Optimality of estimators for misspecified semi-Markov models

For i.i.d. observations, Daniels Da61 [6] and Huber Hu67 [20] show that the maximum likelihood estimator of a misspecified parametric model estimates the parameter that maximizes the Kullback–Leibler (KL) information, and determine its asymptotic distribution. Weaker conditions are given by Pollard Po85 [33]. For applications see also White Wh82 [35], Müller Mu07 [30], and Doksum, Ozeki, Kim and Neto DOKN [7]. Analogous results are obtained for parametric Markov chain models by Ogata Og80 [31], for parametric time series by Hosoya Ho89 [19] and by Andrews and Pollard AP94 [1], and for parametric diffusion models by McKeague Mc84 [28] and Kutoyants Ku88 [25]. We refer also to the monograph of Kutoyants Ku04 [26]. Applications to time series models in econometrics are studied by White Wh84 [36] and Sin and White SW96 [34], and in the monograph of White Wh94 [37].

Greenwood and Wefelmeyer GW97 [15] prove that the maximum likelihood estimator of a misspecified parametric Markov chain model is efficient in a nonparametric sense. Related efficiency results for misspecified parametric time series are in Dahlhaus and Wefelmeyer DW96 [5]. Here we outline corresponding results for semi-Markov processes. We consider both parametric and semiparametric misspecified models. The arguments are heuristic; sufficient regularity conditions can be obtained as in the above references.

Suppose we observe a semi-Markov process $Z_{t}$, $t\geq 0$, with values in an arbitrary measurable space $E$, on a time interval $0\leq t\leq n$. Let $(X_{0},T_{0}),(X_{1},T_{1}),\dots$ denote the embedded Markov renewal process. Its transition distribution factors as

where $Q(x,dy)$ is the transition distribution of the embedded Markov chain $X_{0},X_{1},\dots$, and $R(x,y,du)$ is the conditional distribution of the inter-arrival time $U_{j}=T_{j}-T_{j-1}$ given $X_{j-1}=x$ and $X_{j}=y$.

We assume that the embedded Markov chain is stationary. We write $P_{1}(dx)$, $P_{2}(dx,dy)$ and $P_{3}(dx,dy,du)$ for the stationary laws of $X_{j-1}$, $(X_{j-1},X_{j})$ and $(X_{j-1},X_{j},U_{j})$, respectively. Of course, $P_{2}=P_{1}\otimes Q$ and $P_{3}=P_{2}\otimes R=P_{1}\otimes Q\otimes R$. Set $N=\max\{j:T_{j}\leq n\}$. We note that studying a semi-Markov process is equivalent to studying the embedded Markov renewal process. The latter is a Markov chain. Observing the semi-Markov process up to time $n$ is equivalent to observing the embedded Markov renewal process up to the random time $N$.

Natural estimators for $P_{1}$, $P_{2}$ and $P_{3}$ are the empirical distributions

where $\delta_{x}$ denotes the Dirac measure at a point $x$.

Let $\Theta$ be an open subset of $\mathbb{R}^{d}$. We consider the following three models for the semi-Markov process. In Model Q we assume a parametric form $Q=Q_{\vartheta}$, $\vartheta\in\Theta$, of the transition distribution of the embedded Markov chain. These models are also considered in Greenwood, Müller and Wefelmeyer GMW04 [11]. In Model R we assume a parametric form $R=R_{\vartheta}$, $\vartheta\in\Theta$, of the conditional distribution of the inter-arrival times. In Model S we assume parametric forms $Q=Q_{\vartheta}$ and $R=R_{\vartheta}$, $\vartheta\in\Theta$, for both. Of course, the last model covers the case that $Q$ and $R$ carry different parameters. We assume that $Q_{\vartheta}(x,dy)$ has a density $q_{\vartheta}(x,y)$ with respect to some dominating measure $\mu(dy)$, and $R_{\vartheta}(x,y,du)$ has a density $r_{\vartheta}(x,y,u)$ with respect to some dominating measure $\nu(du)$.

