Efficient estimation for semiparametric semi-markov processes

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1 Efficient estimation for semiparametric semi-markov processes Priscilla E. Greenwood Arizona State University Ursula U. Müller Universität Bremen Wolfgang Wefelmeyer Universität Siegen Abstract We consider semiparametric models of semi-markov processes with arbitrary state space. Assuming that the process is geometrically ergodic, we characterize efficient estimators, in the sense of Hájek and Le Cam, for arbitrary real-valued smooth functionals of the distribution of the embedded Markov renewal process. We construct efficient estimators of the parameter and of linear functionals of the distribution. In particular we treat the two cases in which we have a parametric model for the transition distribution of the embedded Markov chain and an arbitrary conditional distribution of the inter-jump times, and vice versa. Introduction Suppose we observe a semi-markov process Z t, t 0, with embedded Markov renewal process (X 0, T 0 ), (X, T ),..., on a time interval 0 t n. The transition distribution of the Markov renewal process factors as D(x, dy, ds) = Q(x, dy)r(x, y, ds), where Q(x, dy) is the transition distribution of the embedded Markov chain X 0, X,..., and R(x, y, ds) is the conditional distribution of the inter-jump times S j = T j T j given X j = x and X j = y. We assume that the embedded Markov chain is geometrically ergodic. We write P (dx, dy, ds) for the joint stationary law of (X j, X j, S j ), and P (dx) and P 2 (dx, dy) for its marginals. We are interested in estimation of functionals of Q and R. Our results hold also for observations (X 0, T 0 ),..., (X n, T n ) of the embedded Markov renewal process. For discrete state space and the fully parametric or nonparametric

2 cases, the asymptotic distribution of maximum likelihood estimators, Bayes estimators, and empirical estimators has been studied by Taga [3], Pyke and Schaufele [27], Hatori [3], McLean and Neuts [20], Moore and Pyke [22], Ouhbi and Limnios [23, 24] and, with censoring, by Lagakos, Sommer and Zelen [8], Gill [5], Voelkel and Crowley [32] and Phelan [25, 26]. We focus primarily on semiparametric models and on the construction of efficient estimators. The simplest semiparametric models are obtained by specifying a parametric form for one of the factors of D(x, dy, ds) = Q(x, dy)r(x, y, ds). In one case, we assume a parametric model Q for the transition distribution of the embedded Markov chain and leave the conditional distribution of the inter-jump times unspecified (model Q). In a second case, we assume a parametric model R for the conditional distribution of the inter-jump times and leave the transition distribution of the embedded Markov chain unspecified (model R). The estimating problems connected with these two models are specific to the semi- Markov setting; in particular, they have no non-trivial counterpart for Markov chains. To keep the paper readable and short, we concentrate on the two simple models above. More general models, involving possibly infinite-dimensional parameters, perhaps on both factors simultaneously, could be treated along the same lines. We want to estimate and linear functionals of the form Ef(X j, X j, S j ) = P (dx)q(x, dy) R(x, y, ds)f(x, y, s) = P QRf, with Q = Q or R = R parametric. Interesting applications are estimation of probabilities P (X j A, X j B, S j c), P (X j A, X j B) and P (X j A), and of ratios P (S j c X j A, X j B) and P (X j B X j A). We can also treat expectations ES j and conditional expectations E(S j X j A, X j B) and E(X j X j A). Natural estimators for are the maximum likelihood estimators based on the conditional distributions Q or R. We show that they are efficient in our two models. In particular, they are adaptive in the sense that knowing the nonparametric factor of Q(x, dy)r(x, y, ds) cannot give estimators with smaller asymptotic variance. A natural estimator for a linear functional Ef(X j, X j, S j ) is the empirical estimator f(x j, X j, S j ), 2

