Armelle Guillou

Stéphane Loisel

Gilles Stupfler

Estimating the parameters of a seasonal Markov-modulated Poisson process

Keywords: Markov-modulated Poisson process, seasonality, split-time likelihood, strong consistency, asymptotic normality. MSC 2010 Subject Classifications: Primary 62M05, 62F12; Secondary 60F05, 60F15

Motivated by seasonality and regime-switching features of some insurance claim counting processes, we study the statistical analysis of a Markov-modulated Poisson process featuring seasonality. We prove the strong consistency and the asymptotic normality of a maximum split-time likelihood estimator of the parameters of this model, and present an algorithm to compute it in practice. The method is illustrated on a small simulation study and a real data analysis.

Introduction

It is often the case that the insurance claim frequency is impacted by environment variables.

For instance, flood risk is higher in a period of frequent heavy rains, and fire risk is more 1 intense when the weather is particularly dry. Such environment variables may be hidden to some extent to the practitioner: for instance, it is now accepted that the probabilities of severe floods in Australia, strong snowstorms in North America or hurricanes on the East Coast of the United States increase during La Niña episodes (see Neumann et al. [START_REF] Neumann | Tropical Cyclones of the North Atlantic Ocean, 1871-1992[END_REF], Cole and Pfaff [START_REF] Cole | A climatology of tropical cyclones affecting the Texas coast during El Niño/non-El Niño years: 1990-1996[END_REF],

Parisi and Lund [START_REF] Parisi | Seasonality and return periods of landfalling Atlantic basin hurricanes[END_REF] and Landreneau [START_REF] Landreneau | Atlantic tropical storms and hurricanes affecting the United States: 1899-2000[END_REF]). This is now taken seriously by most reinsurers as well as Lloyd's and the UK Met Office [START_REF]Lloyd's Forecasting risk, The value of long-range forecasting for the insurance industry, joint report with the UK Met Office[END_REF]. However, observing and understanding the role of those variables is not easy, which makes it realistic to consider these variables as unobserved so far.

To take such a dependency into account, one may for instance assume that the underlying environment process is a Markov process J in continuous time and that in each state of J, the claim counting process N is a Poisson process. The resulting bivariate process (J, N ) is then called a Markov-Modulated Poisson Process (MMPP). MMPPs have been used in different fields during the past forty years, in particular in data traffic systems (see e.g. Salvador et al. [START_REF] Salvador | Multiscale fitting procedure using Markov modulated Poisson processes[END_REF]), for ATM sources (see Kesidis et al. [START_REF] Kesidis | Effective bandwidths for multiclass Markov fluids and other ATM sources[END_REF]), in manufacturing systems (see e.g. Ching et al. [START_REF] Ching | Circulant preconditioners for Markov-modulated Poisson processes and their applications to manufacturing systems[END_REF]) or even in ecology (see for example Skaug [START_REF] Skaug | Markov modulated Poisson processes for clustered line transect data[END_REF] for applications of MMPPs to clustered line transect data). The MMPP cookbook by Fischer and Meier-Hellstern [START_REF] Fischer | The Markov-modulated Poisson process (MMPP) cookbook[END_REF] sums up the main results and ideas that were behind the rise of MMPP applications. The idea of considering Markov modulation in insurance was first introduced by Asmussen [START_REF] Asmussen | Risk theory in a Markovian environment[END_REF]; the obtained model can capture the fact that the insurance claim frequency may be modified if climatic, political or economic factors change. Such a model has gained considerable attention recently: see for instance Lu and Li [START_REF] Lu | On the probability of ruin in a Markov-modulated risk model[END_REF], Ng and Yang [START_REF] Ng | On the joint distribution of surplus prior and immediately after ruin under a Markovian regime switching model[END_REF], Zhu and Yang [START_REF] Zhu | Ruin theory for a Markov regime-switching model under a threshold dividend strategy[END_REF] and Wei et al. [START_REF] Wei | On the Markov-modulated insurance risk model with tax[END_REF]. The parameters of an MMPP are often estimated using a Maximum Likelihood Estimator (MLE), whose consistency was proved in Rydén [START_REF] Rydén | Parameter estimation for Markov modulated Poisson processes[END_REF]. Various methods have been suggested to compute the MLE; a standard tool is the Expectation-Maximization (EM) algorithm, see Rydén [START_REF] Rydén | An EM algorithm for estimation in Markov-modulated Poisson processes[END_REF] for the implementation of this procedure for the estimation of the parameters of an MMPP. We finally mention that in a recent paper, Guillou et al. [START_REF] Guillou | Estimation of the parameters of a Markov-modulated loss process in insurance[END_REF] introduced a new MMPP-driven loss process in insurance with several lines of business, showed the strong consistency of the MLE and fitted their model to real sets of insurance data using an adaptation of the EM algorithm.

Of course, once the MMPP model is fitted, it is possible to use Bayesian techniques to determine probabilities to be in each state and consequently the average number of events during the next period (see e.g. Scott [START_REF] Scott | Bayesian analysis of a two-state Markov modulated Poisson process[END_REF]). If external information is present, then it is possible to enrich this Bayesian estimation. This is for example the case for some long-tailed non-life insurance businesses, where indices of sectorial inflation can provide useful information. For reinsurance cycles, large claims that may cause a cycle phase change as well as other aspects of competition or adverse development of reserves can sometimes be plugged into the Bayesian estimation process. For some other risks however, even if we feel that a phenomenon might have an impact on the claim frequency, it may be very hard to come up with a measure of such a phenomenon (for instance, the El Niño-La Niña phenomena). In that case our non-Bayesian framework is of interest for actuarial risk assessment. Furthermore, many examples of practical applications in insurance display some sort of seasonal variation. For example, theft in garages are more frequent before Christmas as people tend to store Christmas gifts there, fire risk is more intense in the summer, and hurricanes occur mostly between June and November on the East Coast of the United States. These random, cyclic factors and their impact on insurance risk, which need to be taken into account to carry out a proper regime switching analysis, are yet to be understood and forecasted. In an inhomogeneous context with deterministic intensity function, Lu and Garrido [START_REF] Lu | Doubly periodic non-homogeneous models for hurricane data[END_REF] have fitted double-periodic Poisson intensity rates to hurricane data, for particular parametric forms (like double-beta and sine-beta intensities) to hurricane data. Helmers et al. [START_REF] Helmers | A non-parametric estimator for the doubly periodic Poisson intensity function[END_REF] have provided an in-depth theoretical statistical analysis of such doubly periodic intensities. We aim at carrying out a theoretical statistical analysis in a stochastic intensity framework with seasonality.

An important aspect of pricing in non-life insurance concerns segmentation: thanks to generalized linear models or more sophisticated techniques, the insurer takes into account explanatory variables to adapt the price of the contract and avoid adverse selection (see Ohlsson and Johansson [START_REF] Ohlsson | Non-life insurance pricing with generalized linear models[END_REF]). Besides, individual ratemaking is updated thanks to credibility adjustments in order to take into account the claim history of each policyholder or contract (see Bühlmann and Gisler [START_REF] Bühlmann | A course in credibility theory and its applications[END_REF]). Our approach is only operational and interesting at the aggregated risk management level, and it would be very challenging to try to combine it with regression techniques, from a theoretical point of view as well as the practical point of view since a very large number of data points would be needed to ensure that the estimation is reasonably accurate. The only simple way to combine the two approaches would be to assume that the seasonal and Marko-vian intensities are multiplied for each class of contract by a number that corresponds to the risk of each class and does neither depend on the season nor on the state of the environment.

In other terms, the ratio between claim intensities for classes A and B, say, should remain constant over time. This is of course a strong limitation; in wind-related risk for instance, some buildings are more subject to damage coming from wind effects than others, and consequently the rise of their risk should be sharper during windy episodes.

In this paper, we thus consider an MMPP featuring seasonality, and study estimation issues for this process when the environment process is unobserved. A further motivation for the introduction of this type of process appears in the Solvency II insurance regulation framework.

In this framework, it is required to carry out an ORSA (Own Risk and Solvency Assessment).

The ORSA has both qualitative and quantitative components, including the continuous compliance requirement: the insurer must be able to provide evidence that he/she satisfies solvency requirements at all times (see Vedani and Ramaharobandro [START_REF] Vedani | Continuous compliance: a proxy-based monitoring framework[END_REF]). This leads insurers to define confidence regions or to use a continuous time model. The MMPP with seasonality is one interesting potential tool for this new challenge for some non-life insurance companies.

A problem, however with this type of process is that contrary to when there is no seasonality, the random sequence of the inter-event times is not ergodic. Studying the asymptotic properties of the MLE, as Rydén [START_REF] Rydén | Parameter estimation for Markov modulated Poisson processes[END_REF] and Guillou et al. [START_REF] Guillou | Estimation of the parameters of a Markov-modulated loss process in insurance[END_REF] do, is thus very difficult; furthermore, the particular structure of the likelihood makes it hard to compute the MLE in practice. To tackle this issue, we borrow an idea of Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for hidden Markov models[END_REF][START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF] and we introduce a Split-Time Likelihood (STL).

The logarithm of this quantity is shown to be a sum of ergodic random variables; maximizing the STL then yields a Maximum Split-Time Likelihood Estimator (MSTLE) whose consistency and asymptotic normality can be proven using regenerative theory.

The outline of the paper is as follows. We give a precise definition of our model in Section 2 where we also introduce our estimator. The asymptotic properties are studied in Section 3. We explain how to implement our estimation technique in practice in Section 4 where we illustrate the performance of our estimator on a small simulation study and a real data analysis. Our paper ends by a conclusion and discussion of the results in Section 5. Technical lemmas are given and proven in the appendix.