If Model Q holds, then the transition distribution of the semi-Markov process is semiparametric, $S=Q_{\vartheta}\times R$, with $R$ an infinite-dimensional nuisance parameter. A natural estimator of $\vartheta$ is the partial maximum likelihood estimator $\hat{\vartheta}_{Q}$, which maximizes

Suppose that Model Q is misspecified, and that the true transition distribution of the embedded Markov chain is $Q$. Then $\mathbb{P}_{2}[\log q_{\vartheta}]$ is an empirical version of the KL information $P_{2}[\log q_{\vartheta}]$. Let $K_{Q}(P_{2})$ denote the parameter that maximizes $P_{2}[\log q_{\vartheta}]$. We call $K_{Q}$ a KL functional. Note that the partial maximum likelihood estimator is the empirical version of the KL functional, $\hat{\vartheta}_{Q}=K_{Q}(\mathbb{P}_{2})$. Since Model Q is misspecified, the semi-Markov model is nonparametric. The empirical distribution $\mathbb{P}_{2}$ is efficient for $P_{2}$ in a certain sense. If the KL functional is smooth, i.e. compactly differentiable in an appropriate sense, it follows that $\hat{\vartheta}_{Q}=K_{Q}(\mathbb{P}_{2})$ is efficient for $K_{Q}(P_{2})$. We will not use this approach in this paper. Instead we derive, in Section 3, a stochastic expansion of $\hat{\vartheta}_{Q}$, and determine its influence function. We also show that the KL functional $K_{Q}$ is pathwise differentiable, and determine its canonical gradient. To keep the exposition simple, we do not give regularity conditions for these results. They can be adapted e.g. from those of Greenwood and Wefelmeyer GW97 [15]. It turns out that the canonical gradient equals the influence function of $\hat{\vartheta}_{Q}$. By the characterisation of efficient estimators in Section 2, this shows that $\hat{\vartheta}_{Q}$ is efficient in the nonparametric semi-Markov model. We also show that $\hat{\vartheta}_{Q}$ remains efficient when Model Q is true. The advantage of our approach is that we do not need to check compact differentiability of $K_{Q}$ and a corresponding efficiency property of $\mathbb{P}_{2}$.

The other two models are treated analogously. If Model R holds, then the transition distribution of the semi-Markov process is semiparametric, $S=Q\otimes R_{\vartheta}$, with $Q$ an infinite-dimensional nuisance parameter. A natural estimator of $\vartheta$ is the partial maximum likelihood estimator $\hat{\vartheta}_{R}$, which maximizes

Suppose that Model Q is misspecified, and that the true conditional distribution of the inter-arrival times is $R$. Then $\mathbb{P}_{3}[\log r_{\vartheta}]$ is an empirical version of $P_{3}[\log r_{\vartheta}]$. Again we call the latter KL information. We denote by $K_{R}(P_{3})$ the parameter that maximizes $P_{3}[\log r_{\vartheta}]$, and we call $K_{R}$ a KL functional. Then $\hat{\vartheta}_{R}=K_{R}(\mathbb{P}_{3})$. In Section 4 we derive a stochastic expansion of $\hat{\vartheta}_{R}$ and the canonical gradient of $K_{R}$ and show that $\hat{\vartheta}_{R}$ is efficient in the nonparametric semi-Markov model. We also show that $\hat{\vartheta}_{R}$ remains efficient when Model R is true.

If Model S holds, then the transition distribution of the semi-Markov process is parametric, $S_{\vartheta}=Q_{\vartheta}\otimes R_{\vartheta}$. Set

A natural estimator of $\vartheta$ is the maximum likelihood estimator $\hat{\vartheta}_{S}$, which maximizes

Suppose that Model Q is misspecified, and that the true transition distribution of the embedded Markov renewal process is $S=Q\otimes R$. Then $\mathbb{P}_{3}[\log s_{\vartheta}]$ is an empirical version of $P_{3}[\log s_{\vartheta}]$. Again we call the latter KL information. We denote by $K_{S}(P_{3})$ the parameter that maximizes $P_{3}[\log s_{\vartheta}]$, and we call $K_{S}$ a KL functional. Then $\hat{\vartheta}_{S}=K_{S}(\mathbb{P}_{3})$. In Section 5 we derive a stochastic expansion of $\hat{\vartheta}_{S}$ and the canonical gradient of $K_{S}$ and show that $\hat{\vartheta}_{S}$ is efficient in the nonparametric semi-Markov model. We also show that $\hat{\vartheta}_{S}$ remains efficient when Model S is true. Section 6 contains some additional comments.

