F R Éd

Éric Lavancier
email: frederic.lavancier@univ-nantes.fr

Arnaud Poinas
email: arnaud.poinas@univ-rennes1.fr

Rasmus Waagepetersen

Adaptive estimating function inference for non-stationary determinantal point processes

Keywords: asymptotic normality, determinantal point processes, estimating functions, joint intensities, non-stationary, repulsive

published or not. The documents may come

Adaptive estimating function inference for non-stationary determinantal point processes

Frédéric Lavancier, Arnaud Poinas, Rasmus Waagepetersen

Introduction

A common feature of spatial point process models (except for the Poisson process case) is that the likelihood function is not available in a simple form. Numerical approximations of the likelihood function are available [see e.g. 12, 13, for reviews] but the approaches are often computationally demanding and the distributional properties of the approximate maximum likelihood estimates may be difficult to assess. Therefore much work has focused on establishing computationally simple estimation methods that do not require knowledge of the likelihood function.

1

In this paper we focus on estimation methods for point processes which have known joint intensity functions. This includes many cases of Cox and cluster point process models [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF][START_REF] Illian | Statistical analysis and modelling of spatial point patterns[END_REF]1] as well as determinantal point processes [START_REF] Macchi | The coincidence approach to stochastic point processes[END_REF][START_REF] Soshnikov | Determinantal random point fields[END_REF][START_REF] Shirai | Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties[END_REF][START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF]. These classes of models are quite different since realizations of Cox and cluster point processes are aggregated while determinantal point processes produce regular point pattern realizations.

Knowledge of an nth order joint intensity enables the use of the so-called Campbell formulae for computing expectations of statistics given by random sums indexed by ntuples of distinct points in a point process. Unbiased estimating functions can then be constructed from such statistics by subtracting their expectations. So far mainly the cases of first and second order joint intensities have been considered where the first order joint intensity is simply the intensity function. However, consideration of higher order estimating functions may be worthwhile to obtain more precise estimators or to identify parameters in complex point process models.

Theoretical results have been established in a variety of special cases of first and second order estimating functions for Cox and cluster processes [START_REF] Schoenberg | Consistent parametric estimation of the intensity of a spatial-temporal point process[END_REF][START_REF] Guan | A composite likelihood approach in fitting spatial point process models[END_REF][START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous Neyman-Scott processes[END_REF][START_REF] Guan | A thinned block bootstrap procedure for modeling inhomogeneous spatial point patterns[END_REF][START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF] and for the closely related Palm likelihood estimators [START_REF] Tanaka | Parameter estimation and model selection for Neyman-Scott point processes[END_REF][START_REF] Prokešová | Asymptotic Palm likelihood theory imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018 for stationary point processes[END_REF][START_REF] Prokešová | Two-step estimation procedures for inhomogeneous shot-noise Cox processes[END_REF]. The common general structure of the estimating functions on the other hand calls for a general theoretical set-up which is the first contribution of this paper. Our set-up also covers third or higher order estimating functions and combinations of such estimating functions.

The literature on statistical inference for determinantal point processes is quite limited with theoretical results so far only available in case of minimum contrast estimation for stationary determinantal point processes [START_REF] Biscio | Contrast estimation for parametric stationary determinantal point processes[END_REF]. Based on the general set-up our second main contribution is to provide a detailed theoretical study of estimating function estimators for general non-stationary determinantal point processes.

Specializing to second-order estimating functions, a common approach [START_REF] Guan | A composite likelihood approach in fitting spatial point process models[END_REF][START_REF] Tanaka | Parameter estimation and model selection for Neyman-Scott point processes[END_REF] is to restrict the random sum to pairs of R-close points for some user-specified R ą 0. This may lead to faster computation and improved statistical efficiency. The properties of the resulting estimators depend strongly on R but only ad hoc guidance is available for the choice of R. Moreover, it is difficult to account for ad hoc choices of R when establishing theoretical results. Our third contribution is a simple intuitively appealing adaptive choice of R which leads to a theoretically tractable estimation procedure and we demonstrate its usefulness in simulation studies for determinantal point processes as well as an example of a cluster process.

Estimating functions based on joint intensities

A point process X on R d , d ě 1, is a locally finite random subset of R d . For B Ď R d , we let N pBq denote the random number of points in X X B. That X is locally finite means that N pBq is finite almost surely whenever B is bounded. The so-called joint intensities of a point process are described in Section 2.1. In this paper we mainly focus on determinantal point processes, detailed in Section 3. A prominent feature of determinantal point processes is that they have known joint intensity functions of any order.

imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018

Joint intensity functions and Campbell formulae

For integer n ě 1, the joint intensity ρ pnq of nth order is defined by E ‰ ÿ u1,...,unPX 1 u1PB1,...,unPBn " ż ˆn i"1 Bi ρ pnq pu 1 , . . . , u n qdu 1 ¨¨¨du n (1) for Borel sets B i Ď R d , i " 1, . . . , n, assuming that the left hand side is absolutely continuous with respect to Lebesgue measure on R d . The ‰ over the summation sign means that the sum is over pairwise distinct points in X. Of special interest are the cases n " 1 and n " 2 where the intensity function ρ " ρ p1q and the second order joint intensity ρ p2q determine the first and second order moments of the count variables N pBq, B Ď R d . The pair correlation function gpu, vq is defined as gpu, vq " ρ p2q pu, vq ρpuqρpvq whenever ρpuqρpvq ą 0 (otherwise we define gpu, vq " 0). The product ρpuqgpu, vq can be interpreted as the intensity of X at u given that v P X. Hence gpu, vq ą 1 (ă 1) means that presence of a point at v increases (decreases) the likeliness of observing yet another point at u. The Campbell formula E ‰ ÿ u1,...,unPX f pu 1 , . . . , u n q " ż f pu 1 , . . . , u n qρ pnq pu 1 , . . . , u n qdu 1 ¨¨¨du n follows immediately from the definition of ρ pnq for any non-negative function f : pR d q n Ñ r0, 8r.

A general asymptotic result for estimating functions

Consider a parametric family of distributions tP θ : θ P Θu of point processes on R d , where Θ is a subset of R p . We assume a realization of the point process X with distribution P θ ˚, θ ˚P IntpΘq, is observed on a window W n Ă R d . We estimate the unknown parameter θ by the solution θn of e n pθq " 0 where e n pθq " ¨ř‰ u1,¨¨¨,uq 1 PXXWn f 1 pu 1 , ¨¨¨, u q1 ; θq ´şW q 1 n f 1 pu; θqρ pq1q pu; θqdu . . . ř ‰ u1,¨¨¨,uq l PXXWn f l pu 1 , ¨¨¨, u q l ; θq ´şW q l n f l pu; θqρ pq l q pu; θqdu ‹ ‹ ' for l given functions f i : pR d q qi ˆΘ Ñ R ki such that ř i k i " p.

A basic assumption for the following theorem (verified in Appendix A) is that a central limit theorem is available for e n pθ ˚q (assumption (X3)). In addition to this, a number of technical assumptions (F1) through (F3) (or (F3')), (X1) and (X2) regarding existence and differentiability of joint intensities as well as differentiability of the f i are needed. All the conditions are listed in Appendix A.