2 Our model and estimation procedure

Our model

We consider an irreducible continuous-time Markov process J with generator L on the state space {1, . . . , r}, where r ≥ 2. Consider further a counting process N and real numbers

τ 0 = 0 < τ 1 < • • • < τ k-1 < 1 = τ k such that for every q ∈ N, on the intervals [q + τ s-1 , q + τ s ), if J is in state i, then N is a Poisson process with jump rate λ (s)
i , where λ (s) i ≥ 0. The time interval [q + τ s-1 , q + τ s ) represents season s of the period q + 1. The case k = 1, in which there is actually no seasonality, is just the standard MMPP, which was considered from a statistical point of view by Rydén [START_REF] Rydén | Parameter estimation for Markov modulated Poisson processes[END_REF]; in this paper, we shall focus on the case k ≥ 2. The jump intensity of the counting process N is thus modulated not only by the random switches of the Markov process J but also by switches from one season to another, which happen at nonrandom times.

The context of our work is the following: let us assume that the process N has been observed until time n ∈ N \ {0}, so that the available data consists of 1. the number r of states of J and the times τ 1 , . . . , τ k-1 , 2. the full knowledge of the process N between time 0 and time n.

The goal is to estimate the unknown parameters of the model, namely the elements ℓ ij of the transition intensity matrix L of J and the jump intensities λ (s) i of N . Since the modulating Markov process J is not observed, estimating the parameters of this model is not straightforward. For the sake of shortness, we let Φ be the global parameter of the model: Φ consists of the values ℓ ij , for 1 ≤ i, j ≤ r and i = j and of the λ (s) i , 1 ≤ i ≤ r and 1 ≤ s ≤ k. The distribution of the process with parameter Φ is denoted by P Φ and we let E be the parameter space

E = Φ | L(Φ) is irreducible and min 1≤i≤r min 1≤s≤k λ (s) i (Φ) > 0 .
The space E can be seen as the set of those parameters for which in any state of J and in any season, an event, namely a jump of N , can occur with positive probability, while the irreducibility assumption makes sure that all states of J are visited infinitely often and thus that all parameters may be consistently estimated. Note that an element of E is given by |E| = r(r -1) + rs parameters. Finally, define, for 1 ≤ s ≤ k and 0 ≤ y ≤ τ sτ s-1 the matrix-valued functions

f (y, s, Φ) = exp(y(L(Φ) -Λ (s) (Φ)))Λ (s) (Φ) and F (y, s, Φ) = exp(y(L(Φ) -Λ (s) (Φ))) where Λ (s) (Φ) = diag(λ (s) 1 (Φ), . . . , λ (s) r (Φ)).
The functions f and F , which will be instrumental in defining our procedure, have a nice probabilistic interpretation (see Meier-Hellstern [START_REF] Meier-Hellstern | A fitting algorithm for Markov-modulated Poisson processes having two arrival rates[END_REF]):

for any i, j ∈ {1, . . . , r}, f ij (y, s, Φ) dy := P(T

(s) 1 ∈ dy, J(y) = j | J(0) = i) and F ij (y, s, Φ) := P(T (s) 1 > y, J(y) = j | J(0) = i), where T (s) 1
is the time of the first event of an MMPP whose Markov-modulating process is J, with jump intensities given by Λ (s) . The functions f (•, s, Φ) and F (•, s, Φ) can thus be seen as the probability density and survival functions of the inter-event times for given states of J in season s, respectively.

A Maximum Split-Time Likelihood Estimator

It is impossible to apply any known results on MMPPs here since the jump intensity in a given state of J changes as time goes by; especially, given that J is in state j, N is an inhomogeneous Poisson process. To overcome this issue, we introduce some notation: let W q,s be the number of jumps of N during season s of the period q, and let T (q,s) 1 , . . . , T (q,s) Wq,s be the successive jump times of N during this season. Let Y (q,s) l = T (q,s) l -T (q,s) l-1 be the inter-event times in season s (with T (q,s) 0 = (q -1) + τ s-1 ). It is assumed that the starting distribution of J is its unique stationary distribution a(Φ) on {1, . . . , r}, that is, the only row vector a(Φ) such that a(Φ)L(Φ) = 0 and the sum of the entries of a(Φ) is equal to 1.

Let Z q = (W q,s , Y (q,s) 1 , . . . , Y (q,s) Wq,s ) 1≤s≤k be the random vector representing the information available for period number q. With this notation, the process (Z q ) q≥1 is stationary because J is a stationary Markov process. Besides, given the states of the irreducible Markov chain (J(q)) q∈N , the random variables Z q , q ≥ 1 are independent, so that arguing along the lines of the proof of Lemma 1 in Leroux [START_REF] Leroux | Maximum-likelihood estimation for hidden Markov models[END_REF] shows that the process (Z q ) q≥1 is ergodic. We denote by

Z = (W s , Y (s) 1 , . . . , Y (s)
Ws ) 1≤s≤k a random vector which shares the distribution of the Z q , q ≥ 1.

Let then L 1 (Z, Φ) be the likelihood of the observations over one period, computed under the parameter Φ: if z = (w s , y

(s) 1 , . . . , y (s) ws ) 1≤s≤k , L 1 (z, Φ) = a(Φ) k s=1 ws l=1 f (y (s) l , s, Φ) F τ s -τ s-1 - ws l=1 y (s) l , s, Φ 1 r
where 1 r is the column vector of size r having all entries equal to 1. This quantity can be expanded as:

L 1 (z, Φ) = i0,...,i k+1 a i0 (Φ) k s=1 e ′ is-1 ws l=1 f (y (s) l , s, Φ) × F τ s -τ s-1 - ws l=1 y (s) l , s, Φ e is (2.1)
where e j is the column vector of size r having all entries equal to 0 except the jth which is equal to 1, and e ′ j is the transpose of e j . Following Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF], we let

ST L n (Φ) = n q=1 L 1 (Z q , Φ)
be the split-time likelihood (STL) of the observations. If the random variables Z q , q ≥ 1 were independent, the STL would be the total likelihood of the observations. While these variables are actually not independent, the sequence (Z q ) q≥1 possesses regenerative properties, which implies in particular that it possesses independence properties given a suitable sequence of increasing random times; this is the key idea to the proofs of our main results (see the proof of Theorem 2). An MSTL is then any parameter that maximizes the STL, or equivalently the log-STL

log ST L n (Φ) = n q=1 log L 1 (Z q , Φ)
and an MSTLE is any estimator of an MSTL.

Asymptotic results

We shall write Φ ∼ Φ ′ whenever the distributions of Z under P Φ and P Φ ′ agree. The identifiability of the parameter Φ, or equivalently the determination of the equivalence class of Φ modulo ∼, is a difficult question; for the case of an MMPP, this was discussed in Rydén [START_REF] Rydén | On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions[END_REF]. In the special case when there is a season for which the jump intensities of N are distinct, which shall be the case we consider in our simulation study and real data analysis, this problem does however possess a simple solution:

Proposition 1. Let Φ ∈ E.
Assume that for some s ∈ {1, . . . , k}, the matrix Λ (s) (Φ) possesses distinct diagonal elements. Then the equivalence class of Φ modulo ∼ reduces to those parameters Φ ′ obtained by permutation of the states of the underlying Markov process J.

Proof of Proposition 1. Let Φ ′ be such that Φ ∼ Φ ′ and (J, N ) (resp. (J ′ , N ′ )) have distribution P Φ (resp. P Φ ′ ). By stationarity, the restriction of the process (J, N ) (resp. (J ′ , N ′ )) to the time interval [τ s-1 , τ s ] actually has the same distribution as the restriction of an MMPP

(J , N ) (resp. (J ′ , N ′ )) with parameters (L(Φ), Λ (s) (Φ)) (resp. (L(Φ ′ ), Λ (s) (Φ ′ ))) to the time interval [0, τ s -τ s-1 ].
Using again the stationarity property of the underlying Markov process, it is then enough to show that these two processes are deduced from one another (in the distributional sense) by a permutation of the states. Applying Corollary 1 in Rydén [START_REF] Rydén | On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions[END_REF] completes the proof.

Roughly speaking, in this case, the strong identifiability constraint on season s carries over, in some sense, to the whole process by the stationarity property of J. Back to the general case, the following consistency result holds:

Theorem 1. Let Φ 0 ∈ E be the true value of the parameter and Φ 0 be the equivalence class of Φ 0 modulo ∼. Let K be a compact subset of E such that Φ 0 ∈ K and Φ n be the MSTLE for

Φ 0 on K, computed over n periods. Then if O ⊂ K is an open set containing K ∩ Φ 0 , one has Φ n ∈ O almost surely for n large enough.
Proof of Theorem 1. The proof is similar to that of Theorem 1 in Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF]: let

Φ ∈ E be such that Φ ∼ Φ 0 and G Φ be a neighborhood of Φ as in Lemma 2. If B(Φ, 1/q) denotes the
Euclidean open ball in E with center Φ and radius 1/q, the continuity of the map ϕ → L 1 (Z, ϕ)

yields sup ϕ∈GΦ∩B(Φ,1/q) log L 1 (Z, ϕ) → log L 1 (Z, Φ) as q → ∞.
Noticing that

sup ϕ∈GΦ∩B(Φ,1/q) log L 1 (Z, ϕ) ≤ sup ϕ∈GΦ log L 1 (Z, ϕ) + | log L 1 (Z, Φ)|
the dominated convergence theorem implies

E Φ0 sup ϕ∈GΦ∩B(Φ,1/q) log L 1 (Z, ϕ) → E Φ0 [log L 1 (Z, Φ)] as q → ∞. (3.1)
Since Φ ∼ Φ 0 , the information inequality (see Rao [START_REF] Rao | Linear statistical inference and its applications[END_REF]) gives

E Φ0 [log L 1 (Z, Φ)] + 2ε < E Φ0 [log L 1 (Z, Φ 0 )] (3.2)
for some ε > 0. It is thus a consequence of (3.1) and (3.2) that there exists a (possibly different)

neighborhood G Φ of Φ with E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ) ≤ E Φ0 [log L 1 (Z, Φ 0 )] -ε. (3.3)
Besides, since (Z q ) q≥1 is ergodic,

1 n sup ϕ∈GΦ log ST L n (ϕ) ≤ 1 n n q=1 sup ϕ∈GΦ log L 1 (Z q , ϕ) → E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ) and 1 n log ST L n (Φ 0 ) = 1 n n q=1 log L 1 (Z q , Φ 0 ) → E Φ0 [log L 1 (Z, Φ 0 )] almost surely as n → ∞. Hence lim sup n→∞ 1 n sup ϕ∈GΦ log ST L n (ϕ) ≤ E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ) ≤ E Φ0 [log L 1 (Z, Φ 0 )] -ε almost surely as n → ∞, by (3.3). Finally, remark that the compact set O c ∩ K, where O c is the complement of O, may be covered by a finite number of such neighborhoods G Φi , 1 ≤ i ≤ d; this yields sup ϕ∈O c ∩K {log ST L n (ϕ) -log ST L n (Φ 0 )} ≤ max 1≤i≤d sup ϕ∈GΦ i log ST L n (ϕ) -log ST L n (Φ 0 )
which tends to -∞ almost surely as n → ∞. As a consequence, necessarily Φ n ∈ O for n large enough, which completes the proof.