3 where = max{j : T j n}. Greenwood and Wefelmeyer [8] have shown that this estimator is efficient in the fully nonparametric semi-markov model; see also Greenwood and Wefelmeyer [7] for Markov step processes. We construct better, efficient, estimators for our two semiparametric models Q and R. For our first model, the functional Ef(X j, X j, S j ) can be written Ef(X j, X j, S j ) = P Q Rf = P (dx)q (x, dy)r xy f with R xy f = R(x, y, ds)f(x, y, s). To exploit the structure of the model, we use a plugin estimator, i.e. we replace the conditional expectation Rf by a kernel estimator ˆRf. By what we refer to as the plug-in principle, we expect that P Q ˆRf will converge at the parametric rate n /2 under appropriate conditions on the kernel and the bandwidth, even though the kernel estimator has a slower rate of convergence. In a second step, we replace the parameter by an estimator ˆ. This results in the estimator P ˆQ ˆ ˆRf for Ef(X j, X j, S j ). It is efficient if an efficient estimator ˆ is used for. Related plug-in estimators have been used in other, mainly nonparametric, contexts before. For quadratic functionals of densities with i.i.d. observations see Hall and Marron [2], Bickel and Ritov [2], Eggermont and LaRiccia [4] and the references there. Similar results exist for regression models; see e.g. Goldstein and Messer [6] and Efromovich and Samarov [3]. In semiparametric time series models with independent innovations, the stationary density can be written as a smooth functional of the innovation density and the parameters; n /2 -consistent and efficient plug-in estimators are constructed in Saavedra and Cao [28] and Schick and Wefelmeyer [29, 30]. For our second model, the functional Ef(X j, X j, S j ) can be written Ef(X j, X j, S j ) = P 2 R f = P 2 (dx, dy) R (x, y, ds)f(x, y, s). Here we can estimate the nonparametric part P 2 by the empirical distribution based on the embedded Markov chain. Again we replace by an estimator ˆ. We show that the resulting estimator N n R ˆ(X j, X j, ds)f(x j, X j, s) is efficient if ˆ is efficient. The paper is organized as follows. In Section 2 we state local asymptotic normality for arbitrary semi-markov models and characterize efficient estimators for smooth 3

4 functionals on such models. In Section 3 we construct efficient estimators of and Ef(X j, X j, S j ) for model Q, and in Section 4 for model R. Throughout the paper, the discussion will be informal. 2 Characterization of efficient estimators In this section we consider general semi-markov models described by families of distributions Q(x, dy) and R(x, y, dz). To calculate asymptotic variance bounds and characterize efficient estimators, we fix Q and R and introduce a local model at (Q, R) by perturbing Q as Q nu (x, dy). = Q(x, dy)( + n /2 u(x, y)) and R as R nv (x, y, ds). = R(x, y, ds)( + n /2 v(x, y, s)). Since Q nu and R nv are again conditional distributions, the function u will vary in some subset U 0 of U = {u L 2 (P 2 ) : Q x u = 0}, and the function v will vary in some subset V 0 of V = {v L 2 (P ) : R xy v = 0}. Here Q x u = Q(x, dy)u(x, y) and R xy v = R(x, y, ds)v(x, y, s). Similarly, we will write D x v = D(x, dy, ds)v(x, y, s). The sets U 0 and V 0 are called the tangent spaces for Q and R. For simplicity we take them linear and closed. Note that U 0 and V 0 are orthogonal subspaces of L 2 (P ). The perturbations Q nu. = Q( + n /2 u) and R nv. = R( + n /2 v) are meant in the sense that Q nu and R nv are Hellinger differentiable with derivatives u and v. For appropriate versions in arbitrary Markov step models and in nonparametric semi-markov models see Höpfner, Jacod and Ladelli [4] and Greenwood and Wefelmeyer [8]. We assume that D(x, dy, {0}) = 0, that the mean inter-jump time m = ES j is finite, and that the embedded Markov chain is positive Harris recurrent. Then n m a.s. (2.) Furthermore, the following law of large numbers and martingale central limit theorem hold. For f L 2 (P ) we have f(x j, X j, S j ) P f a.s., (2.2) 4

