A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model

Technical Details Section 2: A Counting Process Approach

The counting process approach is a very useful framework for analyzing duration data since an indicator can be used to denote whether a transition happened or not. Andersen et al. (1993) have provided an excellent survey of counting processes. Less technical surveys have been given by Klein and Moeschberger (1997), Therneau and Grambsch (2000), and Aalen et al. (2009). The main advantage of this framework is that it allows us to express the duration distribution as a regression model with an error term that is a martingale difference. Regression models with martingale difference errors are the basis for inference in time series models with dependent observations. Hence, it is not surprising that inference is much simplified by using a similar representation in duration models.

To start the discussion, we first introduce some notation. A counting process {N(t)|t≥0} is a stochastic process describing the number of events in the interval [0, t] as time proceeds. The process contains only jumps of size +1. For single duration data, the event can only occur once because the units are observed until the event occurs. Therefore we introduce the observation indicator Y(t)=I(T≥t) that is equal to one if the unit is under observation at time t and zero after the event has occurred. The counting process is governed by its random intensity process, Y(t)κ(t), where κ(t) is the hazard in (2). If we consider a small interval (t–dt] of length dt, then Y(t)κ(t) is the conditional probability that the increment dN(t)=N(t)–N(t–) jumps in that interval given all that has happened until just before t. By specifying the intensity as the product of this observation indicator and the hazard rate, we effectively limit the number of occurrences of the event to one. It is essential that the observation indicator only depends on events up to time t.

Usually we do not observe T directly. Instead we observe

The sample paths of the conditioning variables should be up to t–, but because these paths are left continuous we can take them up to t. A fundamental result in the theory of counting processes, the Doob-Meyer decomposition,⁹ allows us to write

where M(t), t≥0 is a martingale with conditional mean and variance given by

The (conditional) mean and variance of the counting process are equal, so the disturbances in (B.2) are heteroscedastic. The probability in (B.1) is zero, if the unit is no longer under observation. A counting process can be considered as a sequence of Bernoulli experiments because if dt is small, (B.3) and (6) give the mean and variance of a Bernoulli random variable. The relation between the counting process and the sequence of Bernoulli experiments given in (B.2) can be considered as a regression model with an additive error that is a martingale difference. This equation resembles a time-series regression model. The Doob-Meyer decomposition is very helpful to the derivation of the distribution of the estimators because the asymptotic behavior of partial sums of martingales is well-known.

Technical Details Section 3: Assumptions 1–4

To simplify the expressions, we use the notation h_i(t, θ)= h_i(t, X_h,I (t), θ).

1. The conditional distribution of T given X(‧) and V has hazard rate

   

   with X(‧) a K covariate bounded stochastic process that is independent of V and such that if the probability of the event

   

   some set S with positive measure and for some constants c₁, c₂, then c₁=c₂=0. For the baseline hazard, 0<lim_t↓₀λ(t, α₀)<∞.

2. For the covariate process X(t), t≥0, we assume that the sample paths are piecewise constant, i.e., its derivative with respect to t is 0 almost everywhere, and left continuous. The hazard that is not conditional on V is

   

   The observation process is Y(t), t≥0 with Y(t)=I9(t≤T)I(t≤C) and we assume

   

   The support of C is bounded.

3. The parameter vector θ=(β′, α′)′ is an M vector with β a K vector and α an L vector. The parameter space Θ is convex. The baseline hazard λ(t, α)>0 and is twice differentiable and the second derivative is bounded in α (in the parameter space) and t.

4. The weight function

   

   is an M vector of bounded and left continuous functions. If

   

   then there are functions μ(u, θ) (an M vector), V_β (u, s, θ) (an M×K matrix), and V_α (u, s, θ) (an M×L matrix) such that

   

   and

   

   and

   

   Define

   

We assume that the M×M matrix [B(θ₀) A(θ₀)] is nonsingular.