Theorem 2.1. Under Assumptions (F1) through (F3) (or (F3')), (X1) and (X2), with a probability tending to one as n Ñ 8, there exists a sequence of roots θn of the estimating equations e n pθq " 0 for which θn

P ÝÑ θ ˚.
Moreover, if (X3) holds true, then

|W n |Σ ´1{2 n H n pθ ˚qp θn ´θ˚q L ÝÑ N p0, I p q.
where Σ n " Varpe n pθ ˚qq, H n pθ ˚q is defined in (F3), and I p is the p ˆp identity matrix.

Second order estimating functions

Referring to the previous section, much attention has been devoted to instances of the case l " 1, q 1 " 2 and k 1 " p. In this case we obtain a second-order estimating function of the form

e n pθq " ‰ ÿ u,vPXXWn f pu, v; θq ´żW 2 n f pu, v; θqρ p2q pu, v; θqdudv. (2)
[5] noted that for computational and statistical efficiency it may be advantageous to use only close pairs of points rather than all pairs of points. Thus in (2) it is common practice to introduce an indicator 1 }u´v}ďR for some constant 0 ă R or choose f so that f pu, vq " 0 whenever }u ´v} ą R. We discuss a method for choosing R in Section 2.4. The general form (2) includes e.g. the score functions of second-order composite likelihood [START_REF] Guan | A composite likelihood approach in fitting spatial point process models[END_REF][START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous Neyman-Scott processes[END_REF] and Palm likelihood functions [START_REF] Tanaka | Parameter estimation and model selection for Neyman-Scott point processes[END_REF][START_REF] Prokešová | Asymptotic Palm likelihood theory imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018 for stationary point processes[END_REF][START_REF] Prokešová | Two-step estimation procedures for inhomogeneous shot-noise Cox processes[END_REF] as well as score functions of minimum contrast object functions based on non-parametric estimates of summary statistics as the K or the pair correlation function. For the second-order composite likelihood of [START_REF] Guan | A composite likelihood approach in fitting spatial point process models[END_REF],

f pu, v; θq " ∇ θ ρ p2q pu, v; θq ρ p2q pu, v; θq ´şW 2 ∇ θ ρ p2q pu, v; θqdudv ş W 2 ρ p2q pu, v; θqdudv while f pu, v; θq " ∇ θ ρ p2q pu, v; θq ρ p2q pu, v; θq
for the second-order composite likelihood proposed in [START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous Neyman-Scott processes[END_REF]. The score of the Palm likelihood as generalized to the inhomogeneous case in [START_REF] Prokešová | Two-step estimation procedures for inhomogeneous shot-noise Cox processes[END_REF] is obtained with [19] also regarded the second-order composite likelihood proposed in [START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous Neyman-Scott processes[END_REF] as a generalization of the stationary case Palm likelihood but the interpretation as a second-order composite likelihood given in [START_REF] Waagepetersen | An estimating function approach to inference for inhomogeneous Neyman-Scott processes[END_REF] is more straightforward. Considering a class of estimating functions of the form (2) a natural question is what is the optimal choice of f ? [START_REF] Deng | Second-order quasi-likelihood for spatial point processes[END_REF] provides a solution to this problem where an approximation of the optimal f is obtained by solving numerically a certain integral equation. This yields a statistically optimal estimation procedure but is computationally demanding and requires specification of third and fourth order joint intensities. When computational speed and ease of use is an issue, there is still scope for simpler methods. Moreover, given several (simple) estimation methods, it is possible to combine them adaptively in order to build a final estimator that achieves better properties than each initial estimator, see [START_REF] Lavancier | A general procedure to combine estimators[END_REF][START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF].

f pu, v; θq " ∇ θ ρ p2q

Adaptive version

Consider second-order composite likelihood using only R close pairs. The weight function f is then of the form

f R pu, v; θq " 1 }u´v}ďR ∇ θ ρ p2q pu, v; θq ρ p2q pu, v; θq . ( 3
)
The performance of the parameter estimates can depend strongly on the chosen R. Simulation studies such as in [START_REF] Prokešová | Two-step estimation procedures for inhomogeneous shot-noise Cox processes[END_REF] and [START_REF] Deng | Second-order quasi-likelihood for spatial point processes[END_REF] usually compare results for several values of R corresponding to different multiples of some parameter associated with 'range of correlation'. For a cluster process this parameter could e.g. be the standard deviation of the distribution for dispersal of offspring around parents. For a determinantal point process the parameter would typically be a correlation scale parameter in the kernel of the determinantal point process, see Section 3. In practice these parameters are not known and among the quantities that need to be estimated. [START_REF] Guan | A composite likelihood approach in fitting spatial point process models[END_REF] suggested to choose an R that minimizes a goodness of fit criterion for the fitted point process model while [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF] suggested to choose R by inspection of a non-parametric estimate of the pair correlation function.

Both approaches imply extra work and ad hoc decisions by the user and it becomes very complex to determine the statistical properties of the resulting parameter estimates. A typical behaviour of many pair correlation functions is that gpu, v; θq converges to a limiting value of 1 when }u ´v} increases and |gpu, v; θq ´1| ď |gpu, u; θq ´1| where the upper bound does not depend on u. If gpu, v; θq " 1 for }u ´v} ą r 0 then counts of points are uncorrelated when they are observed in regions separated by a distance of r 0 . Following the idea that R should depend on some range property of the point process we therefore suggest to replace the constraint }u ´v} ă R in [START_REF] Biscio | Contrast estimation for parametric stationary determinantal point processes[END_REF]

)
where w is some weight function of bounded support r´1, 1s. Later on, when establishing asymptotic results, we will also assume that w is differentiable. A common example of admissible weight function is wprq " e 1{pr 2 ´1q for ´1 ď r ď 1, while wprq " 0 otherwise. The user needs to specify a value of ε but in contrast to the original tuning parameter R, ε has an intuitive meaning independent of the underlying point process. We choose ε " 1%. In the simulation study in Section 4.1 we also consider ε " 5% in order to investigate the sensitivity to the choice of ε.

Asymptotic results for determinantal point processes

A point process X is a determinantal point process (DPP for short) with kernel K : R d ˆRd Ñ R if for all n ě 1, the joint intensity ρ pnq exists and is of the form ρ pnq pu 1 , . . . , u n q " detrKspu 1 , . . . , u n q for all tu 1 , . . . , u n u Ă R d , where rKspu 1 , . . . , u n q is the matrix with entries Kpu i , u j q.

The intensity function is thus ρpuq " Kpu, uq, u P R d . If a determinantal point process with kernel K exists it is unique. General conditions for existence are presented in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF].