5 and for w L 2 (P ) with D x w = 0 we have n /2 w(x j, X j, S j ) m /2 L, (2.3) where L is a normal random variable with mean zero and variance P w 2. M (n) uv Now write M (n) for the distribution of Z t, 0 t n, if Q and R are in effect, and if Q nu and R nv are. Similarly as in Höpfner, Jacod and Ladelli [4] and Greenwood and Wefelmeyer [8], and using orthogonality of U 0 and V 0, we obtain local asymptotic normality: For u U 0 and v V 0, where log dm (n) uv dm (n) = H n 2 σ2 (u, v) + o p (), (2.4) ( H n = n /2 u(xj, X j ) + v(x j, X j, S j ) ), σ 2 (u, v) = m (P 2 u 2 + P v 2 ), and H n is asymptotically normal with mean zero and variance σ 2. We want to estimate functionals of (Q, R). A real-valued functional ϕ(q, R) is said to be differentiable at (Q, R) with gradient (g, h) if g U, h V, and the functional has a linear approximation in terms of the inner product from the LAN-norm, n /2 (ϕ(q nu, R nv ) ϕ(q, R)) m (P 2 (ug) + P (vh)), u U 0, v V 0. (2.5) The projection (g 0, h 0 ) of (g, h) onto U 0 V 0 is called the canonical gradient of ϕ. An estimator ˆϕ is called regular for ϕ at (Q, R) with limit L if L is a random variable such that n /2 ( ˆϕ ϕ(q nu, R nv )) L under M (n) uv, u U 0, v V 0. (2.6) The convolution theorem of Hájek [] and Le Cam [9] says that L is distributed as the convolution of a normal random variable with mean zero and variance σ 2 (g 0, h 0 ) = m (P 2 g P h 2 0) with another random variable. This justifies calling ˆϕ efficient if it has this asymptotic variance. An estimator ˆϕ is called asymptotically linear with influence function (a, b) if a U, b V, and ( n /2 ( ˆϕ ϕ(q, R)) = n /2 a(xj, X j ) + b(x j, X j, S j ) ) + o p (). (2.7) 5

6 With these definitions, ˆϕ is regular and efficient if and only if it is asymptotically linear with influence function equal to the canonical gradient: ( n /2 ( ˆϕ ϕ(q, R)) = n /2 g0 (X j, X j ) + h 0 (X j, X j, S j ) ) + o p (). (2.8) A reference for this characterization in the i.i.d. case is in Bickel, Klaassen, Ritov and Wellner []; for semi-markov processes parametrized by D see Greenwood and Wefelmeyer [8]. We point out that the orthogonality of U 0 and V 0 implies that functionals of one of the factors of Q(x, dy)r(x, y, ds) can be estimated adaptively with respect to the other factor in the following sense. Suppose ϕ(q, R) depends only on Q. Then (2.5) holds with h = 0, and the canonical gradient is of the form (g 0, 0). Suppose now that ˆϕ is efficient in a model with R completely unspecified. Then it will remain efficient for any submodel for R, in particular when R is known. The same holds with interchanged roles of Q and R. We apply this observation to estimation of in models Q and R, Sections 3 and 4. We will also need a version of the central limit theorem (2.3) for functions that are not conditionally centered. Suppose that the embedded Markov chain is geometrically ergodic in the L 2 sense. For k L 2 (P 2 ) define (Ak)(x, y) = i=0 (Q i yk Q i+ x ). Set f 0 (x, y, s) = f(x, y, s) R xy f. Then n /2 ( f(xj, X j, S j ) P QRf ) ( = n /2 ARf(Xj, X j ) + f 0 (X j, X j, S j ) ) + o p (). (2.9) Note that Q x Ak = 0 for k L 2 (P 2 ). For Markov chains, the above martingale approximation goes back to Gordin [9] and Gordin and Lifšic [0]; see Meyn and Tweedie [2], Section 7.4. For semi-markov processes we refer to Greenwood and Wefelmeyer [8]. From (2.3) we obtain that the above standardized sum is asymptotically normal with variance m (P 2 (ARf) 2 + P f0 2 ). 6