We now wish to obtain an asymptotic normality result for our estimator. In what follows, we pick i 0 ∈ {1, . . . , r} and we let ω k be the successive times when the Markov chain (J(q)) reaches i 0 :

ω 1 = min{q ≥ 1 | J(q) = i 0 } and ∀k ≥ 1, ω k+1 = min{q > ω k | J(q) = i 0 }. Let further P i0 Φ (•) = P Φ (• | J(0) = i 0 )
be the probability measure deduced from P Φ given that J starts at i 0 .

Note that for every Φ ∈ E, since L(Φ) is the generator of an irreducible continuous-time Markov chain on a finite state space, then 0 is an eigenvalue of the transpose L ′ (Φ) of L(Φ) with multiplicity 1 and related normalized eigenvector a ′ (Φ). Since the map ϕ → L ′ (ϕ) is infinitely continuously differentiable in a neighborhood of Φ, a straightforward extension of Theorem 8 in Chapter 9 of Lax [START_REF] Lax | Linear Algebra and its applications[END_REF] shows that the map ϕ → a ′ (ϕ) is infinitely continuously differentiable in a neighborhood of Φ. The function ϕ → log L 1 (Z, ϕ) is thus infinitely continuously differentiable in a neighborhood of the true parameter Φ 0 ; if E i0 Φ denotes the expectation under the measure P i0 Φ , we can set for Φ close enough to Φ 0

h i (z, Φ) = ∂ log L 1 ∂ϕ i (z, Φ), A ij (Φ) = E i0 Φ ω1 q=1 h i (Z q , Φ)h j (Z q , Φ) , V ij (Φ) = E i0 Φ ω1 p,q=1 h i (Z p , Φ)h j (Z q , Φ) .
We assume that the matrix

A(Φ) = (A ij (Φ)) is invertible for Φ = Φ 0 and we let V (Φ) = (V ij (Φ)), C(Φ) = [a i0 (Φ)] -1 A -1 (Φ)V (Φ)A -1 (Φ) for Φ in a neighborhood of Φ 0 .
Our asymptotic normality result for Φ n is as follows:

Theorem 2. Let Φ 0 ∈ E be the true value of the parameter and Φ 0 be the equivalence class of Φ 0 modulo ∼. Assume that Φ 0 lies in the interior of E and let K be a compact subset of E,

whose interior contains Φ 0 , such that K ∩ Φ 0 = {Φ 0 }. Let Φ n be the MSTLE computed on K over n periods. Then √ n( Φ n -Φ 0 ) d -→ N (0, C(Φ 0 )) as n → ∞.
Proof of Theorem 2. We start as in the proof of Theorem 4 in Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF]. Condition

K ∩ Φ 0 = {Φ 0 } ensures that Φ n → Φ 0 almost surely as n → ∞.
Especially, with probability 1, one has Φ n ∈ K for n large enough by Theorem 1. For such n, a Taylor expansion of the i-th partial derivative of Φ → log ST L n (Φ) at Φ 0 implies that

T 1,n := 1 √ n n q=1 h i (Z q , Φ 0 ) = T 2,n + T 3,n (3.4)
where

                   T 2,n := - |E| j=1 √ n( Φ n,j -Φ 0,j ) 1 n n q=1 ∂ 2 log L 1 ∂ϕ j ∂ϕ i (Z q , Φ 0 ) T 3,n := - 1 2 |E| j=1 √ n( Φ n,j -Φ 0,j )   |E| k=1 ( Φ n,k -Φ 0,k ) 1 n n q=1 ∂ 3 log L 1 ∂ϕ k ∂ϕ j ∂ϕ i (Z q , Φ n )   ,
with Φ n some point on the line connecting Φ 0 and Φ n . We start by dealing with the righthand side of equality (3.4). To this end, use Lemma 4 and the fact that the process (Z q ) q≥1 is ergodic to obtain

T 2,n = - |E| j=1 √ n( Φ n,j -Φ 0,j )E Φ0 ∂ 2 log L 1 ∂ϕ j ∂ϕ i (Z, Φ 0 ) (1 + o P (1)).
Moreover, since

∂ 2 log L 1 ∂ϕ j ∂ϕ i = 1 L 1 ∂ 2 L 1 ∂ϕ j ∂ϕ i - 1 L 2 1 ∂L 1 ∂ϕ i ∂L 1 ∂ϕ j ,
it is a consequence of Lemma 4 and of a differentiation under the expectation sign that

T 2,n = |E| j=1 √ n( Φ n,j -Φ 0,j )E Φ0 (h i (Z, Φ 0 )h j (Z, Φ 0 ))(1 + o P (1)) = a i0 (Φ 0 ) |E| j=1 √ n( Φ n,j -Φ 0,j )E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 )h j (Z q , Φ 0 ) (1 + o P (1)), (3.5)
by Lemma 6. Besides, Lemma 4, the ergodicity of (Z q ) q≥1 and the consistency of Φ n entail

T 3,n = o P   |E| j=1 √ n( Φ n,j -Φ 0,j )   (3.6)
as n → ∞. Collecting (3.5) and (3.6), we obtain

T 2,n + T 3,n = a i0 (Φ 0 ) |E| j=1 √ n( Φ n,j -Φ 0,j )E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 )h j (Z q , Φ 0 ) (1 + o P (1)) (3.7)
as n → ∞. We now examine the asymptotic properties of T 1,n : notice that (Z q ) q≥1 is regenerative with associated cycle lengths (C q = ω qω q-1 ) q≥1 (where we set ω 0 = 0 for convenience), that is:

• for each l ≥ 2, the random process (C l+1+q , Z ω l +q ) q≥1 is independent of the ω j , 1 ≤ j ≤ l -1 and its distribution does not depend on l;

• given (ω q ) q≥1 , q ≥ 1, the random vectors Z j , ω q-1 ≤ j ≤ ω q -1 are independent.

We may thus apply a law of large numbers similar to the one proposed in Theorem 2 in Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF] (see also Asmussen [START_REF] Asmussen | Applied probability and queues[END_REF]): this leads to

1 √ n T 1,n = 1 n n q=1 h i (Z q , Φ 0 ) P -→ a i0 (Φ 0 )E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 ) as n → ∞
Besides, thanks to Lemma 4, we may once again differentiate under the integral sign to obtain E Φ0 (h i (Z, Φ 0 )) = 0. Lemma 4 and the ergodicity of (Z q ) q≥1 thus yield

1 √ n T 1,n = 1 n n q=1 h i (Z q , Φ 0 ) → 0 almost surely as n → ∞.
Combining these last two convergences entails

E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 ) = 0. Finally E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 )
2 < ∞, see Lemmas 4 and 5; applying a central limit theorem similar to the one proposed in Theorem 3 in Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF] then gives The interesting point about MSTLEs is that confidence regions for Φ 0 may be computed fairly easily, see Lemma 7: it is enough to compute the MSTLE Φ n with a numerical method, to generate random copies of the process under the law P Φn and to apply the strong law of large numbers to get an approximation of C( Φ n ) and hence of C(Φ 0 ).

T 1,n = 1 √ n n q=1 h i (Z q , Φ 0 ) -a i0 (Φ 0 )E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 ) d -→ N (0, a i0 (Φ 0 )V (Φ 0 )) as n → ∞. ( 3
We close this section by noting that Lemma 6 makes it possible to give further insight on the non-singularity of A(Φ 0 ). If A(Φ 0 ) were singular then, by Lemma 6, we would have

∃j ∈ {1, . . . , |E|}, ∀i ∈ {1, . . . , |E|}, E Φ0 (h i (Z, Φ 0 )h j (Z, Φ 0 )) = |E| ℓ=1 ℓ =j γ ℓ E Φ0 (h i (Z, Φ 0 )h ℓ (Z, Φ 0 ))
where the γ ℓ are real constants. It is then straightforward that

E Φ0      hj(Z, Φ 0 ) - |E| ℓ=1 ℓ =j γ ℓ h ℓ (Z, Φ 0 )    2    = 0
so that the partial derivatives of log L 1 (•, Φ 0 ) are linearly dependent. The structure of the log-likelihood (over one period) being highly non-linear, one may therefore expect that A(Φ 0 ) will be invertible in the vast majority of cases.

4 Simulation study and real data analysis

Practical implementation

The MSTLE is typically computed using a (quasi-)Newton algorithm. Such an algorithm being iterative, we give a method to start it in practice. Let w q,s be the number of jumps of N during season s of the period q, t (q,s) 1

, . . . , t (q,s) wq,s be the successive jump times of N during this season and let u

(q,s) l = t (q,s) l -τ s-1 -(q -1)(1 -(τ s -τ s-1 )), 1 ≤ l ≤ w q,s .
For all 1 ≤ s ≤ k, we treat the data D s = (u (q,s) l ) q,l as if it were as a sample of a univariate MMPP with parameters

Φ (s) = (L, λ (s) 1 , . . . , λ (s) r
). An EM algorithm is then used to provide a first estimate of Φ (s) ; the initial estimation procedure is a straightforward adaptation of the one in Guillou et al. [START_REF] Guillou | Estimation of the parameters of a Markov-modulated loss process in insurance[END_REF].