In particular, if K admits the form Kpu, vq " a ρpuqρpvqCpu ´vq for a function C : R d Ñ R with Cp0q " 1, then a sufficient condition for existence of a DPP with kernel K is that ρ is bounded and that C is a square integrable continuous covariance function with spectral density bounded by 1{}ρ} 8 . The normalization Cp0q " 1 ensures that ρ is the intensity of the DPP. We now consider a parametric family of DPPs on R d with kernels K θ where θ P Θ and Θ Ď R p [see 8, 2, for examples of such families]. Henceforth, we assume that K θ is symmetric and the DPP with kernel K θ exists for all θ P Θ.

[8] provide an expression for the likelihood of a DPP on a bounded window and discuss likelihood based inference for stationary DPPs. However, the expression depends on a spectral representation of K which is rarely known in practice and must be approximated numerically. Letting n denote the number of observed points, the likelihood further requires the computation of an n ˆn dense matrix which can be time consuming for large n. As an alternative, [2] consider minimum contrast estimation based on the pair correlation function or Ripley's K-function, but only for stationary DPPs. In the following, we consider general non-stationary DPPs and the estimator θn obtained by solving e n pθq " 0 where e n is given by (2).

We establish in Section 3.1 using Theorem 2.1 the asymptotic properties of the estimate θn where e n is given by (2) for a wide class of test functions f . In Section 3.2, we focus on a particular case of the DPP model, where the parameter θ " pβ, ψq can be separated into a parameter β only appearing in the intensity function and a parameter ψ only appearing in the pair correlation function. Following [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF], it is natural to consider a two-step estimation procedure where in a first step β is estimated by a Poisson likelihood score estimating function, and in a second step the remaining parameter ψ is estimated by a second order estimating function as in (2), where β is replaced by βn obtained in the first step. The asymptotic properties of this two-step procedure again follow as a special case of Theorem 2.1.

Second order estimating functions for DPPs

We assume a realization of a DPP X with kernel K θ ˚, θ ˚P IntpΘq, is observed on a window W n Ă R d . We estimate the unknown parameter θ ˚by the solution θn of e n pθq " 0 where e n pθq is given by (2) for a given R p -valued function f . Therefore, we are in a special case of the set-up in Section 2.2 with l " 1, q 1 " 2, k 1 " p and we assume that f 1 " f satisfies the assumptions (F1) through (F3) (or (F3')) listed in Appendix A. The condition (F1) in this case demands that θ Þ Ñ f pu, v; θq is twice continuously differentiable in a neighbourhood of θ ˚and for θ in this neighbourhood, the derivatives are bounded with respect to pu, vq uniformly in θ. Moreover, from (F2), there exists R ą 0 such that for all θ in a neighbourhood of θ ˚, f pu, v; θq " 0 if }u ´v} ą R.

(

) 5
Concerning (F3) (or (F3')), this condition controls the asymptotic behaviour of the matrix H n pθq given by

H n pθq " 1 |W n | ż W 2 n f pu, v; θq∇ θ ρ p2q pu, v; θq T dudv,
where we recall that in this setting

ρ p2q pu, v; θq " K θ pu, uqK θ pv, vq ´Kθ pu, vq 2 . ( 6
)
The assumptions (F3) and (F3') are technical and needed for the identifiability of the estimation procedure. When H n is a symmetric matrix, assumption (F3) seems simpler to verify than (F3'). As an important example, when f is defined as in (4), we prove in Lemmas 3.2 and 3.3 that (F3) is generally satisfied even if X is not stationary.

Finally, as shown in the proof of Theorem 3.1 below, the assumptions (X1) through (X3) in Theorem 2.1 become: (D1) θ Þ Ñ K θ pu, vq is twice continuously differentiable in a neighborhood of θ ˚, for all u, v P R d . Moreover, the first and second derivative of K θ with respect to θ are bounded with respect to u, v P R d uniformly in θ in a neighborhood of θ ˚.

(D2) The kernel K θ ˚satisfies, for some ε ą 0, sup }u´v}ąr K θ ˚pu, vq " opr ´pd`εq{2 q.

(D3) lim inf n λ min p|W n | ´1Σ n q ą 0 where Σ n :" Varpe n pθ ˚qq.

(W) Dε ą 0 s. Let us briefly comment on these assumptions. (D1) is a standard regularity assumption. Condition (D2) is not restrictive since all standard parametric kernel families satisfy sup }u´v}ąr K θ pu, vq " Opr ´pd`1q{2 q, including the most repulsive stationary DPP [see 8 , 2]. Condition (D3) ensures that the asymptotic variance in the central limit theorem below is not degenerated. Finally, Assumption (W) makes specific the fact that W n is not too irregularly shaped and tends to infinity in all directions. It is for instance fulfilled if W n is a Cartesian product of d intervals whose lengths tends to infinity. Theorem 3.1. Under Assumptions (D1) and (D2), if assumptions (F1) through (F3) (or (F3')) are satisfied for f 1 " f , with a probability tending to one as n Ñ 8, there exists a sequence of roots θn of the estimating equations e n pθq " 0 for which

θn P ÝÑ θ ˚.
If moreover (W) and (D3) holds true, then

|W n |Σ ´1{2 n H n pθ ˚qp θn ´θ˚q L ÝÑ N p0, I p q.
Proof. We deduce from ( 6) that (D1) implies (X1). Moreover, it was shown in [START_REF] Poinas | Mixing properties and central limit theorem for associated point processes[END_REF] that (X2) is a consequence of (D2) and that (X3) is a consequence of (D2), (D3) and (W). Thus, we can conclude by applying Theorem (2.1) in the case l " 1 and q 1 " 2.

In the case of a stationary X and f given by (4), the following lemma shows that (F3) is satisfied under mild assumptions that are violated only in degenerate cases. For instance, if p " 1, the main assumption boils down to ∇ θ ρ p2q p0, t; θ ˚q ‰ 0 for some t ‰ 0 such that |K θ ˚ptq| ą ? εK θ ˚p0q. Proof. By definition of w and (D2), there exists R ą 0 such that hptq " 0 when }t} ě R. By Lemma A.1, H n pθ ˚q converges towards the positive semi-definite matrix Hpθ ˚q " ş }t}ăR hptqdt. In this case, proving (F3) is equivalent to showing that φ T Hpθ ˚qφ " 0 only if φ " 0. For this, let A be the set of t such that |K θ ˚ptq| ą ? εK θ ˚p0q, φ P R p and note that since wpεK θ ˚p0q 2 {K θ ˚ptq 2 q ą 0 for t P A and hptq is continuous and positive semi-definite, φ T Hpθ ˚qφ " 0 ô @t P A, φ T hptqφ " 0 ô @t P A, ∇ θ ρ p2q p0, t; θ ˚qT φ " 0 ô φ P ´spant∇ θ ρ p2q p0, t; θ ˚q : t P Au ¯K .

By assumption spant∇ θ ρ p2q p0, t; θ ˚q : t P Au " R p whereby φ " 0, which concludes the proof.