7 To calculate gradients of linear functionals Ef(X j, X j, S j ), we need the following perturbation expansion due to Kartashov [5, 6, 7]: n /2 (P nu Q nu k P Qk) P 2 (kbu) = P 2 (uak), k L 2 (P 2 ), (2.0) where B is the adjoint of A. We will not need the explicit form of B. The perturbation expansion implies that ϕ(q, R) = P f = P QRf is differentiable for f L 2 (P ), n /2 (P nu Q nu R nv f P QRf) P 2 (uarf) + P (vf 0 ), u U 0, v V 0. (2.) Here we have used that U and V are orthogonal. For a proof of (2.) we refer to Greenwood and Wefelmeyer [8]. Note that there we do not factor D and have local parameters h(x, y, s) which here are written u(x, y) + v(x, y, s). 3 Model Q In this section we consider model Q, in which we have a parametric family Q for Q and leave R unspecified. For simplicity we assume that is one-dimensional. A natural estimator for is the maximum likelihood estimator based on Q. Suppose Q (x, dy) has density q (x, y) with respect to some dominating measure ν Q (x, dy), and that q has derivative q with respect to. Write λ = q /q for the score function. The maximum likelihood estimator ˆ solves the estimating equation λ (X j, X j ) = 0. A stochastic expansion of ˆ is now obtained by the usual arguments. First we recall two well-known relations for λ and λ, namely 0 = (ν Q q ) = ν Q q = Q λ, 0 = (Q λ ) = ν Q (λ q ) = ν Q (λ q + λ q ) = Q (λ 2 + λ ). Write P 2 = P Q and let I = P 2 λ 2 denote Fisher information. We obtain from the second relation that I = P 2 λ. From the law of large numbers (2.2) and (2.) we obtain by Taylor expansion that ˆ is asymptotically linear with influence function (mi λ, 0): n /2 ( ˆ ) = mi n /2 7 λ (X j, X j ) + o p (). (3.)

8 From the martingale central limit theorem (2.3) we conclude that n /2 ( ˆ ) is asymptotically normal with variance mi. To prove semiparametric efficiency of ˆ, we must interpret as a functional of (Q, R) through ϕ(q, R) =. The local model for Q is obtained by perturbing as na = + n /2. a and Q as Q na = Q ( + n /2 aλ ). Hence the tangent space U 0 for Q consists of all functions of the form aλ, a R. The canonical gradient (g 0, h 0 ) of is therefore of the form (a 0 λ, 0), where a 0 is determined from (2.5) by a = m P 2 (aλ a 0 λ ) = aa 0 m I, a R. This gives a 0 = mi and g 0 = mi λ. Since ˆ has influence function (mi λ, 0) by (3.), it is efficient by characterization (2.8). Note that ˆ is adaptive with respect to R in the sense that it remains efficient even if we know R. Now we consider estimation of a linear functional Ef(X j, X j, S j ) = P 2 Rf with f L 2 (P 2 R). A natural estimator is the empirical estimator f(x j, X j, S j ). We have ARf U and f 0 V. From (2.9) we obtain that the empirical estimator is asymptotically linear with influence function (marf(x, y), mf 0 (x, y, s)) and asymptotic variance m(p 2 (ARf) 2 + P 2 Rf 2 0 ). If nothing were known about Q, the empirical estimator would be efficient; see Greenwood and Wefelmeyer [8]. Since we have assumed a parametric model Q, we can construct better estimators exploiting the structure of the model. We assume that the state space is the real line, and that P has Lebesgue density p. Let p and q be the densities of P and Q. Then p 2 (x, y) = p (x)q (x, y) is the density of P 2. We write Rf = a/p 2 with a(x, y) = p(x, y, s)ds f(x, y, s) and estimate Rf by ˆRf = â/ˆp 2 with kernel estimators â(x, y) = ˆp 2 (x, y) = ( x b k Xj 2 b ( x b k Xj 2 b 8, y X ) j f(x, y, S j ), b, y X ) j, b

9 where k is a mean zero density and b = b n is a bandwidth that tends to zero at a rate to be determined later. Our estimator for P 2 Rf is P 2 ˆ ˆRf with ˆ a n /2 -consistent estimator of. We prove that it is asymptotically linear if f is differentiable. Under appropriate smoothness assumptions on p, a modified proof will cover discontinuous f, in particular indicator functions. To calculate the influence function of P 2 ˆ ˆRf, we write ˆRf = Rf + â a ˆp 2 ˆp 2 p 2 Rf. ˆp 2 Then our estimator is approximated as. P 2 ˆ ˆRf = P2 ˆRf + dxdy(â(x, y) a(x, y)) dxdy (ˆp 2 (x, y) p 2 (x, y))r xy f. Let b = n /4. Since the kernel k integrates to one and has mean zero, a change of variables u = (x X j )/b and v = (x X j )/b and a Taylor expansion give Similarly, dxdy â(x, y) = dxdy ˆp 2 (x, y)r xy f = dudv k(u, v)f(x j + bu, X j + bv, S j ) = f(x j, X j, S j ) + o p (n /2 ). (3.2) dudv k(u, v) R(X j + bu, X j + bv, ds)f(x j + bu, X j + bv, s) = R Xj,X N j f + o p (n /2 ). (3.3) n With the notation f 0 (x, y, s) = f(x, y, s) R xy f, these two expansions lead to P 2 ˆ ˆRf = P2 ˆRf + f 0 (X j, X j, S j ) + o p (n /2 ). (3.4) It remains to expand P 2 ˆRf. With Q na. = Q ( + n /2 aλ ) and the perturbation expansion (2.0) for u = aλ and a = n /2 ( ˆ ), a Taylor expansion gives P 2 ˆRf = P 2 Rf + P 2 (λ ARf)(ˆ ) + o p (n /2 ). (3.5) 9