We assume that the embedded Markov chain is positive Harris recurrent and geometrically ergodic in $L_{2}(P_{2})$. We make the usual assumption that the conditional distribution of the inter-arrival times does not charge zero. We also assume that the mean inter-arrival time $m=EU_{j}$ is finite. Then

For a function $f\in L_{2}(P_{3})$ we have the strong law of large numbers

For a function $f\in L_{2}(P_{3})$ with $Sf=0$ we have the martingale central limit theorem

where $Y$ denotes a standard normal random variable.

In order to characterize efficient estimators for functionals of semi-Markov models, we consider a family $Q_{\delta}$, $\delta\in\Delta$, of transition distributions of the embedded Markov chain, and a family $R_{\delta}$, $\delta\in\Delta$, of conditional distributions of the inter-arrival time. Here $\Delta$ is a possibly infinite-dimensional set, the parameter space. We fix $\delta\in\Delta$ and set $Q=Q_{\delta}$, $R=R_{\delta}$ and

Note that $V$ and $W$ can be viewed as orthogonal subspaces of $L_{2}(P_{3})$. We assume that the parametrization is smooth in the following sense. There is a linear space $K$, the tangent space of $\Delta$, and a bounded linear operator $D=(D_{Q},D_{R}):K\to V\times W$, and for each $k\in K$ there is a sequence $\delta_{nk}$ in $\Delta$ such that $Q_{nk}=Q_{\delta_{nk}}$ is Hellinger differentiable at $Q$ with derivative $D_{Q}k\in V$,

and $R_{nk}=R_{\delta_{nk}}$ is Hellinger differentiable at $R$ with derivative $D_{R}k\in W$,

Now write $M_{n}$ for the distribution of $Z_{t}$, $0\leq t\leq n$, if $Q$ and $R$ are in effect, and $M_{nk}$ if $Q_{nk}$ and $R_{nk}$ are. By Taylor expansion and 2 and 3, we obtain local asymptotic normality:

and

For Markov chains, different proofs are in Penev Pe91 [32], Bickel Bi93 [2] and Greenwood and Wefelmeyer GW95 [13]; see also Bickel and Kwon BK01 [4]. For Markov step processes see Höpfner, Jacod and Ladelli HJL90 [18] and Höpfner Ho93a [16, 17]. A proof for nonparametric semi-Markov models is in Greenwood and Wefelmeyer GW96 [14].

We want to estimate a $d$-dimensional functional $\varphi:\Delta\to\mathbb{R}^{d}$ of the parameter $\delta$. We call $\varphi$ differentiable at $\delta$ with gradient $(v_{\varphi},w_{\varphi})$ if $v_{\varphi}\in V^{d}$, $w_{\varphi}\in W^{d}$, and

The canonical gradient $(v_{\varphi}^{*},w_{\varphi}^{*})$ of $\varphi$ is the componentwise projection of $(v_{\varphi},w_{\varphi})$ onto the closure of $(DK)^{d}$ in $(L_{2}(P_{3}))^{d}$. If $DK$ is closed in $L_{2}(P_{3})$, we can write $(v_{\varphi}^{*},w_{\varphi}^{*})=(D_{Q}k_{\varphi},D_{R}k_{\varphi})$ for some $k_{\varphi}\in K$. This will be the case in Sections 3–5.

An estimator $\hat{\varphi}$ is called regular for $\varphi$ at $\delta$ with limit $L$ if $L$ is a $d$-dimensional random vector such that

The convolution theorem says that

with $Y_{d}$ a $d$-dimensional standard normal random vector, and $A$ a $d$-dimensional random vector independent of $Y_{d}$. This justifies calling $\hat{\varphi}$ efficient for $\varphi$ at $\delta$ if $n^{1/2}(\hat{\varphi}-\varphi(\delta))$ is asymptotically normal under $M_{n}$ with covariance matrix $m^{-1}(P_{2}[v_{\varphi}^{*}v_{\varphi}^{*\top}]+P_{3}[w_{\varphi}^{*}w_{\varphi}^{*\top}])$.

An estimator $\hat{\varphi}$ is called asymptotically linear for $\varphi$ at $\delta$ with influence function $(a,b)$ if $a\in V^{d}$, $b\in W^{d}$, and

We have the following characterization. An estimator $\hat{\varphi}$ is regular and efficient for $\varphi$ at $\delta$ if and only if it is asymptotically linear with influence function equal to the canonical gradient,

For proofs of the convolution theorem and the characterization we refer to Bickel, Klaassen, Ritov and Wellner BKRW98 [3].