Similarly, we can show that even in the non-stationary case, condition (F3) is satisfied for the function in (4) but under slightly stronger assumptions on ∇ θ ρ p2q pu, v; θ ˚q. Namely, we demand that all functions v Þ Ñ ∇ θ ρ p2q pu, v; θ ˚q are not contained in a single hyperplane of R p nor confined around 0. This is similar in essence to what we have assumed in the previous corollary but with the need of a uniform condition with respect to u. Functions that do not satisfy these requirements are arguably degenerate. Lemma 3.3. Assume (W), (D2) and that K θ ˚is bounded. Let f be as in [START_REF] Deng | Second-order quasi-likelihood for spatial point processes[END_REF]

@v P A, |φ T ∇ θ ρ p2q pu, v; θ ˚q| ą δ then (F3) is satisfied.
Proof. By definition of w, (D2) and the fact that K θ ˚is bounded, there exists R ą 0 such that hpu, vq " 0 when }v ´u} ě R. The integral in (F3) writes By (W), we have

H n pθ ˚q " ż W 2 n hpu,
}ε n } |W n | ď |W n zW aR n | |W n | ż }v´u}ăR sup uPR d }hpu, vq}dv ď |BW n ' R| |W n | ż }v´u}ăR sup uPR d }hpu, vq}dv Ñ 0,
and for all φ,

φ T ˆżW aR n ż Wn hpu, vq1 }u´v}ďR dudv ˙φ " ż W aR n ˜ż}u´v}ďR φ T hpu, vqφdv ¸du.
By our assumption on ∇ θ ρ p2q , there exists a set A of positive Lebesgue measure such that

@v P A, ∇ θ ρ p2q pu, v; θ ˚q P spantφu X Bp0, δq C and w ˆεK θ ˚pu, uqK θ ˚pv, vq K θ ˚pu, vq 2 ˙ą inf xPr0,1{µs
wpxq.

Hence for }φ} " 1,

1 |W n | φ T ˆżW aR n ż Wn hpu,

Two-step estimation for a separable parameter

We consider a family of kernels

K θ pu, vq " a ρpu; βqCpu, v; ψq a ρpv; βq,
where θ :" pβ T , ψ T q T P Θ Ă R p`q with β P R p and ψ P R q , ρp.; βq are non-negative functions, and Cp¨, ¨; ψq are correlation functions, in particular Cpu, u; ψq " 1 for any ψ. Note that in this case the DPP with kernel K θ has intensity ρp.; βq and its pair correlation function is gpu, v; ψq " 1 ´C2 pu, v; ψq.

As in the preceding section, we assume a DPP X with kernel K θ ˚, θ ˚P IntpΘq, is observed on a window W n Ă R d . In the spirit of [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF], we estimate θ ˚in two steps. First, β ˚is estimated as the solution βn of s n pβq " 0 where

s n pβq " ÿ uPXXWn ∇ β ρpu; βq ρpu; βq ´żWn ∇ β ρpu; βqdu
is the score function for a Poisson point process. Then, we estimate ψ ˚by the solution ψn of u n pp βn , ψq " 0 where

u n pθq " ‰ ÿ u,vPXXWn f pu, v; θq ´żW 2 n f pu, v; θqρ p2q pu, v; θqdudv
for a given R q -valued function f and where ρ p2q pu, v; θq " ρ p2q pu, v; β, ψq " ρpu; βqρpv; βqp1Ć 2 pu, v; ψqq in this case. Here and in the following for convenience of notation e.g. identify u n pβ, ψq with u n pθq when θ " pβ T , ψ T q T . This two-step procedure is a particular estimating equation procedure, since θn :" p βT n , ψT n q T is obtained as the solution of e n pθq " 0 where e n pθq " ps n pβq T , u n pβ, ψq T q T . Thus, this is a particular case of the setting in Section 2.2 where l " 2, q 1 " 1, q 2 " 2, f 1 " ∇ β ρpu; βq{ρpu; βq and f 2 " f . We assume in the following theorem the same conditions on the DPP X as in the previous section. Similarly, we assume that (F1) through (F3) (or (F3')) are satisfied for f 1 and f 2 . In this particular case, the matrix H n involved in (F3) simply writes

H n pβ, ψq " ˆH1,1 n pβ, ψq 0 H 2,1 n pβ, ψq H 2,2 n pβ, ψq ẇhere H 1,1 n pβq " 1 |W n | ż Wn ∇ β ρpu; βq∇ β ρpu; βq T ρpu; βq du, H 2,1 n pβ, ψq " 1 |W n | ż W 2 n f pu, v; β, ψq∇ β ρ p2q pu, v; β, ψq T dudv, H 2,2 n pβ, ψq " 1 |W n | ż W 2 n f pu, v; β, ψq∇ ψ ρ p2q pu, v; β, ψq T dudv.
Since it is a non symmetric matrix, condition (F3') is more applicable than (F3). Mild conditions ensuring (F3') in the stationary case are provided in Lemma 3.5. Theorem 3.4. Under Assumptions (D1) and (D2), if assumptions (F1) through (F3) (or (F3')) are satisfied for f 1 " ∇ β ρpu; βq{ρpu; βq and f 2 " f , then with a probability tending to one as n Ñ 8, there exists a sequence of solutions θn " p βT n , ψT n q T to the estimating equation ps n pβq T , u n pβ, ψq T q T " 0 for which

θn P ÝÑ θ ˚.
imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF] If moreover (W) and (D3) hold true, then

|W n |Σ ´1{2 n H n pθ ˚qp θn ´θ˚q L ÝÑ N p0, I p`q q.
Proof. The proof follows the same lines as the proof of Theorem 3.1.

The next lemma is similar to Lemma 3.2. The main technical condition is not restrictive. When q " 1 it boils down to ∇ ψ p1 ´C2 p0, t; ψ ˚qq ‰ 0 for some t such that Cp0, t; ψ ˚q ě ? εCp0, 0; ψ ˚q. If h is integrable at the origin and if spant∇ ψ p1´C 2 p0, t; ψ ˚qq : Cp0, t; ψ ˚q ě ? εCp0, 0; ψ ˚qu " R q , then (F3') is satisfied under (W), (D1) and (D2).

Proof. By definition of w and (D2), there exists R ą 0 such that hptq " 0 when }t} ě R. Since K θ pu, vq and f are invariant by translation then H n pθq converges by Lemma A.1.

In particular, we have

H 1,1 n pβq Ñ 1 β , H 2,2 n pβ, ψq Ñ β 2 ż }t}ďR hpt; ψqdt, H 2,1 n pβ, ψq Ñ 2β ż }t}ďR w ˆε Cp0, 0; ψq 2 Cp0, t; ψq 2 ˙∇ψ p1 ´C2 p0, t; ψqqdt.
The limit of H n pθq is continuous by (D1). In this case, proving (F3') is equivalent to showing that the limit of H n pθ ˚q is invertible. Since this matrix is block triangular and β ą 0 then it is invertible if and only if the limit of H 2,1 n pθ ˚q is invertible. This is done the same way as in Lemma 3.2.

imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018

Simulation study

In this section we use simulation studies to investigate the performance of our adaptive estimating function and to compare two-step estimation with simultaneous estimation.

