We consider a jump Markov process , with values in a state space . We suppose that the corresponding infinitesimal generator , hence the law of , depends on a parameter . We prove that several models (filtered or not) associated with are linked, by their regularity according to a certain scheme. In particular, we show that the regularity of the model is equivalent to the local regularity of .
1 Introduction and main results
Jump Markov processes, have found application in Bayesian statistics, chemistry, economics, information theory, finance, physics, population dynamics, speech processing, signal processing, statistical mechanics, traffic modeling, thermodynamics, and many others . Regularity plays a significant role in the classical asymptotic statistics for parametric statistical models for jump Markov processes; see [2,3,4] for recent developments. Asymptotic normality or Bernstein-von Mises-type theorems impose several regularity conditions so that their results hold rigorously. In this article, we focus on the regularity conditions of several statistical models associated with a jump Markov process with values being an arbitrary space state, endowed with a -field . Let be the canonical space of piecewise constant functions , right continuous for the discrete topology. Let be the canonical process, the canonical filtration, and . Let and be the sequence of the jump times of , which are almost surely increasing to . To each and , we associate
We assume that, under , the process is Markovian, starts from , is non-exploding, and admits the infinitesimal generator
The existence of the probabilities is guaranteed by the boundedness of the function for instance. The following facts clarify our focus on the different statistical models that will be presented later on:
under , and conditionally to , the distribution of is exponential with parameter ;
is the transition probability of the embedded Markov chain ;
, is the distribution of under ;
The associated sub-Markovian transition kernels satisfy the backward Kolmogorov equations:(1)
The Markov process is simple if is a Markov kernel, i.e., for every , is a probability measure on . In this case, the transition functions are also Markov and satisfy the Chapman-Kolmogorov equation
cf.  for more account.
The multivariate point process associated with the process is
and its compensator, under , is
The model E is not a proper statistical model since is not a probability measure. Nevertheless, the extension of the notion of regularity to models associated with families of finite positive measures is also feasible and is described as follows. Let be a family of finite positive measures in . For , we denote by a measure that dominates , and , and by , and , be Radon-Nikodym derivatives, respectively, of and according to and of according to . The Lebesgue decomposition of , with respect to , is given by the pair ,
We start by recalling the notion of “error functions” which was introduced in  as follows.
A function is called an error function, if . More generally, an error function is any positive function such that
(Regularity of non-filtered models). The model is regular at , if the random function
is differentiable at , i.e., there exists a random vector and an error function , such that
Note that if the regularity of the model holds, then is necessarily -square integrable. Furthermore, if is a family of probability measures, then The Hellinger integral of order between the measures and , is defined by
and is independent of the dominating measure The regularity of the model is equivalent to one of these two assertions:
There exist an error function and a random vector (the same as before), such that(7)
Let be a sample space endowed with a filtration , and a family of probability measures coinciding on . The regularity of the statistical filtered model
mimics the one in Definition 2 and is expressed in terms of likelihood processes [8,7]. For a clear presentation, we need to introduce the likelihood process of with respect to , defined in Jacod and Shiryaev’s book , by
The process is a positive -supermartingale and is a martingale if
For any probability measure , locally dominating and , the -martingales
and the stopping times
provide this version of :
As for non-filtered models, we have the following definition.
(Regularity of filtered models). Let be a stopping time relative to . The model is said to be regular (or differentiable) at time and at , if the model is regular in the sense of Definition 2. That means that there exists an -measurable, -square-integrable, and centered random vector and two error functions , such that
is the Hellinger integral of order , at time , and which is independent of the choice of the dominating probability measure. The Fisher information matrix of the model is then
It is worth noting that if the regularity at a time implies the regularity at any stopping time . In particular, if the regularity holds along a sequence , increasing to infinity, then there exists a local martingale , locally square-integrable, null at zero, such that if for some , then is a version of the random variable in (10). In particular, if (9) and (10) are satisfied for all , then is a square-integrable martingale, null at 0, cf. [7, Corollary 3.16].
We are now able to introduce the notion of local regularity, which is less restrictive than the preceding one.
(Local regularity of filtered models). A sequence of stopping times is called a localizing sequence if it is -almost surely increasing to . A localizing family is a sequence formed by the pair , where is a localizing sequence and is a sequence of stopping times, satisfying
The model (8) is said to be locally regular (or locally differentiable at ), if there exists a right continuous, left limited process on , such that, for all satisfying
there exists a localizing family , satisfying
Note that if the model is regular along a localizing sequence, then it is necessarily locally regular. By Theorem [7, Theorem 4.6], the process is a locally square-integrable ( )-local martingale and the Fisher information process , at , is defined as the predicable quadratic covariation of :
The local regularity does guarantee the integrability of ; however, it is the minimal condition we require to obtain the property of local asymptotic normality (LAN) for statistical models. In this case, the Fisher information quantities provide the lower bound of the variance of any estimator of the unknown parameters intervening in the models, see [10,11,12] for instance.