To prove asymptotic linearity of estimators in misspecified models, we need the following martingale approximation. Set $L_{2,0}(P_{2})=\{f\in L_{2}(P_{2}):P_{2}[f]=0\}$. The potential $G$ of the embedded Markov chain is defined by

For $f\in L_{2}(P_{2})$ set

Then $QAf=0$ and

Let $f\in L_{2}(P_{3})$ and set $f_{0}=f-Rf$. Then we obtain the stochastic expansion

Note that $QARf=0$ and $Sf_{0}=0$. Hence $ARf(X_{j-1},X_{j})$ and $f_{0}(X_{j-1},X_{j},U_{j})$ are orthogonal martingale increments. For discrete-time processes, the martingale approximation 7 is due to Gordin Go69 [9] and Gordin and Lifšic GL78 [10]. It was discovered independently by Maigret Ma78 [27], Dürr and Goldstein DG86 [8] and Greenwood and Wefelmeyer GW95 [13]. See also Section 17.4 in the monograph of Meyn and Tweedie MT93 [29]. The martingale approximation 7 and the martingale central limit theorem 3 imply that

To calculate canonical gradients of functionals in misspecified models, we need the following perturbation expansion, due to Kartashov Ka85a [21, 22, 23],

Here $P_{2nk}$ denotes the distribution of $(X_{j-1},X_{j})$ if $Q_{nk}$ is in effect. This pathwise version of the perturbation expansion suffices for our purposes. Greenwood and Wefelmeyer GW95 [13] show that it follows also from the martingale approximation 7.

In Model Q we assume a parametric model $q_{\vartheta}$, $\vartheta\in\Theta\subset\mathbb{R}^{d}$, for the $\mu$-density of the transition distribution of the embedded Markov chain, and consider the conditional inter-arrival time distribution as unknown. Suppose the model is misspecified, and the true transition distribution is $Q$. Then the KL functional $K_{Q}(P_{2})$ maximizes $P_{2}[\log q_{\vartheta}]$, and the partial maximum likelihood estimator $\hat{\vartheta}_{Q}$ maximizes $\mathbb{P}_{2}[\log q_{\vartheta}]$. Write

for the $d$-dimensional vector of partial derivatives of $\log q_{\vartheta}(x,y)$. Then $K_{Q}(P_{2})$ solves $P_{2}[\chi_{\vartheta}]=0$, and $\hat{\vartheta}_{Q}$ solves $\mathbb{P}_{2}[\chi_{\vartheta}]=0$. Heuristically, by Taylor expansion,

Here $\dot{\chi}_{\vartheta}(x,y)$ is the $d\times d$ matrix of partial derivatives of $\chi_{\vartheta}(x,y)$. With 1 and 2 we obtain

If Model Q is correctly specified and $Q=Q_{\vartheta}$, then $K_{Q}(P_{2})=\vartheta$. We also have the following relations, which are well-known in the i.i.d. case,

In particular, the partial Fisher information matrix for Model Q is $I_{\vartheta}=-P_{2}[\dot{\chi}_{\vartheta}]=P_{2}[\chi_{\vartheta}\chi_{\vartheta}^{\top}]$. Hence, for the correctly specified model, the partial maximum likelihood estimator $\hat{\vartheta}_{Q}$ has the stochastic expansion

This means that $\hat{\vartheta}_{Q}$ is asymptotically linear with influence function $mI_{\vartheta}^{-1}(\chi_{\vartheta},0)$, and $n^{1/2}(\hat{\vartheta}_{Q}-\vartheta)$ is asymptotically normal with covariance matrix $mI_{\vartheta}^{-1}$.

If the model is misspecified, then $\chi_{K_{Q}(P_{2})}$ is not in $V^{d}$. We apply the martingale approximation 7 to 10 and see that $\hat{\vartheta}_{Q}$ is asymptotically linear with influence function $-m(P_{2}[\dot{\chi}_{K_{Q}(P_{2})}])^{-1}(A\chi_{K_{Q}(P_{2})},0)$. Hence $n^{1/2}(\hat{\vartheta}_{Q}-K_{Q}(P_{2}))$ is asymptotically normal with covariance matrix