10 Suppose now that ˆ is efficient. Then it has influence function (mi λ, 0) by (3.). Together with (3.4) and (3.5) we obtain that n /2 (P 2 ˆ ˆRf P2 Rf) ( = mn /2 I P 2(λ ARf)λ (X j, X j ) + f 0 (X j, X j, S j ) ) + o p (). Hence by (2.3) our estimator is asymptotically normal with variance m ( I (P 2(λ ARf)) 2 + P 2 Rf 2 0 ). Note that by the Cauchy Schwarz inequality, I (P 2(λ ARf)) 2 P 2 (ARf) 2. Since the empirical estimator has asymptotic variance m(p 2 (ARf) 2 + P 2 Rf 2 0 ), our estimator is better unless ARf is proportional to λ. Now we prove that our estimator P 2 ˆ ˆRf is efficient. By the characterization (2.8) of efficient estimators, we must show that the influence function of P 2 ˆ ˆRf equals the canonical gradient of the functional ϕ(q, R) = P 2 Rf. Let na = + n /2 a and R nv. = R( + n /2 v). Then Q na. = Q ( + n /2 aλ ), and the perturbation expansion (2.) implies n /2 (P 2na R nv f P 2 Rf) ap 2 (λ ARf) + P 2 R(vf 0 ), a R, v V. Since R is unspecified and hence the tangent space V 0 for R is V, the canonical gradient of P 2 Rf is of the form (a 0 λ, mf 0 ), where a 0 is determined from (2.5) by ap 2 (λ ARf) = aa 0 m I, a R. This gives a 0 = mi P 2(λ ARf). Hence P 2 ˆ ˆRf is efficient by characterization (2.8). We end this section with some comments. If we set f(x, y, s) = s, we obtain an efficient estimator for P 2 Rf = ES j = m, the mean inter-jump time. If the inter-jump time distribution does not depend on the states, then our estimator is asymptotically equivalent to the empirical estimator Nn S j. If the state space is discrete, we can replace ˆR = â/ˆp 2 by the simpler estimator 0

11 Rf = a/p 2 with a(x, y) = (X j = x, X j = y)f(x, y, S j ), p 2 (x, y) = (X j = x, X j = y). The analysis of P 2 ˆRf then simplifies in (3.2) and (3.3). Some examples would be estimation of P (a, b, (, c]), P 2 (a, b), P (a) and of ratios R(a, b, (, c]) and Q(a, b). 4 Model R In this section we consider model R, in which we have a parametric family R for R and leave Q unspecified. Again we assume that is one-dimensional. We proceed as in Section 3. A natural estimator for is the maximum likelihood estimator based on R. We assume that R (x, y, ds) has density r (x, y, s) with respect to some dominating measure ν R (x, y, ds), and write µ = ṙ /r for the score function. We have R µ = 0 and R (µ 2 + µ ) = 0. In particular, the Fisher information J = P 2 R µ 2 equals P 2R µ. The maximum likelihood estimator solves the estimating equation µ (X j, X j, S j ) = 0. As in Section 3 we obtain that ˆ is asymptotically linear, now with influence function (0, mj µ ): n /2 ( ˆ ) = mj n /2 µ (X j, X j, S j ) + o p (). (4.) Hence n /2 ( ˆ ) is asymptotically normal with variance mj. To prove efficiency of ˆ, we interpret as a functional of (Q, R) through ϕ(q, R ) =. The local model for R is described by perturbing as na = + n /2 a and R as. = R ( + n /2 aµ ). So the tangent space for R consists of all functions of the R na form aµ, a R, and the canonical gradient (g 0, h 0 ) of is of the form (0, a 0 µ ), where a 0 is determined from (2.5) by a = m P 2 R (aµ a 0 µ ) = aa 0 m J, a R.