Let ( L (s) , λ (s) 1 , . . . , λ (s)
r ) be the EM estimations obtained with this procedure. The EM algorithms involved may have switched some of the parameters: we thus note, if M = (m ij ) is a square matrix having size r and σ is a permutation of {1, . . . , r}, M • σ = (m σ(i)σ(j) ). Let, for such permutations σ 1 , . . . , σ k ,

L (σ1,...,σ k ) = 1 k k s=1 L (s) • σ s and Φ (σ1,...,σ k ) = ( L (σ1,...,σ k ) , ( λ (s)
σs(1) , . . . , λ

σs(r) ) 1≤s≤k ).

We then run the optimization algorithm starting from each of these values: in doing so, we obtain estimates Φ (σ1,...,σ k ) of the parameters. Finally, we compute

(σ 0 1 , . . . , σ 0 k ) = arg max σ1,...,σ k ST L n ( Φ (σ1,...,σ k ) )
and our MSTLE is Φ = Φ (σ 0 1 ,...,σ 0 k ) .

Simulation study

In this section, we examine how our estimator behaves on a finite sample situation. We are motivated by climate-related statistical problems such as the impact of the El Niño-La Niña cycle on hurricane risk. We choose reasonable parameters for a simple illustration, but cannot claim that those parameters are estimated on a real data set, simply because it is very hard to understand the impact of this effect on hurricane frequency. We consider the situation r = 2, that is, the underlying Markov process J is a two-state continuous-time Markov process. We further choose k = 2 and τ 1 = 1/2, so that any period is divided into two seasons of equal length. The following cases are considered:

• Case 1:

ℓ 12 = 1, ℓ 21 = 2, λ (1)
1 = 1, λ (1)
2 = 5, λ (2)
1 = 5, λ (2)
2 = 25.

• Case 2:

ℓ 12 = 3, ℓ 21 = 10, λ (1)
1 = 1, λ (1)
2 = 5, λ (2)
1 = 5, λ (2)
2 = 25.

• Case 3:

ℓ 12 = 1, ℓ 21 = 2, λ (1)
1 = 1/2, λ (1)
2 = 5, λ (2)
1 = 5/2, λ (2)
2 = 25.
For each of these cases, we consider the situations n = 50 and n = 100. Our estimation procedure is carried out on S = 100 replications of the considered process. In each case, we compute the mean and median L 1 -error related to each parameter. We also computed asymptotic 95% confidence intervals (ACI) with the help of Theorem 2 and Lemma 7; we display the mean and median half-length of these intervals as well as the estimated coverage probabilities in Tables 123.

Following the intuition, the jump intensities are estimated more accurately when the ratio of the claim frequencies in different states is further from 1. This is because identifying the hidden states of J is easier when both regimes of the process are further apart, and thus more information on the process can be recovered in such cases. Besides, estimating the elements of L is harder than estimating the jump intensities since J is unobserved, which is confirmed by our results. Coverage probabilities are globally satisfying; although they are fairly low in case 2 for n = 50, they improve to reasonable levels for n = 100.

From this simulation study, one can see that in practice, if an insurance company had around 50 or 100 years of data, which would be an ideal situation, for intensities like the ones we chose (as far as the El Niño-La Niña-related problem on hurricane risk is concerned, 5 times as many hurricanes during the hurricane season as in the other one, and 5 times as many hurricanes during a bad El Niño-La Niña phase as in a favorable one), it is feasible to estimate the parameters with an average relative error of around 10 to 20%. Of course, we have taken here a quite favorable case, as we have only two states of the environment and two seasons with fixed intensities. Considering 5 or more states and 4 seasons would require a period of observation too long for the approach to be reasonable. However, in most hidden Markov models in insurance and finance, the number of states is 2 or 3. In the near future, one could probably refine these estimations thanks to partial observations of the environment process, as climatologists understand better and better the El Niño-La Niña phenomenon and its impact on insurance perils.

Real data analysis

In this section, we focus on extreme climate events (large bushfires, cyclones, floods, hail, storms, tornados, tsunamis) experienced by Australia from March 1967 to March 2012. During this timeframe, 193 such extreme events were observed. Their frequency is largely influenced by a seasonality effect; the first "season" we consider is fall-winter, which in Australia roughly corresponds to the end of March until the end of September, and the second one is springsummer, namely from the beginning of October to mid-March. Moreover, extreme climate events tend to be influenced by external effects such as the El Niño-La Niña phenomenon or even global phenomena like climate change, which motivates the use of our technique on this data set. Our unit of time is the standard calendar year; our data is represented in Fig. 1 and our estimates are given in Table 4. A reconstruction of the underlying process J, carried out by a straightforward adaptation of the Viterbi algorithm (see Viterbi [START_REF] Viterbi | Error bounds for convolutional codes and an asymptotically optimum decoding algorithm[END_REF]) to our case, is given in Fig. 2.

The algorithm seems to identify an underlying process J with very small transition rates. A consequence of this is that the confidence intervals for the elements of L are poor. This is not the case however for the jump intensities, and the algorithm confirms that in a given state of J, the intensities in both seasons are significantly different, thus emphasizing the importance of taking into account the seasonality effect. Furthermore, the frequency of extreme climate events is significantly higher in the spring-summer season, which was expected at least as far as bushfires, cyclones and storms were concerned. We underline that, due to the fact that there are very few transitions of J, the Markov assumption on this process is questionable; nevertheless, as already observed in Guillou et al. [START_REF] Guillou | Estimation of the parameters of a Markov-modulated loss process in insurance[END_REF] on real sets of non-life and life insurance data, the algorithm correctly matches a point where the slope of N sharply increases with a breakpoint in the data set where the hidden process switches from state 2 to state 1, thus leading to higher frequencies of jumps of the counting process. We thus believe that our approach is superior to a naive model using seasonal versions of several different Poisson processes since such an approach would not be helpful in identifying potential unknown breakpoints in the global structure of the model. It is also superior to the use of a classical MMPP approach as in Guillou et al. [START_REF] Guillou | Estimation of the parameters of a Markov-modulated loss process in insurance[END_REF] which would not be able to handle the seasonal components of the data.

Conclusion and discussion

In this paper, we introduce an MMPP featuring seasonality for which we discuss estimation issues when the environment process is unobserved. After establishing the main asymptotic properties of the resulting estimator, we illustrate its performance on a small simulation study in which we observe that the number of states is crucial. We conclude our numerical section by a real data analysis on extreme climate events in Australia on which our developed methodology seems to perform well. In the real data study, the Markov assumption is of course questionable, as well as the absence of impact of climate change on the parameters. Extending the number of states in order to get a Markov process is feasible in theory, but would lead to estimation problems that would not be tractable in practice. Another problem, when considering climaterelated problems such as the motivating hurricane risk example, is that in different states of the environment, the seasons may be longer or shorter, as conditions are more or less met for events to happen. This is not considered here and would require further theoretical analysis; linked to this concern, if one had further details about the events and access to meteorological data, it would of course be relevant to distinguish between categories of extreme events and to carry out a more detailed analysis. Having said that, even if ENSO values were available, their measure would not be precise enough and their link with claim intensity and frequency would still be too loose to use credibility adjustments at this stage, although it is likely to be possible in the near future. Finally, note that we focus here on frequency of extreme events, but of course for the insurer the claim amounts are important. Nevertheless, our approach accounts for periods of time when there are significantly more events, which is very important for reinsurance and risk management purposes.

Appendix: technical lemmas and proofs

The first result is a necessary step to obtain the strong consistency of our estimator.

Lemma 1. For every ϕ ∈ E and z = (w s , y

(s) 1 , . . . , y (s)
ws ) 1≤s≤k , one has

m(rm) k+2 (rm) w1+•••+w k ≤ L 1 (z, ϕ) ≤ M (rM ) k+2 (rM ) w1+•••+w k
where

M := max 1, max j,s λ (s) j (ϕ) < ∞ and if K s := [0, τ s -τ s-1 ], m := min min i a i (ϕ), min i,j,s min y∈Ks f ij (y, s, Φ), min i,j,s min y∈Ks F ij (y, s, Φ) > 0.
Proof of Lemma 1. Start by remarking that

F αβ (y, s, ϕ) = [exp(y(L(ϕ) -Λ (s) (ϕ)))] αβ = P(J(y) = β, N (s) (y) = 0 | J(0) = α) ≤ 1 when (J, N (s)
) is an MMPP with transition intensity matrix L(ϕ) and jump intensity matrix Λ (s) (ϕ), see Meier-Hellstern [START_REF] Meier-Hellstern | A fitting algorithm for Markov-modulated Poisson processes having two arrival rates[END_REF]. Therefore, for every y ≥ 0, one has F αβ (y, s, ϕ) ≤ M and f αβ (y, s, ϕ) ≤ M . Using (2.1), we immediately obtain

L 1 (z, ϕ) ≤ M (rM ) k+2 (rM ) w1+•••+w k .
Besides, the compactness of K s and the continuity of the maps involved entail min i,j,s min y∈Ks f ij (y, s, Φ) > 0 and min i,j,s min y∈Ks F ij (y, s, Φ) > 0 so that m > 0. Hence, using (2.1), the inequality

L 1 (z, ϕ) ≥ m(rm) k+2 (rm) w1+•••+w k which completes the proof.
A second pivotal tool is the following technical lemma:

Lemma 2. For every Φ ∈ E, there exists a neighborhood G Φ of Φ in E such that

E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ) < ∞.
Proof of Lemma 2. Let G Φ be a neighborhood of Φ, A = sup ϕ∈GΦ L 1 (Z, ϕ) ≤ 1 and write

E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ) = E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ)1l A c -E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ)1l A ≤ E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ)1l A c -E Φ0 [log L 1 (Z, Φ)1l A ] . (B.1)
The goal is to show that the quantity in the right-hand side of this inequality is finite for

a suitable choice of the neighborhood G Φ of Φ in E. z = (w s , y (s) 1 , . . . , y (s)
ws ) 1≤s≤k and G Φ be a compact neighborhood of Φ such that ϕ → a(ϕ) is continuous on G Φ and M := sup ϕ∈GΦ max 1, max j,s λ (s) j (ϕ) < ∞. Since the logarithm function is increasing, using Lemma 1 we deduce that

E Φ0 sup ϕ∈GΦ log L 1 (Z, ϕ)1l A c ≤ log M + k + 2 + k s=1 E Φ0 (W s ) log rM < ∞ (B.2)
since E Φ0 (W s ) < ∞ for all s. Furthermore, using Lemma 1, we have

-E Φ0 [log L 1 (Z, Φ)1l A ] ≤ | log m| + k + 2 + k s=1 E Φ0 (W s ) | log rm| < ∞. (B.3)
for some m > 0. Using together (B.1), (B.2) and (B.3) concludes the proof.