Let us now prove efficiency of $\hat{\vartheta}_{Q}$, first for the correctly specified model. For $c\in\mathbb{R}^{d}$ set $\vartheta_{nc}=\vartheta+n^{-1/2}c$. Assume that $q_{nc}=q_{\vartheta_{nc}}$ is Hellinger differentiable at $\vartheta$,

Let $\mathcal{R}$ denote the set of all conditional inter-arrival distributions. For $w\in W$ choose a sequence $R_{nw}$ in $\mathcal{R}$ that is Hellinger differentiable at $R$,

Then the assumptions of Section 2 hold with $\Delta=\Theta\times\mathcal{R}$, $K=\mathbb{R}^{d}\times W$, $D_{Q}(c,w)=c^{\top}\chi_{\vartheta}$, $D_{R}(c,w)=w$. The functional to be estimated is $\varphi(\vartheta,R)=\vartheta$. By orthogonality of $V$ and $W$, its canonical gradient is obtained from 6 as $(c_{\vartheta}^{\top}\chi_{\vartheta},0)$ with $d\times d$ matrix $c_{\vartheta}$ determined by

i.e. $c_{\vartheta}=mI_{\vartheta}^{-1}$. Hence the canonical gradient of $\vartheta$ is $mI_{\vartheta}^{-1}(\chi_{\vartheta},0)$ and equals the influence function of $\hat{\vartheta}$, which is therefore efficient for the correctly specified model.

Suppose now that the model is misspecified, and let $\mathcal{Q}$ be the set of all transition distributions of the embedded Markov chain. Let $Q$ denote the true transition distribution. For $v\in V$ choose a sequence $Q_{nv}$ in $\mathcal{Q}$ that is Hellinger differentiable at $Q$,

Then the assumptions of Section 2 hold with $\Delta=\mathcal{Q}\times\mathcal{R}$, $K=V\times W$, $D_{Q}(v,w)=v$, $D_{R}(v,w)=w$. The functional to be estimated is $\varphi(Q,R)=K_{Q}(P_{2})$. Heuristically,

With $P_{2nv}[\dot{\chi}_{K_{Q}(P_{2})}]\to P_{2}[\dot{\chi}_{K_{Q}(P_{2})}]$ we obtain

The perturbation expansion 8 yields

Hence

and the canonical gradient of $K_{Q}$ is obtained from 6 as $-m(P_{2}[\dot{\chi}_{K_{Q}(P_{2})}])^{-1}(A\chi_{K_{Q}(P_{2})},0)$ and equals the influence function of $\hat{\vartheta}_{Q}$, which is therefore efficient for the misspecified model.

Model R is completely analogous to Model Q, with interchanged roles of the transition distribution $Q$ of the embedded Markov chain, and the conditional inter-arrival time distribution $R$. Specifically, in Model R we assume a parametric model $r_{\vartheta}$, $\vartheta\in\Theta\subset\mathbb{R}^{d}$, for the $\nu$-density of the conditional inter-arrival time, and consider the transition distribution of the embedded Markov chain as unknown. Suppose the model is misspecified, and the true conditional inter-arrival time distribution is $R$. Then the KL functional $K_{R}(P_{3})$ maximizes $P_{3}[\log r_{\vartheta}]$, and the partial maximum likelihood estimator $\hat{\vartheta}_{Q}$ maximizes $\mathbb{P}_{3}[\log r_{\vartheta}]$. Write

for the $d$-dimensional vector of partial derivatives of $\log r_{\vartheta}(x,y,u)$. Then $K_{R}(P_{3})$ solves $P_{3}[\varrho_{\vartheta}]=0$, and $\hat{\vartheta}_{R}$ solves $\mathbb{P}_{3}[\varrho_{\vartheta}]=0$. Heuristically, by Taylor expansion,

Here $\dot{\varrho}_{\vartheta}(x,y,u)$ is the $d\times d$ matrix of partial derivatives of $\varrho_{\vartheta}(x,y,u)$. With 1 and 2 we obtain

If Model R is correctly specified and $R=R_{\vartheta}$, then $K_{R}(P_{3})=\vartheta$. We also have the following relations,

In particular, the partial Fisher information matrix for Model R is $J_{\vartheta}=-P_{3}[\dot{\varrho}_{\vartheta}]=P_{3}[\varrho_{\vartheta}\varrho_{\vartheta}^{\top}]$. Hence, for the correctly specified model, the partial maximum likelihood estimator $\hat{\vartheta}_{R}$ has the stochastic expansion