Performance of adaptive estimating function

In order to assess the adaptive test function (4) against the truncated test function [START_REF] Biscio | Contrast estimation for parametric stationary determinantal point processes[END_REF] with a prescribed R, we consider a DPP model in R 2 with a Bessel-type kernel Kpu, vq " a ρpuqρpvq J 1 p2}u ´v}{αq }u ´v}{α ,

where J 1 denotes the Bessel function of the first kind, ρ is the intensity and α controls the range of interaction of the DPP. For existence, ρ and α must satisfy

α 2 }ρ} 8 ď 1 π . ( 7
)
This relation shows the tradeoff between the expected number of points and the strength of repulsiveness that we can obtain. This model is a particular instance of the Bessel-type DPP introduced in [2]. It covers a large range of repulsiveness, from the Poisson point process (when α is close to 0) to the most repulsive DPP (when α " 1{ a π}ρ} 8 ). For this model, we consider three constant values of ρ, ρ P t50, 100, 1000u, corresponding to homogeneous DPPs, and an inhomogeneous situation where ρpuq " ρpx, yq " 20 expp4xq when u P r0, 1s 2 . The latter case corresponds to a log-linear intensity function involving two parameters. For each ρ, three values of α are considered: a small one, a medium one, and a last one close to the maximal possible value satisfying [START_REF] Illian | Statistical analysis and modelling of spatial point patterns[END_REF]. Examples of point patterns simulated on r0, 1s 2 are displayed in Figure 1. All simulations are carried out using R [START_REF] Core | R: A Language and Environment for Statistical Computing R Foundation for Statistical Computing[END_REF], in particular the library spatstat [1].

We estimate ρ and α by a two-step procedure as studied in Section 3.2 from realisations of the DPP on W " r0, 1s 2 . The alternative global approach of Section 3.1 is discussed in the next section. In the first step, the parameters arising in ρ are estimated by the score function for a Poisson point process. This gives ρ " N pX X W q{|W | in the homogeneous cases. In the second step, we consider the estimating equation based on (3) where θ is α in this setting and when R P t0.05, 0.1, 0.25u, and based on the adaptive test function (4) with ε " 0.01 and the weight function w given at the end of Section 2.4. This yields four different estimators of α. The root mean square errors (RMSEs) of these estimators and the mean computation time estimated from 1000 replications are summarised in Table 1. Boxplots are displayed in Figure S1 in the supplementary material. Note that the codes have not been optimised, but the same computational strategy has been used for all methods, making the comparison of the mean computation time meaningful.

The Bessel-type kernel and the aforementioned test functions used in the two-step estimation procedure fulfill the assumptions of Theorem 3.4 and Lemma 3.5 (for the homogeneous case), ensuring nice asymptotic properties of the estimators considered in this section. This is confirmed by the estimated RMSE's reported in Table 1, that decrease when the intensity ρ increases [which mimics the effect of an increasing window since rescaling the window by a factor 1{k is equivalent to change ρ into k 2 ρ and α into α{k, see (2.4) in 8]. Moreover, these RMSE's show that the best choice of R in the test function [START_REF] Biscio | Contrast estimation for parametric stationary determinantal point processes[END_REF] clearly depends on the range of interaction of the underlying process. This emphasizes the importance of a data-driven approach to choosing R since the range is unknown in practice. Fortunately, the performance of the adaptive method is, except for the case ρ " 100, α " 0.01, always better than the worst choice of R and very close to the best R. For the exceptional case, the small differences in performance can be explained by Monte Carlo error. Further, use of the adaptive method implies only little or no extra computional effort. In presence of many points, the adaptive version is in fact much faster to compute than the estimator based on (3) with the choice of a too large R, see for instance the results for ρ " 1000 and R " 0.25.

Table S2 in the supplementary material shows the root mean square errors of the adaptive estimator using ε " 0.05. The RMSEs obtained with ε " 0.05 are bigger than those obtained with ε " 0.01. Nevertheless, the adaptive method with ε " 0.05 still performs well in the sense that it usually performs better than the worst R and usually almost as good as the best R. Because the above estimation methods sometimes fail to converge, we also report in Table S1 in the supplementary material the percentages of times each method has converged in our simulation study. These percentages are similar for all methods. Note that the results in Table 1 and in Figure S1 are based on 1000 simulations where all four methods have converged.

Two-step versus simultaneous

Most models used in spatial statistics involve a separable parameter θ " pβ, ψq where β only appears in the intensity function and ψ only appears in the pair correlation function. This makes the two-step procedure described in Section 3.2 available, as exploited in the previous simulation study. However a simultaneous second order estimating equation approach might be a better alternative. It is not easy to compare the respective performance of the two approaches through the asymptotic variances obtained in Section 3.1 and Section 3.2. In this section, we show through an example why the two-step procedure seems preferable.

We consider a stationary model with parameter θ " pρ, ψq, where ρ is the intensity and the pair correlation function writes gpu, v; θq " gpr; ψq with r " }u ´v}. In this case the two-step procedure, based on the observation X X W and using the adaptive test function [START_REF] Deng | Second-order quasi-likelihood for spatial point processes[END_REF] q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ " 50, α " 0.02 ρ " 50, α " 0.04 ρ " 50, α " 0.07 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ " 100, α " 0.01 ρ " 100, α " 0.03 ρ " 100, α " 0.05 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ " 1000, α " 0.005 ρ " 1000, α " 0.01 ρ " 1000, α " 0.015 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q ρ " ρpuq, α " 0.005 ρ " ρpuq, α " 0.01 ρ " ρpuq, α " 0.015 1. Estimated root mean square errors (ˆ10 3 ) and mean computation time (in seconds) of α for a Bessel-type DPP on r0, 1s 2 , for different values of ρ and α. The 3 first estimators use the test function (3) with R " 0.05, R " 0.1 and R " 0.25 respectively, while the last estimator is the adaptive version based on (4). The standard errors of the RMSE estimations are given in parenthesis. The last column gives the averages of "practical ranges" (i.e. maximal solution to |gprq ´1| " 0.01) used for the adaptive estimator, along with their standard deviations in parenthesis. For each value of ρ and α, these quantities are computed from 1000 simulations where all four estimation methods have converged.

Here F denotes the cumulative distribution function of R " }U ´V } where U and V are independent variables uniformly distributed on W and tr ij u is the set of all pairwise distances of X X W . On the other hand, by a simultaneous procedure using the same test function, we get that ψ is the root of

. ( 10
)
The more complicated expression of ( 9) in comparison with [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] implies that epψq can be highly irregular in ψ. Figure S2 in the supplementary material shows an example for one realisation of a DPP with a Gaussian kernel with range ψ. For this example epψq exhibits many different roots, although the dataset contains a fairly large number of points (about 1000). The consequence is an extreme sensitivity to the initial parameter when we try to solve epψq " 0. In contrast e 2 pψq " 0 has one clear solution. This advocates the use of the two-step approach.