The next result shows that for every i 0 ∈ {1, . . . , r}, the survival function of the random time

ω 1 = min{q ≥ 1 | J(q) = i 0 } converges to 0 geometrically fast.
Lemma 3. For every i 0 ∈ {1, . . . , r}, there exists a neighborhood G of Φ 0 and a constant c ∈ (0, 1) such that ∀k ∈ N, sup

Φ∈G P i0 Φ (ω 1 > k) ≤ c k . In particular, E i0 Φ0 (ω k 1 ) < ∞ for every k ≥ 1.
Proof of Lemma 3. The result is obviously true for k = 0. Pick k ≥ 1 and note that

P i0 Φ (ω 1 > k) = j =i0 P i0 Φ (J(1) = i 0 , . . . , J(k -2) = i 0 , J(k -1) = j, J(k) = i 0 ) = j =i0 P i0 Φ (J(1) = i 0 , . . . , J(k -2) = i 0 , J(k -1) = j)P j Φ (J(1) = i 0 ). (B.4)
Set, for i, j ∈ {1, . . . , r}, P ij (Φ) :

= P i Φ (J(1) = j) = [exp(L(Φ))] ij .
In particular, the maps Φ → P ij (Φ) are continuous. Moreover, since the Markov process J is irreducible, it holds that P ij (Φ) > 0 for all i and j. Consequently

0 < c = max j =i0 sup Φ∈G P j Φ (J(1) = i 0 ) < 1.
Using (B.4) entails

sup Φ∈G P i0 Φ (ω 1 > k) ≤ c sup Φ∈G P i0 Φ (ω 1 > k -1)
which gives the desired result by induction on k.

A key element in the proof of Theorem 2 is to show the following technical result:

Lemma 4. There exists a neighborhood G of Φ 0 in E and positive constants C, C ′ such that for any i, j, ℓ:

sup ϕ∈G max 1 L 1 (Z, ϕ) , L 1 (Z, ϕ), ∂L 1 ∂ϕ i (Z, ϕ) , ∂ 2 L 1 ∂ϕ i ∂ϕ j (Z, ϕ) , ∂ 3 L 1 ∂ϕ i ∂ϕ j ∂ϕ ℓ (Z, ϕ) ≤ C exp C ′ k s=1 W s .
Especially, there exist (possibly different) positive constants C, C ′ such that for any i, j, ℓ:

sup ϕ∈G max |log L 1 (Z, ϕ)| , ∂ log L 1 ∂ϕ i (Z, ϕ) , ∂ 2 log L 1 ∂ϕ i ∂ϕ j (Z, ϕ) , ∂ 3 log L 1 ∂ϕ i ∂ϕ j ∂ϕ ℓ (Z, ϕ) ≤ C exp C ′ k s=1 W s ,
the right-hand side of the above inequality defines a random variable with finite P Φ0 -moments and for all i 1 , i and j,

E i1 Φ0 sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)| < ∞. Proof of Lemma 4. Assume that G is a neighborhood of Φ 0 such that ϕ → a(ϕ) is continuous on G and inf ϕ∈G min j,s λ (s) j (ϕ) > 0. If K s = [0, τ s -τ s-1
] then for any ϕ ∈ G, the ideas of the proof of Lemma 1 yield

m := inf ϕ∈G min i a i (ϕ), min y∈Ks min i,j,s f ij (y, s, Φ), min y∈Ks min i,j,s F ij (y, s, Φ) > 0.
For all ϕ ∈ G and every z = (w s , y

(s) 1 , . . . , y (s) ws ) 1≤s≤k , Lemma 1 entails L 1 (z, ϕ) ≥ m(rm) k+2 (rm) w1+•••+w k . (B.5) If moreover G is a neighborhood of Φ 0 with sup ϕ∈G max i,j ℓ ij (ϕ) < ∞
max y∈Ks ∂ 3 f np ∂ϕ ℓ ∂ϕ q ∂ϕ r (y, s, ϕ) , ∂ 3 F np ∂ϕ ℓ ∂ϕ q ∂ϕ r (y, s, ϕ)
and note that M = max{M 1 , M 2 , M 3 , M 4 } < ∞. Equation (2.1), inequality (B.5) and tedious computations may then be used to show that one may find positive constants C, C ′ such that

sup ϕ∈G max 1 L 1 (Z, ϕ) , L 1 (Z, ϕ), ∂L 1 ∂ϕ i (Z, ϕ) , ∂ 2 L 1 ∂ϕ i ∂ϕ j (Z, ϕ) , ∂ 3 L 1 ∂ϕ i ∂ϕ j ∂ϕ ℓ (Z, ϕ) ≤ C exp C ′ k s=1 W s .
Consequently

sup ϕ∈G max |log L 1 (Z, ϕ)| , ∂ log L 1 ∂ϕ i (Z, ϕ) , ∂ 2 log L 1 ∂ϕ i ∂ϕ j (Z, ϕ) , ∂ 3 log L 1 ∂ϕ i ∂ϕ j ∂ϕ ℓ (Z, ϕ) ≤ C exp C ′ k s=1 W s ,
for (possibly different) positive constants C, C ′ . Finally, since for every s ∈ {1, . . . , k}, W s is a Poisson distributed random variable, one has E Φ0 (x Ws ) < ∞ for all s and x > 0, which proves that the right-hand side of the inequality above defines a random variable having finite P Φ0 -moments. The second part of the lemma is thus a straightforward consequence of the above inequality.

The next result shall be used to check a couple of integrability conditions used in the proof of Theorem 2.

Lemma 5. Assume that ψ is a Borel measurable nonnegative function. We consider the random variable U = ω1 q=1 ψ(Z q ).

• If E Φ0 (ψ(Z)) < ∞ then for every i 0 ∈ {1, . . . , r}, E i0 Φ0 (U ) < ∞.

• If E Φ0 (ψ 2 (Z)) < ∞ then for every i 0 ∈ {1, . . . , r}, E i0 Φ0 (U 2 ) < ∞.

Proof of Lemma 5. We start by remarking that for any l ≥ 1, if E Φ0 (ψ l (Z)) < ∞ then for any i 1 :

E i1 Φ0 |ψ l (Z)| ≤ E Φ0 |ψ l (Z)| a i1 (Φ 0 ) < ∞ since a i1 (Φ 0 ) > 0.
To prove the first statement, we then write

E i0 Φ0 (U ) = ∞ N =1 E i0 Φ0 N q=1 ψ(Z q )1l {ω1=N } = ∞ q=1 E i0 Φ0 (ψ(Z q )1l {ω1≥q} ).
Since {ω 1 ≥ q} = q-1 l=0 {J(l) = i 0 }, the hidden Markov structure of (Z q ) yields

E i0 Φ0 (U ) = ∞ q=1 i1 =i0 E i1 Φ0 (ψ(Z))P i0 Φ0 (ω 1 ≥ q, J(q) = i 1 ) ≤ E i0 Φ0 (ω 1 ) max i1 E i1 Φ0 (ψ(Z)).
Since the Markov chain (J(q)) is irreducible on {1, . . . , r} and has stationary distribution a(Φ 0 ), one has E i0 Φ0 (ω 1 ) = 1/a i0 (Φ 0 ) < ∞, from which we deduce that the right-hand side is finite.

We now turn to the second part of the lemma. Note that

E i0 Φ0 (U 2 ) = ∞ N =1 E i0 Φ0 N p,q=1 ψ(Z p )ψ(Z q )1l {ω1=N } = ∞ N =1 N q=1 E i0 Φ0 ψ 2 (Z q )1l {ω1=N } + 2 ∞ N =1 E i0 Φ0    N p,q=1 p<q ψ(Z p )ψ(Z q )1l {ω1=N }    .
We already know from the first statement of the lemma that

∞ N =1 N q=1 E i0 Φ0 (ψ 2 (Z q )1l {ω1=N } ) = E i0 Φ0 ω1 q=1 ψ 2 (Z q ) < ∞.
Further, for all Borel nonnegative functions f 1 , . . . , f N and all i 1 , . . . , i N , the hidden Markov structure of (Z q ) entails

E i0 Φ0 f 1 (Z 1 ) • • • f N (Z N )1l {J(1)=i1,..., J(N )=iN } = N q=1 E i0 Φ0 (f q (Z q ) | J(q -1) = i q-1 , J(q) = i q ) P i0 Φ0 N q=1 {J(q) = i q } = N q=1 E iq-1 Φ0 (f q (Z 1 ) | J(1) = i q ) P i0 Φ0 N q=1 {J(q) = i q } .
Since we may write

{ω 1 = N } =   j1,...,jN-1 =i0 N -1 q=1 {J(q) = j q }   ∩ {J(N ) = i 0 } it is straightforward that ∞ N =1 E i0 Φ0    N p,q=1 p<q ψ(Z p )ψ(Z q )1l {ω1=N }    ≤ max i1,i2 E i1 Φ0 (ψ(Z) | J(1) = i 2 ) 2 E i0 Φ0 (ω 2 1 )
which is finite since E i0 Φ0 (ω 2 1 ) < ∞, see Lemma 3.