The restriction on the baseline hazard in Assumption A1 ensures identification (see Section 3) and guarantees that the semiparametric information bound is nonsingular (see below). Assumption A2 states that the covariates and the observation indicator are predetermined. Assumption A4 is about smoothness: Suppose that one censors all the data at u=τ+ψ then the expressions in equation (30) and (31) do not change if the value of ψ varies. The derivation of the asymptotic distribution of the LR estimator follows the proof in Tsiatis (1990). Tsiatis requires that the density of U₀ is bounded. For the MPH model, this density is

If E(V)=∞, this density is not bounded at u₀=0. Inspection of Tsiatis’ proof shows that this does not change the result, and we do not need to impose the restriction that E(V) is finite. The transformed durations are observed up to τ with τ<∞ such that for some ψ,η>0

Pr[min (U₀, C) > τ+ψ]≥η.

In the MPH model, this is just an assumption on the distribution of C because for U₀ it is satisfied for all τ<∞.

Technical Details Section 4: Lemma 2–3

Lemma 2: If the derivative of κ is bounded on [0, τ] then for ε>0 with

and

we have

for u₁, u₂ with 0<u₁<u₂<τ.

If Y_h,N(t) is bounded away from zero on [0, τ] for large N, then (B.14) and (B.15) imply that if b_N=N^–c for

We do not observe the transformed duration

Lemma 3: The kernel K is positive and bounded on [–1, 1] (and zero elsewhere) and satisfies a Lipschitz condition on this interval. The covariate process X(t) is bounded on [0, τ] and so is

uniformly for 0≤u≤τ, θ∈N(θ₀) and H has derivatives that are bounded for 0≤u≤τ, θ∈N(θ₀). Then for ε>0 such that

we have

Proof: See below.

Note that the conditions on b_N are determined in Lemma 2 and that a bandwidth proportional to N^–1/5 and

At this point we consider the condition in (B.18) more closely. With

Y^U (u, θ)=I(h(T, θ)≥u)‧I(h(C, θ)≥u).

If the censoring time and the duration are conditionally independent given the history up to t, i.e.,

then

If N(θ₀) is an open neighborhood of θ₀, X_i and C_i are i.i.d., and

then

and by the uniform law of large numbers

uniformly for θ∈N(θ₀) and 0≤u≤τ. Because by (B.23) the limit is bounded away from zero, we have

uniformly for θ∈N(θ₀) and 0≤u≤τ with

Because h(T,θ₀)=U₀, (B.19) holds for θ=θ₀ if κ₀(u) is bounded for 0≤u≤τ. From the expression for κ_U (u, θ) in (9), a sufficient condition for κ_U(u, θ) to be bounded for all θ in a neighborhood of θ₀ and 0≤t≤τ is that λ(t, α)>0 for all t and on a neighborhood of α₀. In the same way, (B.20) holds if the hazard of C is bounded and λ(t, α) is bounded away from zero in a neighborhood around α₀.

Proof of Lemma 1

This relates the hazard of the distribution of U(θ) to that of U₀

Because h(h^–1(u, θ), θ)=u, we have

The derivatives of κ_U(u, θ) with respect to θ are

where the last equality follows from a change of variables in the integral. In the same way, we obtain with a change of variable in the integral

The proof consists of checking the conditions for asymptotic linearity of S_N(θ) in Tsiatis (1990) and a computation of the coefficients in the linear approximation. In Tsiatis’ proof the covariate in the estimating equation is X_i. We have

The assumptions that λ(t,α) is bounded away from zero for all t≥0 and α in the parameter space, that

Next we linearize S_N(θ). Because

we have if |θ–θ₀| is small

The second term is after substitution of (B.29), and (B.30)

The normalized vectors of coefficients converge to (B.12) and (B.13) if (B.10) and (11) hold. This proves the lemma.

Proof of Theorem 1

By van der Vaart (1998) Theorem 5.45, we have from Lemma 1

with M₀ the martingale associated with the counting process N₀ for U₀. By the central limit theorem for integrals of predetermined functions with respect to a martingale, [see e.g., Anderson et al. (1993)], the sum on the right-hand side converges to a normal distribution with the variance matrix in (24).

Proof of Lemma 2 and 3

We have

We first consider the second term. Because K is Lipschitz this is bounded by

Moreover by the mean value theorem, we have that for some intermediate

Because X_i(t) is bounded on [0, τ] and so is

Because the estimator

Next we consider the first term in (B.34). By subtraction and addition of expected values, this term is bounded by

The first and second terms converge to 0 in probability if

This expression is bounded (both H and K are bounded) by

The first term goes to 0 in probability if