotonic transformation of the duration variable, we have
. In this case the efficiency bound explodes. This is interesting because covariates are often used to demonstrate the identification of models with unobserved heterogeneity. Although the presence of observed covariates is not always a necessary condition for identification, the efficiency bounds indicate a key role for observed covariates to obtain an
-consistent estimator.
5. CONCLUSION
This paper has introduced, to our knowledge, the first
-consistent and efficient semiparametric maximum likelihood estimator for a class of duration models with unobserved heterogeneity. There are several directions for extending this research. First, we wish to allow for various censoring mechanisms. Second, we wish to generalize the results to competing risks and other multivariate models. Third, we wish to refine the results for more extensive specification testing and diagnostic checking.
APPENDIX A
Throughout the Appendixes, Assumptions P and K are assumed to hold. A subscript q on a vector (matrix) denotes its qth (lqth) (arbitrary) element. The expectations operator, conditional on zi, is indicated by
. We first note several results that will be useful in the derivations to follow.
Remark A1. Λi and the covariates are statistically independent because, from equation (2.4), f (Λi(β)|xi) = f (Λi(β)).
Remark A2. By the chain rule of calculus,
so that c(ti,xi)(1) = (1/λi)(d/dti)c(ti,xi). In particular, ([dtri ]Λi)(1) = (1/λi)[dtri ]λi and ([dtri ]2Λi)(1) = (1/λi)[dtri ]2λi.
Remark A3. Put D = f + 2γε′/5(1 − χ2(f)) and note that, if f < γε′/5, then 1 − χ2(f) > ½ and D = D1[f ≥ γε′/5] + D1[f < γε′/5] ≥ f 1[f ≥ γε′/5] + 2γε′/5(1 − χ2(f))1[f < γε′/5] ≥ γε′/5.
Remark A4. For any Δ < ∞, χ1(f)(1 − χ2(f)) = O(N−Δ) uniformly in f ≥ 0. To see this note that, for large enough N, γε/5/2 > 3γε′/5/2 and
for any finite Δ.
Remark A5. From the definition of
,
, and the positivity of f, we have
and because
it follows that
, uniformly on z,β. Similarly
, and because
,
Remark A6. For any
, any Δ < ∞,
. To see this, note that
and χ1(0)N−Δ → 0 for any Δ < ∞.
Remark A7. Note that (1 − χ1(f))χ1(f) and (1 − χ2(f))χ2(f) are simply logistic density functions and
Remark A8. Put
. From the mean value theorem
where f is a mean value. From Remark A5 and the triangle inequality
Remark A9. From the definition of
and the dominating conditions it follows that
Put
Here N1i(β) and N2i(β) are, to first order, the expected values, conditional on zi, of
and
.
For the following two lemmas, all functions are evaluated at β = β0. Put Di = fi + 2γε′/5(1 − χ2(fi)),
,
,
.
LEMMA A1.
Proof. We can decompose
Let
From Corollary C4 in Appendix C and the dominating conditions,
The qth (arbitrary) element of the first term on the right-hand side of (A.13) has quadratic mean
From Remarks A4 and A8 and because 0 ≤ χ1 ≤ 1,
Using Corollary C4 and because the variance of
is bounded by ((Nγ)−1)C
Now, with the exception of
and
, each of the terms in B is a function only of Λ1,…, ΛN. Thus
Therefore,
. From Corollary C4,
and
. The variances of
and
are of the usual orders O(1/Nγ) and O(1/Nγ3). Consequently, using Remark A8 and the assumptions on the window width, D2N is oP(N−1/2). It is trivial that
Applying Lemma B1 in Appendix B to D4N yields the stated result. █
LEMMA A2.
Proof. From Lemma A1 we have
Note that
, Qi = χ1i fi /Di, and 0 < Qi ↑ 1 uniformly on fi. From random sampling and dominated convergence
█
LEMMA A3. Let
denote the Hessian matrix
, evaluated at (possibly different) points βlq, l,q = 1,…, k. Suppose ∥βlq − β0∥ = OP(N−1/2) for all l,q. Then,
.
Proof. We write the l,qth (arbitrary) element of
as
. Using a second-order Taylor series we have
where
is between βlq and β0. We will first show that the second and third terms in this Taylor series are oP(1). We will then derive the probability limit of the first term. Noting that the Hessian involves differentiating λi,
, and the smoothing functions it is convenient to decompose the summands in
where
Note that the first two terms in the Taylor series in equation (A.21) are evaluated at β = β0. Consider first
. The rth element of
can be written as
and
By Cauchy–Schwarz
Therefore
and
. Similarly, using Proposition C7 in Appendix C, we have
, j = 2,3,4. Regarding the third term in the Taylor series, the term
is a function of the first four derivatives of Λi(β), Λj(β), and Kij. The former are Bi-dominated; the latter are bounded by assumption. It is tedious but straightforward using Remark A9 to show that
so that, because ∥βlq − β0∥ = OP(N−1/2),
. To derive the probability limit of
it is useful to first decompose
By applying Cauchy–Schwarz and Propositions C5 and C6 in Appendix C, it follows that each of the last three terms on the right-hand side has an average that is oP(1). Further, because
where fi is a mean value, it follows that
with
. By Remark A9, the dominating conditions, and the fact that fi ≤ Di,
By Cauchy–Schwarz and because 0 ≤ χ1′ ≤ 1,
. Note that χ1′ is simply the logistic density function. Let μ(f) denote the density function of the random variable f. Using the change of variables, ω = (f − γε/5)/γε/4, we have
We therefore see that the right-hand side of (A.30) is bounded by γε/5γε/8C. We thus see that the expected value of |Ai| is bounded by γε/5γε/8γ−ε/4C = O(γ3ε/40) = o(1). Therefore, the average of the Ai's converges to zero in probability, and hence
. Similarly,
,
, and
so that
where
These terms can be bounded simply so that
Therefore, by the law of large numbers and Remark B1 in Appendix B we have that
█
Proof of Theorem 1. Using standard substitutions,
where
is the Jacobian of
with arguments evaluated at (possibly different) points between
and β0. Rearranging this we have
It is sufficient to show that (a)
and (b)
and
.