Lemma 6 is the key to the conclusion of the proof of Theorem 2.

Lemma 6. Assume that ψ is a Borel measurable function such that

E Φ0 |ψ(Z)| < ∞. Then E Φ0 (ψ(Z)) = a i0 (Φ 0 )E i0 Φ0 ω1 q=1 ψ(Z q ) .
Proof of Lemma 6. The ergodicity of the process (Z q ) q≥1 entails

1 n n q=1 ψ(Z q ) → E Φ0 (ψ(Z)) almost surely as n → ∞.
Besides, Lemma 5 yields E| ω1 q=1 ψ(Z q )| < ∞, so that the regenerative properties of (Z q ) and a law of large numbers similar to the one proposed in Theorem 2 in Rydén [START_REF] Rydén | Consistent and asymptotically normal parameter estimates for Markov modulated Poisson processes[END_REF] give

1 n n q=1 ψ(Z q ) P -→ a i0 (Φ 0 )E i0 Φ0 ω1 q=1 ψ(Z q )
as n → ∞ from which the result follows.

Lemma 7 below is the essential result needed to construct confidence regions for the estimator.

Lemma 7. If Φ n → Φ 0 as n → ∞, one has C(Φ n ) → C(Φ 0 ) as n → ∞.
Proof of Lemma 7. We shall show that if Φ n → Φ 0 , then A(Φ n ) → A(Φ 0 ) as n → ∞. Let n be so large that Φ n belongs to a (compact) neighborhood G of Φ 0 as in Lemma 4. Using the definition of A, one has:

|A ij (Φ n ) -A ij (Φ 0 )| ≤ E i0 Φn ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) -E i0 Φ0 ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) + E i0 Φ0 ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) -E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 )h j (Z q , Φ 0 ) .
Applying Lemma 6 and differentiating under the expectation sign yields for any

Φ ∈ G E i0 Φ0 ω1 q=1 h i (Z q , Φ)h j (Z q , Φ) = 1 a i0 (Φ 0 ) E Φ0 - ∂ 2 log L 1 ∂ϕ i ∂ϕ j (Z, Φ) .
Lemma 4 and the dominated convergence theorem thus entail

E i0 Φ0 ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) -E i0 Φ0 ω1 q=1 h i (Z q , Φ 0 )h j (Z q , Φ 0 ) → 0 as n → ∞.
Afterwards, notice that for every Φ ∈ E

E i0 Φ ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) = i1 =i0 ∞ q=1 E i1 Φ (h i (Z, Φ n )h j (Z, Φ n ))P i0 Φ (ω 1 ≥ q, J(q -1) = i 1 )
so that

E i0 Φn ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) -E i0 Φ0 ω1 q=1 h i (Z q , Φ n )h j (Z q , Φ n ) ≤ r 1,n + r 2,n with r 1,n = C i1 =i0 q≥1 P i0 Φn (ω 1 ≥ q, J(q -1) = i 1 ) -P i0 Φ0 (ω 1 ≥ q, J(q -1) = i 1 ) and r 2,n = max i1 E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )) q≥1 P i0 Φn (ω 1 ≥ q)
where we set

C = max i1 E i1 Φ0 sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|
which is finite by Lemma 4. We start by controlling r 1,n . Set P ij (Φ) = P i Φ (J(1) = j) as in the proof of Lemma 3. For all q ≥ 1 and j = i 0 ,

P i0

Φn (ω 1 ≥ q, J(q -1) = j) = i1,...,iq-2 =i0

P i0,i1 (Φ n ) • • • P iq-2,j (Φ n ) → i1,...,iq-2 =i0 P i0,i1 (Φ 0 ) • • • P iq-2,j (Φ 0 ) as n → ∞ = P i0 Φ0 (ω 1 ≥ q, J(q -1) = j).
Besides, Lemma 3 entails that there exists a constant c ∈ (0, 1) such that for all j = i 0 and q ≥ 1:

sup n≥1 |P i0 Φn (ω 1 ≥ q, J(q -1) = j) -P i0 Φ0 (ω 1 ≥ q, J(q -1) = j)| ≤ 2c q-1
so that the dominated convergence theorem gives r 1,n → 0 as n → ∞. To control r 2,n , use

Lemma 3 to get for all n r 2,n ≤ 1 1 -c max i1 E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )) .
To prove that the right-hand side of this inequality converges to 0 as n → ∞, pick p ≥ 1 and let W = k s=1 W s be the total number of events over one period. For any i 1 , it holds that

E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )) ≤ E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )1l { W ≤p} ) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )1l { W ≤p} ) + E i1 Φn sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} + E i1 Φ0 sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} .
Since the state space of J is finite, it is enough to prove that the right-hand side of this inequality converges to 0 as n → ∞. Using Lemma 4, we may find positive constants

C 1 , C 2 such that E i1 Φn sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} + E i1 Φ0 sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} ≤ C 1 E i1 Φn (exp(C 2 W )1l { W >p} ) + E i1 Φ0 (exp(C 2 W )1l { W >p} ) . Pick now ε > 0. Since sup ϕ∈G max i,j λ (i) j (ϕ) < ∞, one has sup Φ∈G E i1 Φ (C W 3 exp(C 4 W )) < ∞.
Consequently, one can choose p ∈ N \ {0} such that for every n

E i1

Φn sup

ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} + E i1 Φ0 sup ϕ∈G |h i (Z, ϕ)h j (Z, ϕ)|1l { W >p} ≤ 2ε/3.
It is then enough to prove that for all fixed q ≥ 0,

E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )1l { W =q} ) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )1l { W =q} ) → 0
as n → ∞. Note that Z may be written Z = (W, Y ) where W = (W 1 , . . . , W k ) and Y is the list of inter-event times in year one. Therefore, if L 1 (w, y, Φ 0 | i 1 ) is the likelihood of (w, y)

given that J(0) = i 1 , we may write By Lemma 4 and the irreducibility of J, there exists a positive constant C q such that 2 sup ϕ∈G |h i (w, y, ϕ)h j (w, y, ϕ)|1l { w =q} sup ϕ∈G L 1 (w, y, ϕ | i 1 ) ≤ C q .

E i1 Φn (h i (Z, Φ n )h j (Z, Φ n )1l { W =q} ) -E i1 Φ0 (h i (Z, Φ n )h j (Z, Φ n )1l { W =q} ) ≤ w =q
Finally, for every t > 0, the simplex (y 1 , . . . , y ws ) ws i=1 y i ≤ t and ∀i, y i ≥ 0 is contained in the hypercube ws i=1 {y i | 0 ≤ y i ≤ t}, whose volume is t ws < ∞. The dominated convergence theorem consequently gives A(Φ n ) → A(Φ 0 ) as n → ∞. The convergence V (Φ n ) → V (Φ 0 ) as n → ∞ is obtained in the same way.

. 8 )

 8 Collecting (3.4), (3.7) and (3.8) concludes the proof.

  sup ϕ∈G |h i (w, y, ϕ)h j (w, y, ϕ)|1l { w =q} |L 1 (w, y, Φ n | i 1 ) -L 1 (w, y, Φ 0 | i 1 )|dywhere, for a given w = (w 1 , . . . , w k ), the integral in the right-hand side is on the y ws ) 1≤s≤k whose components are nonnegative and are such that for every s ∈ {1, . . . , k}, ws l=1 y (s) l ≤ τ sτ s-1 . Since L 1 is a continuous function of the parameters, the integrand goes to 0 everywhere as n → ∞. Furthermore, this function is bounded from above by the function (w, y) → 2 sup ϕ∈G |h i (w, y, ϕ)h j (w, y, ϕ)|1l { w =q} sup ϕ∈G L 1 (w, y, ϕ | i 1 ).

Figure 1 :Figure 2 :

 12 Figure 1: Counting process N of our real data set.

  and sup

									ϕ∈G	max j,s	λ (s) j (ϕ) < ∞
	then similarly
	M 1 := sup ϕ∈G	max i	a i (ϕ), max i,j,s	max y∈Ks	f ij (y, s, Φ), max i,j,s	max y∈Ks	F ij (y, s, Φ) < ∞.
	Let further
	M 2 = sup ϕ∈G	max n,p,s	max ℓ	max y∈Ks	∂f np ∂ϕ ℓ	(y, s, ϕ) ,	∂F np ∂ϕ ℓ	(y, s, ϕ) ,
	M 3 = sup ϕ∈G	max n,p,s	max ℓ,q	max y∈Ks	∂ 2 f np ∂ϕ ℓ ∂ϕ q	(y, s, ϕ) ,	∂ 2 F np ∂ϕ ℓ ∂ϕ q	(y, s, ϕ)
	and M 4 = sup ϕ∈G	max n,p,s	max ℓ,q,r

Table 1 :

 1 Mean and median L 1 -errors associated to the estimators in case 1.

		ℓ 12	ℓ 21	λ (1) 1	λ (1) 2	λ (2) 1	(2) λ 2
	Case 1
	n = 50
	Mean L 1 -error Median L 1 -error	0.303 0.683 0.381 0.866 0.774 2.506 0.247 0.544 0.239 0.586 0.708 1.893
	Mean half-length of the 95% ACI	0.818 1.740 0.791 2.177 1.611 5.380
	Median half-length of the 95% ACI	0.766 0.837 0.886 2.082 1.570 5.184
	Coverage probabilities of the 95% ACI	0.93	0.93	0.92	0.95	0.92	0.88
	n = 100
	Mean L 1 -error Median L 1 -error	0.307 0.547 0.248 0.657 0.487 1.629 0.234 0.423 0.176 0.510 0.384 1.259
	Mean half-length of the 95% ACI	0.632 1.208 0.561 1.529 1.194 3.704
	Median half-length of the 95% ACI	0.613 1.097 0.551 1.460 1.155 3.582
	Coverage probabilities of the 95% ACI	0.88	0.92	0.94	0.91	0.93	0.94