12 This gives a 0 = mj function (0, mj to Q. and h 0 = mj µ. Since ˆ is asymptotically linear with influence µ ), it is efficient by characterization (2.8) and adaptive with respect To estimate Ef(X j, X j, S j ) = P 2 R f, we can again use the empirical estimator. However, a better estimator is ˆP 2 R ˆf = Here ˆP 2 stands for the empirical distribution R ˆ(X j, X j, ds)f(x j, X j, s). δ (Xj,X j )(dx, dy), where δ (Xj,X j ) is the one-point distribution on (X j, X j ). With R na. = R ( + n /2 aµ ) and a = n /2 ( ˆ ), a Taylor expansion gives ˆP 2 R ˆf = P 2 R f + ˆP 2 R ˆf P 2 R ˆf + P 2 R ˆf P 2 R f = P 2 R f + R (X j, X j, ds)f(x j, X j, s) P 2 R f +P 2 R (µ f)(ˆ ) + o p (n /2 ). Since R µ = 0, we have P 2 R (µ f) = P 2 R (µ f 0 ) and hence n /2 ( ˆP 2 R ˆf P 2 R f) ( = mn /2 R (X j, X j, ds)f(x j, X j, s) P 2 R f) +P 2 R (µ f 0 )(ˆ ) + o p (). Suppose that ˆ is efficient for. Then ˆ is asymptotically linear with influence function (0, mj µ ); see (4.). From the martingale approximation (2.9) we see that ˆP 2 R ˆf then has influence function (mar f, mj P 2R (µ f 0 )µ ). Hence ˆP 2 R ˆf is asymptotically normal with variance m ( P 2 (AR f) 2 + J (P 2R (µ f 0 )) 2). Note that by the Cauchy Schwarz inequality, J (P 2R (µ f 0 )) 2 P 2 R f

13 Hence our estimator is better than the empirical estimator unless f 0 is proportional to µ. Now we prove that ˆP 2 R ˆf is efficient. Let na = +n /2 a and Q nu. Then R na = R ( + n /2 aµ ), and (2.) implies. = Q(+n /2 u). n /2 (P 2nu R na f P 2 R f) P 2 (uar f) + ap 2 R (µ f 0 ), u U, a R. Since Q is unspecified and hence the tangent space U 0 for Q is U, the canonical gradient of P 2 R f is of the form (mar f, a 0 µ ), where a 0 is determined by ap 2 R (µ f 0 ) = aa 0 m J, a R. This gives a 0 = mj P 2R (µ f 0 ). Hence ˆP 2 R ˆf is efficient by characterization (2.8). For example, if we set f(x, y, s) = s, we obtain an efficient estimator of the mean inter-jump time m = ES j. R ˆ(X j, X j, ds)s It is better than the empirical estimator Nn S j unless s R (x, y, ds)s is proportional to µ (x, y, s). If the inter-jump time distribution does not depend on the states, i.e. R (x, y, ds) = R (ds), then our estimator is equivalent to the simpler estimator R ˆ(ds)s, which is better than the empirical estimator Nn S j unless s R (x, y, ds)s is proportional to µ (s), i.e. if the inter-jump time distribution R is exponential with scale parameter. Acknowledgment. Work supported by NSERC, Canada. References [] Bickel, P. J.; Klaassen, C. A. J.; Ritov, Y.; Wellner, J. A., Efficient and Adaptive Estimation for Semiparametric Models. Springer: New York, 998. [2] Bickel, P. J.; Ritov, Y., Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhya Ser. A, 50 (988) [3] Efromovich, S.; Samarov, A., Adaptive estimation of the integral of squared regression derivatives. Scand. J. Statist., 27 (2000) [4] Eggermont, P. P. B.; LaRiccia, V. N., Maximum Penalized Likelihood Estimation, Vol I, Density Estimation. Springer: New York, 200. [5] Gill, R. D., Nonparametric estimation based on censored observations of a Markov renewal process. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 53 (980)

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