Due to the aforementioned very strong sensitivity to the initial value of ψ, conclusions from comparison of the simultaneous estimate of ψ with the two-step estimate of ψ can be quite arbitrary. However, we report in Figure S3 in the supplementary material the distribution of estimates of ρ from 1000 simulations of a Bessel-type DPP with ρ " 1000 and ψ " α " 0.01, using either [START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF] from the simultaneous approach or ρ " N pX XW q{|W | from the two-step approach. For the simultaneous method we either chose the true value α " 0.01 as the starting point for the numerical solution of epαq " 0 to get α, or fixed α at the true value, i.e. α " 0.01, in [START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF]. The estimate ρ " N pX X W q{|W | is unequivocally better than [START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF] in terms of root mean square error, even when the true value of α is used for α in [START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF]. This confirms our recommendation.

The simultaneous estimation approach in this example is covered by our theoretical results in Sections 3.1 and 3.2. It shows that while our consistency result guarantees the existence of a consistent sequence of parameter estimates (roots) there could also exist other non-consistent sequences.

Discussion

In this paper we provide a very general asymptotic framework for estimating function inference for spatial point processses with known joint intensities. Specific asymptotic results are obtained for determinantal point processes.

The performance of second order estimating functions depends strongly on a tuning parameter that controls which pairs of points are used in the estimation. Our adaptive choice of this tuning parameter is intuitively appealing, easy to implement and performs well in the simulation studies considered. It moreover seamlessly integrates with the asymptotic results where the use of the adaptive method poses no extra theoretical difficulties. Though we focus in this paper on determinantal point processes, the adaptive method is applicable for any spatial point process with known pair correlation function.

As an example we provide in Section 3 of the supplementary material a simulation study in case of a cluster process.

Appendix A: Assumptions and proof of Theorem 2.1

Our general Theorem 2.1 depends on a number of assumptions. The setting is the same as in Section 2.2. We moreover define diampxq as the largest distance between two coordinates of x. The assumptions (F1) through (F3) are mainly related to the test functions f i , while for X we assume (X1) through (X3). (F1) For all i " 1, . . . , l and for all x P pR d q qi , θ Þ Ñ f i px; θq is twice continuously differentiable in a neighbourhood of θ ˚. Moreover, the first and second derivative of f i with respect to θ are bounded with respect to x P pR d q qi uniformly in θ belonging to this neighbourhood. (F2) There exists a constant R ą 0 such that for all θ in a neighbourhood of θ ˚, all functions x Þ Ñ f i px; θq vanish when diampxq ą R.

Define the matrices H n pθq by

H n pθq " ¨H1 n pθq . . . H l n pθq ‹ ',
where for all i H i n pθq :"

1 |W n | ż W q i n f i px; θq∇ θ ρ pqiq px; θq T dx. (F3) The matrices H n pθ ˚q satisfy lim inf nÑ8 ˆinf }φ}"1 φ T H n pθ ˚qφ ˙ą 0. ( F3
') There exists a neighbourhood of θ ˚such that for all n high enough and all θ in this neighbourhood, H n pθq is invertible and }H n pθq ´1} is uniformly bounded with respect to n and θ, where } ¨} stands for any matrix norm.

(X1) For all θ in a neighbourhood of θ ˚and all q i , i " 1, . . . , l, the intensity functions x Þ Ñ ρ pqiq px; θq are well-defined and bounded. Moreover, θ Þ Ñ ρ pqiq px; θq is twice continuously differentiable in a neighbourhood of θ ˚, for all x P pR d q qi . Finally, the first and second derivative of ρ pqiq with respect to θ are bounded with respect to x P pR d q qi uniformly in θ belonging to this neighbourhood. (X2) For all q i , i " 1, . . . , l, the intensity functions ρ pqiq p¨; θ ˚q, ¨¨¨, ρ p2qiq p¨; θ ˚q of X are well-defined. Moreover, the intensity functions ρ pqiq p¨; θ ˚q, ¨¨¨, ρ p2qi´1q p¨; θ ˚q are bounded and for all bounded sets W Ă R d there exists a constant C 0 ą 0, so that ş W ϕ i px 1 qdx 1 ă C 0 , i " 1, . . . , l where ϕ i is the function

ϕ i : x 1 Þ Ñ sup diampxqăR sup diampyqăR sup y1PW ρ p2qiq px 1 , x 2 , ¨¨¨, x qi , y 1 , ¨¨¨, y qi ; θ ˚q ´ρpqiq px 1 , x 2 , ¨¨¨, x qi ; θ ˚qρ pqiq py 1 , ¨¨¨, y qi ; θ ˚q with R coming from (F2). (X3) X satisfies the central limit theorem Σ ´1{2 n e n pθ ˚q L ÝÑ N p0, I p q,
where e n is defined in Section 2.2 and Σ n " Varpe n pθ ˚qq.

Assumptions (F1) and (F2) are basic regularity conditions on the f i 's. Similarly (X1) and (X2) ensure that the intensity functions of X exist and are sufficiently regular. The technical assumptions are in fact (F3) (or (F3')) and (X3). While the latter strongly depends on the underlying point process (see [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF] for Cox processes and [START_REF] Poinas | Mixing properties and central limit theorem for associated point processes[END_REF] for DPPs), the former can be simplified in some cases. For example, if H n pθ ˚q are symmetrical matrices for all n then (F3) writes lim inf n λ min pH n pθ ˚qq ą 0 where λ min pH n pθ ˚qq denotes the smallest eigenvalue of H n pθ ˚q. If the matrices H n pθ ˚q are not symmetrical, Assumption (F3') will be preferred since (F3) does not translate well for non-symmetrical matrices. Furthermore, if X is stationary and all f i 's are invariant by translation, then H n pθq converges towards a matrix Hpθq explicitly given in Lemma A.1 below, thus Assumption (F3) simply becomes inf }φ}"1 φ T Hpθ ˚qφ ą 0 and (F3') is satisfied whenever Hpθ ˚q is invertible by continuity of Hpθq.

Lemma A.1. Assume (W), (X1), (F2) and let θ P R p . Suppose that all ρ pqiq p¨; θq's and f i p¨; θq's are invariant by translation, i.e. f i pu 1 , u; θq " f i p0, u ´u1 ; θq where we denote by u the vector pu 2 , ¨¨¨, u qi q. If u Þ Ñ f i p0, u; θq is integrable for all i such that q i ě 2, then H n pθq converges to a matrix Hpθq. In particular, for all i we have

lim nÑ8 H i n pθq " ż }t}ďR f i p0, t; θq∇ θ ρ pqiq p0, t; θq T dt.
This lemma is verified in Section B. We now turn to the proof of the theorem. To prove the consistency of θn and get its rate of convergence we apply the following result, where }.} stands for any matrix norm.

Theorem A.2 ([23]

). Suppose that e n pθq is continuously differentiable with respect to θ and define J n pθq :" ´d dθ T e n pθq :" ´ˆB Bθ j e n pθq i ˙1ďi,jďp .

Suppose that for all α ą 0 sup

θPM α n pθ ˚q › › › › 1 |W n | pJ n pθq ´Jn pθ ˚qq › › › › P ÝÑ 0, (11)
where M α n pθ ˚q :"

# θ P Θ : }θ ´θ˚} ď α a |W n | + ,
and suppose that there exists l ą 0 such that

P ˆ1 |W n | inf }φ}"1 φ T J n pθ ˚qφ ă l ˙Ñ 0. ( 12
)
Assume, moreover, that the class of random vectors for a sufficiently large n.