Table 2 :

 2 Mean and median L 1 -errors associated to the estimators in case 2.

		ℓ 12	ℓ 21	λ (1) 1	λ (1) 2	λ (2) 1	(2) λ 2
	Case 2
	n = 50
	Mean L 1 -error Median L 1 -error	1.593 4.228 0.692 2.448 1.207 5.341 1.372 3.504 0.472 1.773 1.030 4.182
	Mean half-length of the 95% ACI	3.528 9.221 1.219 4.549 2.510 11.258
	Median half-length of the 95% ACI	3.036 7.936 1.034 3.682 2.362 10.473
	Coverage probabilities of the 95% ACI	0.79	0.83	0.84	0.79	0.86	0.91
	n = 100
	Mean L 1 -error Median L 1 -error	1.304 3.145 0.483 1.715 0.847 3.599 0.816 2.184 0.262 1.272 0.654 2.980
	Mean half-length of the 95% ACI	2.893 7.323 0.882 3.115 1.875 8.269
	Median half-length of the 95% ACI	2.429 6.393 0.764 2.788 1.749 7.860
	Coverage probabilities of the 95% ACI	0.9	0.95	0.86	0.87	0.92	0.93

Table 3 :

 3 Mean and median L 1 -errors associated to the estimators in case 3.

		ℓ 12	ℓ 21	λ (1) 1	λ (1) 2	λ (2) 1	(2) λ 2
	Case 3
	n = 50
	Mean L 1 -error Median L 1 -error	0.272 0.551 0.255 0.812 0.370 1.671 0.213 0.490 0.197 0.611 0.289 1.435
	Mean half-length of the 95% ACI	0.715 1.487 0.593 1.972 1.073 4.707
	Median half-length of the 95% ACI	0.720 1.452 0.594 1.963 1.059 4.709
	Coverage probabilities of the 95% ACI	0.94	0.96	0.91	0.92	0.98	0.96
	n = 100
	Mean L 1 -error Median L 1 -error	0.197 0.410 0.194 0.568 0.298 1.392 0.169 0.332 0.173 0.505 0.238 1.200
	Mean half-length of the 95% ACI	0.479 0.978 0.421 1.386 0.757 3.240
	Median half-length of the 95% ACI	0.476 0.951 0.416 1.371 0.753 3.221
	Coverage probabilities of the 95% ACI	0.91	0.92	0.9	0.9	0.96	0.96

Table 4 :

 4 Estimates of the parameters on our real data set.

	Parameters Estimates	95% ACI
	ℓ 12	0.0771	[-1.162, 1.316]
	ℓ 21	0.0230	[-0.345, 0.391]
	λ (1) 1	5.434	[2.329, 8.540]
	λ (1) 2	0.762	[0.142, 1.381]
	λ (2) 1	10.659	[6.567, 14.750]
	λ (2) 2	5.361	[3.945, 6.776]

Acknowledgments

The authors thank the associate editor and two anonymous referees for their helpful comments which led to significant improvements of this article.


Christophe Révillion

Thibault Catry

Monitoring and mapping forest fires in the south-western Indian Ocean using very high resolution optical imagery

come

Monitoring and mapping forest fires in the southwestern Indian Ocean using very high resolution optical imagery.

Christophe REVILLION (1) , Thibault CATRY (1) SEAS-OI, UMR 228 Espace-Dev (IRD, UM2, UAG, UR), 40 Avenue de Soweto, CS70651, 97447 Saint-Pierre Cedex, La Réunion, France 4) This study was part of the operational mission of the SEAS-OI station, gathering local actors in the management of forest fire crisis in the south-western Indian Ocean. In particular, we demonstrate here the interest of the satellite approach in the management of natural disasters on remote territories like Europa island, and in protected area where a rapid response is needed. Thanks to the combination of optical SPOT 5 and PLEAIDES high resolution products, SEAS-OI was able to provide operational products for the management of these fires, and the monitoring of .

1) The South-western Indian Ocean is characterized by various heterogeneous insular territories. Many of these islands are biodiversity hotspots and in order to preserve this heritage, some of them are protected.

www.seas-oi.org

Contacts : christophe.revillion@ird.fr, contact@seas-oi.org, http://www.espace.ird.fr/

  burned area Barren land area after the fire Recovery in progress by the wet vegetation Extent of the fire on the St-Paul pond Burned area on Europa island The Study areas, Europa and La Reunion (A) and focus on Saint-Paul (B) 3) The monitoring of the post-fire recovery was realized using Pléiades images. Very high spatial resolution images allowed us to understand more precisely how different plant groupings recovered after the fire. On the pond of Saint-Paul vegetation dominated by Cyperus Papyrus (in dark green on map) is almost back to its prefire condition in six months, however areas with Typha domingensis began to regenerate but much more slowly. 08/27/2012 01/05/2014 2) In 2013, the monitoring and mapping of the fires of Etang-Saint-Paul (July, La Reunion) and Europa (June) were carried out by the SEAS-OI (Survey of Environment Assisted by Satellites in the Indian Ocean), in collaboration with the ONF (Office National des Forêts) of La Reunion and the SDIS (Service Départemental d'Incendie et de Secours), using high resolution SPOT 5 imagery. These maps allowed the estimation of the fire extent, to about 4 and 70 hectares for Europa and Etang-Saint-Paul respectively, and to support field intervention during the crisis. 2013), Distribution Astrium services / Spot images S.A., France, all rights reserved.


Martha Hajiw

Alain Valtz

Elise El Ahmar

Christophe Coquelet
email: christophe.coquelet@mines-paristech.fr

APPARENT HENRY'S LAW CONSTANTS OF FURAN IN DIFFERENT N-ALKANES AND ALCOHOLS AT TEMPERATURES FROM 293 TO 323 K

Keywords: furan, alkane solvent, alcohol solvent, apparent Henry's law constant

Apparent Henry's law constants of furan in different n-alkanes and alcohols at temperatures from 293 to 323 K

INTRODUCTION

With the growing population and the economical development of non-OECD nations (ie. outside the Organization for Economic Cooperation and Development), the world energy consumption is expected to increase by 48% until 2040 [START_REF]International Energy Outlook 2016 -With Projections to 2040[END_REF]. To fulfil the demand, mankind is facing various challenges: depletion of easily accessible oil fields, exploitation of deepwater or unconventional fields, carbon dioxide emissions. Therefore, part of research is now dedicated to the development of high value compounds. The most promising raw material is lignocellulose, which is, according to [START_REF] Lange | Furfural A Promising Platform for Lignocellulosic Biofuels[END_REF] [START_REF] Lange | Furfural A Promising Platform for Lignocellulosic Biofuels[END_REF], cheaper, more abundant and sustainable. It comes from plant biomass (wood residues, cereal straw and forestry waste), which is made up of lignin and cellulose [START_REF] Harmsen | Literature Review of Physical and Chemical Pretreatment Processes for Lignocellulosic Biomass[END_REF]. After pyrolysis of lignocellulose, a pyroligneous liquid is obtained, containing numerous oxygenated molecules such as aromatics, esters, phenols.... The thermodynamic behaviour of oxygenated molecules is not perfectly predicted because thermodynamic models and equations of state used in process design traditionally come from oil and gas engineering, and thus fail for these compounds. Indeed, strong association interactions like hydrogen bonding or dimerization are not taken into account in oil and gas engineering models. Thus, it is important to have accurate predictive models with limited parameters to describe their behaviour, since it is of a great importance for the chemical industry especially for the design of new production processes from lignocellulosic material [START_REF] Lugo | Molecular-Based Predictive Models Applied to High Value Oxygen-Bearing Molecules and their Mixtures[END_REF]. For this purpose, the MEMOBIOL project (Molecular Scale Modeling for Lignocellulosic Biorefineries financed by the French National Research Agency (ANR-09-CP2D-10-04)) has been done in order to evaluate different predictive tools for calculating the thermodynamic phase equilibria and properties required by the chemical industry. The CTP was one of the academic partners involved in this project, in which new Vapour-Liquid and Liquid-Liquid equilibrium experimental data of different binary and ternary systems containing high added-value oxygenated molecules and different solvents have been determined ( [START_REF] Nala | Vapour-Liquid Equilibrium (VLE) for the Systems Furan plus n-Hexane and Furan plus Toluene. Measurements, Data Treatment and Modeling Using Molecular Models[END_REF][START_REF] Auger | Equilibrium Data and GC-PC-SAFT Predictions for Furanic Extraction[END_REF])

In this paper, we focused on furan as a derived molecule obtained from the pyrolysis of the lignocellulosic biomass. It intervenes in the acid-catalyzed hydrolysis of xylan-type hemicelluloses [START_REF] Besson | Conversion of Biomass into Chemicals over Metal Catalysts[END_REF]. One application is the production of the Tetrahydrofuran (THF) solvent, widely used in chemical and technological processes [START_REF] Alkhaldi | Densities, Ultrasonic Speeds and Refractive Indices of Phenetole with N-Methyl-2-Pyrrolidone, N,N-Dimethylformamide and Tetrahydrofuran Binary Mixtures at Different Temperatures[END_REF]. It can be also used as a precursor for the synthesis of pharmaceutical molecules ( [START_REF] Joule | Heterocyclic Chemistry[END_REF]). Apparent Henry's law constants of furan in different alkane and alcohol solvents have been measured using a gas inert stripping method. Activity coefficients at infinite dilution have been deduced. These properties are useful to evaluate the deviation from ideality and then to determine the most suitable solvent of extraction.

EXPERIMENTAL SECTION

Chemicals

Chemicals used in this study are presented in Table 1.