(a) follows from Lemmas A1 and A2 and from Lemma B2 in Appendix B. (b) follows from Lemma A3. █
APPENDIX B
LEMMA B1.
.
Proof. It is sufficient to show that
. From the i.i.d. sampling assumption
Put
so that
where
, j ≠ i. From the i.i.d. sampling assumption
Next,
It is straightforward that
and
.
Put
It is useful to decompose
whereby
Were a standard, symmetric kernel being used, a31* would have mean zero, conditional on z3. With the boundary modifications, slightly more delicate arguments are required to show that this is o(γ). We have
Using integration by parts and Remark A2
Because Q ↑ 1, note that, for any Λ3, ∫(1/γ)K13Q(Λ1) dΛ1 → 1 and
Therefore,
. Substituting (B.7), (B.8), and (B.16) into (B.6) we see that B = o(N − 1). Further substituting (B.5) and (B.17) into (B.1) yields the desired result. █
The following lemma shows that using a nonparametric estimate of the density of Λ in the likelihood trimming function has no effect on the asymptotic distribution of the score function.
LEMMA B2.
Proof. Second-order Taylor series expansions of
around the points fi, i = 1,…, N, yields
where
and where fi, i = 1,…, N, are mean values and we note that |φi| ≤ 1. The proof consists of showing that
,
, and
.
Because
and
are functions of Λi,i = 1,…, N, by iterated expectations and random sampling,
Noting that
and
it follows that
. The qth (arbitrary) element of bi is written
Now, because χ1i(1 − χ1i) ≤ 1, Di ≥ γε′/5, and
Let fi denote a mean value.
so
Applying the Cauchy–Schwarz inequality to the first term
We see then that
. Because
converges in mean absolute value to zero,
. Next,
and
so that
and the result holds. █
Put
Remark B1.
. By differentiation, integration by parts, and Remark A2
Noting that we can write
allows us to confirm the result.
APPENDIX C
For a function c(zi,β), put
, cf (Λi(β)) = c(Λi(β)) f (Λi(β)) and
. In this Appendix all functions are evaluated at β = β0.
Lemma C1. Suppose that cf (Λi) is differentiable to order s with respect to Λi on
and these derivatives are bounded. Then for some finite
.
Proof. The density function of Λj is f (Λj), and using an sth-order Taylor series we can write
where
and Λi = Λi − κwγ κ ∈ (0,1) is a mean value. The result follows from the moment conditions on K* and the boundedness of cf(s). █
Lemma C2. Suppose that cf (Λi) is differentiable with respect to Λi on
to order s + 1, these derivatives are bounded, and cf (0) = cf (∞) = 0. Then, for some finite C ≥ 0: (a)
and (b)
.
Proof. To prove (a), using the proof of Lemma C1 we have
To prove (b), note that cf (∞) = 0, so that using integration by parts we have
. The rest of the steps are as in the proof of Lemma C1. █
Lemma C3. Suppose that cf (Λi) is differentiable with respect to Λi on
to order s + 2, these derivatives are bounded, cf (0) = cf (∞) = 0, and cf(1)(0) = cf(1)(∞) = 0. Then, for some finite C ≥ 0: (a)
, (b)
, and (c)
.
Proof. Follow the same steps as used in the proof of Lemma C2 except that, in proving (b), integration by parts is used twice. █
Corollary C4. For some finite C ≥ 0: (a)
, (b)
, and (c)
.
Proof. Part (a) follows directly from Lemma C1. To prove (b) note that
and apply Lemma C2 and the dominating conditions on [dtri ]Λi. To prove (c) note that
and apply Lemma C3 and the dominating conditions on [dtri ]Λi and [dtri ]2Λi. █
Proposition C5.
.
Proof. Using iterated expectations and the random sampling assumption
█
Proposition C6.
.
Proof. Without loss of generality we consider the qth (arbitrary) element of
. As in Proposition C5
The first term on the right-hand side can be bounded as follows:
where the last inequality follows from the dominating conditions and a standard change of variables. Inserting this last expression into the previous equation gives the desired result. █
Proposition C7. Let
denote an arbitrary third-order derivative of
. Then
.
Proof. By random sampling
, say, where μi = μ(Λi /γ) and ω = (Λi − Λj)/γ.
By inspection,
. Using integration by parts as in the proof of Lemma C2,
, r = 1,2,3. The Ar functions are simply products of the derivatives of Λi and conditional expectations of these derivatives. The former factor out of the preceding integral. The latter, multiplied by f (Λi) and its derivatives, are bounded, and
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