We now verify the assumptions of Theorem A.2. There is no loss in generality by assuming that all f i are symmetric functions. Otherwise we can just replace f i pxq by its symmetrized version pq i !q ´1 ř uPπpxq f i puq where πpxq denotes the set of all vectors obtained by permuting the components of x. This does not change the value of e n pθq and each symmetrized function still satisfies Assumptions (F1) through (F3). We will use at several places the following result.

Lemma A.3. Let X be a point process satisfying Assumption (X2). Consider any i P t1, ¨¨¨, lu, any bounded set W Ă R d , and any symmetric bounded function g : pR d q qi Ñ R ki vanishing when two of its components are at a distance greater than R for a given constant R ą 0. Then Proof. Since g is a symmetric function, then gpx 1 , ¨¨¨, x qi q does not depend on the order of the x i . Thus, for any set of q i points S " tx 1 , ¨¨¨, x qi u, we can write gpSq for the value of g at an arbitrary order of the points in S and we write

› › › › › › Var ¨‰ ÿ x1,¨¨¨,xq i PXXW gpx 1 , ¨¨¨, x qi q '› › › › › › " Op|W
‰ ÿ x1,¨¨¨,xq i PXXW gpx 1 , ¨¨¨, x qi q " q i ! ÿ SĂXXW gpSq1 |S|"qi .
We start by expanding E "`ř

SĂXXW gpSq1 |S|"qi ˘`ř SĂXXW g T pSq1 |S|"qi ˘‰ as qi ÿ k"0 E » - - - ÿ S,T ĂXXW |S|"|T |"qi,|SXT |"k gpSqg T pT q fi ffi ffi fl " qi ÿ k"0 E » - - - ÿ U ĂXXW |U |"2qi´k ÿ S 1 ĂSĂU |S 1 |"k,|S|"qi gpSqg T pS 1 Y pU zSqq fi ffi ffi fl " qi ÿ k"0 1 p2q i ´kq! ż W 2q i ´k ÿ S 1 ĂSĂtx1,¨¨¨,x 2q i ´k u |S 1 |"k,|S|"qi gpSqg T pS 1 Y ptx 1 , ¨¨¨, x 2qi´k uzSqqρ p2qi´kq px; θ ˚qdx " qi ÿ k"0 `qi k ˘`2qi´k qi p2q i ´kq! ż W 2q i ´k gpx 1 , ¨¨¨, x qi qg T px 1 , ¨¨¨, x k , x qi`1 , ¨¨¨, x 2qi´k qρ p2qi´kq px; θ ˚qdx. ( 14
)
By Assumption (X2), the functions ρ pqiq , ¨¨¨, ρ p2qi´1q are all bounded. Moreover, as a consequence of our assumptions on g, each component of each term for k ě 1 in ( 14) is bounded by

1 q i !pq i ´kq! ˆqi k ˙ż W 2q i ´k }g} 2
8 }ρ p2qi´kq } 8 1 t0ď|xi´x1|ďR, @iu dx which is Op|W |q. However, for k " 0, the term is Op|W | 2 q. Instead of controlling this term alone, we consider its difference with the remaining term in the variance we are looking at, that is To apply Theorem A.2 under Assumptions (F1) through (F3), it remains to show the following lemma.

1 pq i !q 2 ż W 2q i gpxqg T pyqρ p2qiq px, y; θ ˚qdxdy ´E « ÿ SĂXXW gpSq1 |S|"qi ff E « ÿ SĂXXW gpSq1 |S|"qi ff T " 1 pq i !q 2 ż W 2q i gpxqg T pyqpρ
Lemma A.5. Under Assumptions (F1) through (F3), (X1) and (X2) we have for all α ą 0, sup

θPM α n pθ ˚q › › › › 1 |W n | pJ n pθq ´Jn pθ ˚qq › › › › P ÝÑ 0. ( 15
)
where M α n pθ ˚q is defined as in Theorem A.2 and there exists l ą 0 such that

P ˆ1 |W n | inf }φ}"1 φ T J n pθ ˚qφ ă l ˙Ñ 0.
Proof. We write It remains to prove that there exists l ą 0 such that (A.5) holds. By Assumption (F3) choose ε ą 0 so that lim inf nÑ8 φ T H n pθ ˚qφ ą ε and let l " ε{2. In the case where θ " θ ˚, the second term in ( 16) is just the expectation of the first one and the third term is equal to |W n |H i n pθ ˚q which is deterministic. Thus when θ " θ ˚, the L 2 norm of the first two terms in [START_REF] Shirai | Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties[END_REF]

J
› › › › B Bθ j H n pθq ´1› › › › " Op a W n q
where the boundedness of BH n pθq ´1{Bθ j for each j was noted above and the boundedness of E ´sup θPM α n pθ ˚q }J n pθq} ¯follows by considerations in the last part of the proof of Lemma A.5 as a consequence of the regularity assumptions imposed on H n pθq by Assumption (F3'). This finishes proving [START_REF] Soshnikov | Determinantal random point fields[END_REF]. The result ( 18) is then a consequence of the fact that H n pθ ˚q´1 J n pθ ˚q converges towards I p when n goes to infinity. Finally, by Lemmas A. For all i, if q i " 1 then H i n pθq is constant. Otherwise, since f i and X are stationary, the integral in (F3) writes

H i n pθq " ż Wn ż W q i ´1 n f i pu 1 , u; θq∇ θ ρ pqiq pu 1 , u; θq T dudu 1 " ż W aR n ż W q i ´1 n f i p0, u ´u1 ; θq∇ θ ρ pqiq p0, u ´u1 ; θq T dudu 1 `εn where ε n " ż WnzW aR n ż W q i ´1 n f i p0, u ´u1 ; θq∇ θ ρ pqiq p0, u ´u1 ; θq T dudu 1 .
By integrability of f i , (W), (X1) and (F2), we have as n Ñ 8. This proves the convergence of H n pθq. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.02 -0.01 0.00 0.01 0.02 0.03 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.02 0.00 0.02 0.04 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 ρ " 50, α " 0.02 ρ " 50, α " 0.04 ρ " 50, α " 0.07 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.010 -0.005 0.000 0.005 0.010 0.015 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.02 -0.01 0.00 0.01 0.02 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 ρ " 100, α " 0.01 ρ " 100, α " 0.03 ρ " 100, α " 0.05 q q q q q q q q q q q q q q q q q q q q q q q -0.002 -0.001 0.000 0.001 0.002 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.0015 -0.0005 0.0005 0.0010 0.0015 0.0020 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.0020 -0.0010 0.0000 0.0005 0.0010 ρ " 1000, α " 0.005 ρ " 1000, α " 0.01 ρ " 1000, α " 0.015 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.004 -0.002 0.000 0.002 0.004 0.006 q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.004 -0.002 0.000 0.002 0.004 q q q q q q q q q q q q q q q q q q q q q -0.004 -0.002 0.000 0.002 ρ " ρpuq, α " 0.005 ρ " ρpuq, α " 0.01 ρ " ρpuq, α " 0.015 1 of the main manuscript, with the addition of the adaptive estimator using ε " 0.05. These quantities are computed from 1000 simulations where all five estimation methods have converged (explaining the differences with Table 1 of the main manuscript).