Equipment

Apparent Henry's Law Constant and Infinite Dilution Activity Coefficient determination was conducted with the gas stripping method using a specially designed apparatus using dilutor and saturator cell. The experimental procedure is fully described by Richon et al. [START_REF] Richon | Infinite Dilution Activity-Coefficients of Linear and Branched Alkanes from C 1 to C 9 in N-Hexadecane by Inert-Gas Stripping[END_REF][START_REF] Richon | Infinite Dilution Henry Constants of Light-Hydrocarbons in Normal-Hexadecane, Normal-Octadecane, and 2,2,4,4,6,8,8-Heptamethylnonane by Inert-Gas Stripping[END_REF] and Krummen et al. [START_REF] Krummen | Measurement of Activity Coefficients at Infinite Dilution in Solvent Mixtures Using the Dilutor Technique[END_REF]. Both cells are inside a liquid bath regulated within 0.01 K. A platinum probe, in contact with the liquid phase of the "dilutor cell" is connected to an electronic display. Temperature accuracy after probe calibration is below 0.1 K. The composition of the gas leaving the cell is periodically sampled and analysed using gas chromatograph (PERICHROM model PR2100, France) equipped with a flame ionization detector connected to a data acquisition system (Winilab III). The analytical column was: PORAPAK Q 80/100 MESH (1/8" Silcosteel tube, length: 1.2 m, ID: 2 mm, Restek, France).

The flow meter (Analyt-MTC, Messtechnik GmbH & co, model 358) was calibrated using bubble flow meter with the carrier gas (Helium). The apparent Henry's law coefficient, H, of solute i can be calculated with Equation 1, obtained from mass balance on the dilutor cell.

𝐻 𝑖 = - 1 𝑡 𝑙𝑛 𝑆 𝑖 𝑆 𝑖 𝑡=0 𝑅𝑇𝑁 𝐷 1 - 𝑃 𝑠𝑜𝑙𝑣 𝑠𝑎𝑡 𝑃 + 𝑉 𝐺 𝑡 𝑙𝑛 𝑆 𝑖 𝑆 𝑖 𝑡=0 (1)
Parameters of Equation 1 are given in Table 2. Vapour pressure (in Pa) has been calculated with the DIPPR equation n°101 (Equation 2) with the corresponding parameters (Table 3).

𝑙𝑛 𝑃 𝑠𝑎𝑡 = 𝐴 + 𝐵 𝑇 + 𝐶𝑙𝑛 𝑇 + 𝐷𝑇 𝐸 (2)
The .

EXPERIMENTAL RESULTS AND DISCUSSION

Apparent Henry's law constant and activity coefficient at infinite dilution

Apparent Henry's law constant is directly determined using the gas stripping method as previously described. The symmetric convention links the apparent Henry's law constant with the activity coefficients at infinite dilution (Equation 5), assuming that the Poynting factor and the fugacity coefficient have a value of approximately unity [START_REF] Scilipoti | Selection of Solvents with A-UNIFAC Applied to Detoxification of Aqueous Solutions[END_REF].

𝛾 𝑖 ∞ = 𝐻 𝑖 𝑇, 𝑃 𝑠𝑜𝑙𝑣 𝑠𝑎𝑡 𝑃 𝑖 𝑠𝑎𝑡 (5)
Experimental data of measurements in n-alkanes are tabulated in Table 4 and the ones in alcohols in Table 5.

To study the temperature dependence, the logarithm of the activity coefficient at infinite dilution of furan in some solvents is drawn as a function of inverse temperature (Figure 1).

From Tables 4 and5 and Figure 1, the following observations can be made:

(a) the limiting activity coefficients values are a decreasing function with the temperature, except for ethanol (close to a monotonic behaviour);

(b) a nonlinear behaviour of ln(γ i ∞ )versus 1/T is observed for n-pentane and n-hexane;

(c) for both alcohols and alkanes (except n-pentane), the γ i ∞ values decrease with increasing number of carbon atoms. A hypothesis for n-pentane behaviour would the difference of volatility between all hydrocarbons and its boiling temperature close to the one of furan;

(d) limiting activity coefficients of furan in alkanes are closer to one (value of ideal mixture) than in alcohols: alkanes would be better solvents for furan than alcohols.

(e) n-pentane has the lowest activity coefficient value at infinite dilution.

From these data, n-pentane seems to be the better solvent to extract furan at temperatures below 300 K.

Chemical engineering criteria

In chemical engineering, different criteria, calculated with activity coefficient at infinite dilution, can help to select a solvent [START_REF] Scilipoti | Selection of Solvents with A-UNIFAC Applied to Detoxification of Aqueous Solutions[END_REF]. One may mention distribution coefficient or selectivity. In our study, we can calculate the solvating power (S p ). It is the criterion representing the ability of a solvent to form strong secondary bonds with solute molecules. It can be calculated with Equation 5.

𝑆 𝑝 = 1 𝛾 𝐴,𝑠𝑜𝑙𝑣 ∞ 𝑀 𝑠𝑜𝑙𝑣 𝑀 𝐴 (5)
where M solv and M A are the molar mass of the solvent and the solute respectively.

Solvating power values of the different solvents presented in this paper and toluene are drawn on Figure 2 at 298K. Based on Figure 2 and previous conclusions, toluene, and to a lesser extent n-pentane, seem to be appropriate solvents for furan extraction.

Phase equilibria data

Vapour-Liquid and Vapour-Vapour-Liquid Equilibrium data have been previously measured for some furan-solvent binary systems (toluene, n-hexane, ethanol and n-octanol). It has been shown that the binary toluene-furan system presents a Raoult's law behaviour [START_REF] Nala | Vapour-Liquid Equilibrium (VLE) for the Systems Furan plus n-Hexane and Furan plus Toluene. Measurements, Data Treatment and Modeling Using Molecular Models[END_REF][START_REF] Auger | Equilibrium Data and GC-PC-SAFT Predictions for Furanic Extraction[END_REF]. This ideal behaviour has been confirmed with excess enthalpy [START_REF] El Ahmar | Experimental and Predicted Excess Molar Volumes and Enthalpies of Binary Systems Containing Furan[END_REF] and excess volume [START_REF] Nala | Vapour-Liquid Equilibrium (VLE) for the Systems Furan plus n-Hexane and Furan plus Toluene. Measurements, Data Treatment and Modeling Using Molecular Models[END_REF] measurements. From these data, two conclusions can be drawn: toluene seems to be the better solvent to extract furan and apparent Henry's law constant measurements appears to be an appropriate setup in terms of time consumption, costs and ease of use.

However, only apparent Henry's law constants do not provide any information about the probability to have an azeotrope. Since no data are available for the n-pentane-furan system in the open literature, has been predicted at atmospheric pressure using the original UNIFAC model [START_REF] Fredenslund | Group-Contribution Estimation of Activity-Coefficients in Nonideal Liquid-Mixtures[END_REF]. As shown on Figure 3, this system has an azeotrope.

Original UNIFAC is also used to predict phase diagram of furan+n-hexane and furan+toluene binary systems at atmospheric pressure. This model seems to predict correctly both phase diagrams (Figure 3). Therefore, even if the limiting activity coefficient analysis presents npentane as a potential solvent for furan extraction, from a process point of view, it is not suitable since it would be difficult to separate the solute and the solvent using distillation column.

CONCLUSION

In this study, apparent Henry's law constant of furan in eight normal alkane (C 5 to C 10 , C 12 and C 16 ) and eight (primary and secondary) alcohol solvents have been measured using the inert gas stripping method. To the best of our knowledge, no apparent Henry's law constant data are available for furan in the open literature. From these data, activity coefficients at infinite dilution have been calculated. If referring to Henry's law constants, n-pentane seems to be an appropriate solvent for furan extraction. However, VLE data are necessary to confirm or not this assumption. Indeed, VLE prediction with the UNIFAC models shows that the npentane-furan binary system presents an azeotrope. Therefore, more thermophysical properties (such as VLE, partition coefficient of the solute/solvent system, density and surface tension) are needed to choose the adequate solvent for furan extraction.

  maximal relative uncertainty of the Apparent Henry's law coefficient U(H i , k=2)/H i is estimated to be 4.7 %. It takes into account the uncertainties related to the flow, solute peak area determination, temperature, pressure and number of moles of the solvent. Equation3reminds the mathematical expression used to calculate the uncertainties. Uncertainties on repeatability (u rep ) are calculated considering the measurements done with at least three different flow values and are about 2%. Equation 4 details the method for the determination of the uncertainty on calculation (u cal ).

			𝑢 𝐻 𝑖 = 𝑢 𝑟𝑒𝑝 𝐻 𝑖	2 + 𝑢 𝑐𝑎𝑙 𝐻 𝑖	2
									(3)
	𝑢 𝑐𝑎𝑙 𝐻 𝑖
	=	𝜕𝐻 𝑖 𝜕𝛼 𝑃,𝐷,𝑇,𝑁,𝑉 𝐺 2	𝑢 𝛼 2 + +	𝜕𝐻 𝑖 𝜕𝑃 𝛼 ,𝐷,𝑇,𝑁,𝑉 𝐺 2 𝜕𝐻 𝑖 𝜕𝑁 𝛼,𝐷,𝑇,𝑃,𝑉 𝐺 𝑢 𝑃 2 + 2 𝑢 𝑁 2 +	𝜕𝐻 𝑖 𝜕𝐷 𝛼,𝑃,𝑇,𝑁,𝑉 𝐺 2 2 𝜕𝐻 𝑖 𝜕𝑉 𝐺 𝛼,𝐷,𝑇,𝑁,𝑃	𝑢 𝐷 2 + 𝑢 𝑉 𝐺 2	2 𝜕𝑇 𝛼 ,𝐷,𝑃,𝑁,𝑉 𝐺 𝜕𝐻 𝑖	𝑢 𝑇 2
									(4)
	with α the slope of the curve -	1 𝑡	𝑙𝑛	𝑆 𝑖 𝑆 𝑖 𝑡=0

ACKNOWLEDGMENTS

The ANR through the project Memobiol (ANR-09-CP2D-10-04 MEMOBIOL) is gratefully acknowledged for the financial support.