According to [7, Theorem 6.2], the local regularity is equivalent to the two following conditions:
and for all ,
is the variation process of .
is a version of the Hellinger process of order between and , i.e., is a predictable nondecreasing process, null at zero, such that(17)
is the predictable nondecreasing process intervening in the Doob-Meyer decomposition of the supermartingale . Since and coincide on , then necessarily and there exists a -local martingale such that
The results that we obtain complete those of Höpfner et al. , who proved that if is regular and if the process satisfies a condition of positive recurrence (resp. null recurrence), then the model localized around the parameter is or locally asymptotically normal or is locally asymptotically mixed normal. The main result is as follows.
The model is locally regular for all , if, and only if, is regular for all .
2 Additional regularity properties
Our notations and the calculus of the Hellinger integrals and of the likelihood processes are borrowed from Höpfner  and Höpfner et al. . For and , the following measures will be used in the sequel.
is a measure dominating , and . Thus, also dominates , , and ;
is a transition probability on dominating , , and ;
is a probability measure, locally dominating , and ;
, and .
If we choose
and if is the probability under which the canonical process , starts from , and has the infinitesimal generator , then we have
With the convention , a version of the likelihood processes of , with respect to and to , is given in  by
With the notations
a version of the likelihood processes of , relative to and , is explicitly given by
The Hellinger integral of order between and is then
and the Hellinger integral of order at time between and relative to the filtration , is expressed by
We also consider the quantities
which are used to define the Hellinger process , of order , between and , and relative to . It is expressed by
Finally, we define the function
where is the Fisher information matrix of the model at , whenever it is regular. Consequently, is expressed by
Observe that the function in (20) is such that the function
is nondecreasing in , satisfies Thus, has the vocation to be an error function.
We can now state a first technical but intuitive result.
Let . Then the following assertions are equivalent.
is regular and is differentiable at ;
is regular at time .
In the three following theorems, we complete our results by studying the regularity of the filtered model , at fixed times , or at the jump times . In this direction, we obtain only partial results appealing to some additional conditions of integrability.
Let . For all , assume the following.
Condition . There exists , an error function and a measurable function , satisfying the following:
Then, is regular at the time , for all .
For all , assume the following. The model is regular, and
Condition . The error function f in (7), associated with the model , satisfies the following. There exists such that
Then, the model is regular for all , and all .
The control in the first integral in condition is exactly the required condition for to be regular. The finiteness of the second integral will ensure integrability conditions in the proof of Theorem 7.
Equivalently, we could replace the error function in condition B( ) by the one in (6). The integrability condition becomes
and the only difference is that we would have to check two inequalities instead of one.
We conclude with our last result.
1. Let . If is regular at a time , then is regular.
2. Furthermore, if is regular at a time , for all , then, is locally regular, for all .
3 Proofs of the theorems
We will sometimes use the notion of isomorphism between two statistical models. Referring to Strasser’s book , we say that two models and are isomorphic if they are randomized of each other. To illustrate this notion, assume for instance that and are, respectively, dominated by and . Then, the model is randomized from , if there exists a Markovian operator , such that
The models and are randomized of each other if they are mutually exhaustive, which is always the case in our study, each time an isomorphism holds, cf. [14, Lemma 23.5 and Theorem 24.11]. When computing expectations, these isomorphisms allow us to handle at our convenience, one of the two likelihoods of the models and . The latter is justified by the fact that they have the same law under the respective probability quotient.
Proof of Theorem 6
: (a) By (7), the regularity of at is equivalent to the existence of a random vector , and of an error function , such that
The latter implies
(b) The implication “ is regular at is differentiable at ” is shown in , using the fact that the differentiability of in implies the differentiability of in . Furthermore, the derivative at of is
(c) Let us define
where the function
Then, we can write
: (a) Under the condition of differentiability of at 0, we obtain
(b) According to (7), the regularity of , at , is equivalent to the existence of a centered vector , and of an error function such that
The vector is defined by
which belongs to and satisfies
(c) Let us define
By (26), we have
With the same arguments as in (c), we retrieve
Consequently, there exists an error function such that
: Let . Since
is the tensorial product of two probability measures, then is statistically isomorphic to , where
The differentiability of , at , is equivalent to the differentiability of the model . The assertion is then a consequence of [15, Corollary I.7.1] in Ibragimov and Has’minskii’s book.
: As in the proceeding implication, observe that is statistically isomorphic to , and the result becomes a simple consequence of [15, Theorem I.7.2].
: This equivalence is deduced from the fact that is the distribution of , then is identified with restricted to the -field . Thus, and are statistically isomorphic.□
Proof of Theorem 5
(1) For the necessity condition, we will check (15) and (16), as it was done for the Markov chains in . For fixed , we choose the dominating probability , the one for which the process has the generator
(1)(a) Using the function in (20), we have
(1)(b) The Doob-Meyer decomposition of the supermartingale asserts that
where is a local martingale and is a predictable nondecreasing process. Since the jump times of the process are totally inaccessible, then is left-quasi continuous and has necessarily continuous paths, cf. [16, Theorem 14]. From the decomposition of the additive functional on the event , into a local martingale , and a process with finite variation (see [5, p. 40]), we may write in the form
Applying Ito’s formula to the semimartingale , we obtain