|ε n,kl | |W n | ď 1 |W n | ż WnzW aR n ż pR d q q i ´1 |f i p0,
imsart-bj ver. 2014/10/16 file: eeGeneral-supplementary.tex date: June 15, 2018 q q q q q q q q q q q q q q q q q q q q q q q 900 950 1000 1050 1100

. Distribution of estimates of ρ obtained from 1000 realisations of the Bessel-type DPP on W " r0, 1s 2 with ρ " 1000 and α " 0.01. Left: the simultaneous estimator as given in equation [START_REF] Lavancier | A tutorial on estimator averaging in spatial point process models[END_REF] with initial value for the numerical solution given by the true value 0.01 of α. Middle: as left but using the true value of α instead of α. Right: ρ " N pX X W q{|W | corresponding to the first step of a two-step procedure.

imsart S3. For the Thomas model, estimated root mean square errors of various estimators of κ and σ (ˆ10 3 ). The 3 first estimators use the test function (3) of the main manuscript with R " 0.05, R " 0.1 and R " 0.25 respectively; the fourth and fifth estimators are the adaptive version based on (4) where ε " 0.01 and ε " 0.05; the three last estimators are from the library spatstat: based on K, on Guan's composite likelihood (clik) and on Palm likelihoodall with default settings. The standard errors of the MSE estimations are given in parenthesis.

adaptive versions are better than any of the fixed R estimators. The adaptive method also has good stable performance compared with the three spatstat methods. In particular, the adaptive method performs much better than Guan's composite likelihood with default settings.

Figure 1 .

 1 Figure 1. Examples of point patterns simulated from a Bessel-type DPP on r0, 1s 2 for different values of ρ and α. For the last row, ρpx, yq " 20 expp4xq.

# 1 a

 1 |W n | e n pθ ˚q : n P N + is stochastically bounded. Then, for all ε ą 0, there exists d ą 0 such that PpD θn : e n p θn q " 0 and |W n |} θn ´θ˚} ă dq ą 1 ´ε (13)

Figure S1 .

 S1 Figure S1. Distribution of α ´α for a Bessel-type DPP on r0, 1s 2 for different values of ρ and α. In each subfigure, the 3 first estimators on the left use the test function (3) of the main manuscript with R " 0.05, R " 0.1 and R " 0.25 respectively, while the last estimator is the adaptive version based on (4).

  If e.g. ε " 1% this means that we only consider pairs of points pu, vq so that the difference between gpu, v; θq and the limiting value 1 is within 1% of the maximal value |gpu, u; θq ´1|. Note that this choice of pairs of points is adaptive in that it depends on θ.We then modify the function f R to be

	|gpu, v; θq ´1| |gpu, u; θq ´1| for a small ε. f adap pu, v; θq " w ˆε gpu, u; θq gpu, v; θq	by the constraint ´1 ˙∇θ ρ p2q pu, v; θq ą ε, ´1 ρ p2q pu, v; θq	(4

  t. |BW n 'pR `εq| " op|W n |q, where B in this context denotes the boundary of a set, R is defined in (5), and |W n | Ñ 8, as n Ñ 8.

  vq1 }u´v}ďR dvdu "

				ż	ż
				n W aR	Wn	hpu, vq1 }u´v}ďR dvdu `εn
	where	ż	ż
	ε n "	n WnzW aR	Wn	hpu, vq1 }u´v}ďR dvdu.

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[START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF]

  , provides ρ " N pX X W q{|W | and ψ is the root of

	e 2 pψq "	ÿ rij	w	ˆε gp0; ψq gpr ij ; ψq ´1 ´1 ˙∇ψ gpr ij ; ψq gpr ij ; ψq
				´N pX X W q 2	ż W	w	ˆε gp0; ψq gpr; ψq	´1 ´1 ˙∇ψ gpr; ψqdF prq. (8)

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[START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF]

  2 for different values of ρ and α. For the last row, ρpx, yq " 20 expp4xq.

	ρ	α		R " 0.05 R " 0.1 R " 0.25 Adaptive	R
	50	0.02	rmse:	5.84	5.83	6.29	5.97	0.047
				(0.15)	(0.17)	(0.19)	(0.18)	(0.020)
			time:	0.43	0.48	0.68	0.64
		0.04 rmse:	15.60	9.18	9.19	9.25	0.106
				(0.44)	(0.20)	(0.22)	(0.21)	(0.037)
			time:	0.48	0.50	0.68	0.73
		0.07 rmse:	13.32	8.25	8.22	8.15	0.147
				(0.33)	(0.23)	(0.24)	(0.24)	(0.050)
			time:	0.50	0.45	0.59	0.72
	100	0.01	rmse:	2.44	2.45	2.58	2.63	0.024
				(0.08)	(0.08)	(0.09)	(0.09)	(0.009)
			time:	0.44	0.57	1.22	0.70
		0.03 rmse:	5.34	5.12	5.28	5.27	0.064
				(0.13)	(0.13)	(0.14)	(0.13)	(0.019)
			time:	0.40	0.47	0.98	0.70
		0.05 rmse:	5.78	4.43	4.50	4.53	0.139
				(0.12)	(0.12)	(0.10)	(0.12)	(0.022)
			time:	0.52	0.56	1.16	0.95
	1000 0.005 rmse:	0.67	0.88	0.83	0.72	0.015
				(0.02)	(0.02)	(0.02)	(0.02)	(0.003)
			time:	3.83	19.04	110.07	9.38
		0.01 rmse:	0.57	0.59	0.61	0.56	0.028
				(0.01)	(0.02)	(0.01)	(0.01)	(0.005)
			time:	2.68	10.40	60.79	6.84
		0.015 rmse:	0.47	0.46	0.52	0.47	0.026
				(0.01)	(0.01)	(0.01)	(0.01)	(0.002)
			time:	2.53	9.81	55.78	7.75
	Inhom	0.005 rmse:	1.58	1.65	1.66	1.61	0.014
				(0.04)	(0.04)	(0.04)	(0.04)	(0.005)
			time:	0.89	2.50	10.30	1.19
		0.01 rmse:	1.34	1.36	1.36	1.32	0.025
				(0.03)	(0.03)	(0.03)	(0.03)	(0.008)
			time:	0.76	1.86	7.66	1.22
		0.015 rmse:	1.43	1.47	1.48	1.40	0.030
				(0.03)	(0.03)	(0.03)	(0.03)	(0.006)
			time:	0.86	1.90	7.46	1.40
	Table

  |q.

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[START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF]

  p2qiq px, y; θ ˚q ´ρpqiq px; θ ˚qρ pqiq py; θ ˚qqdxdy. , x 2 , ¨¨¨, x qi , y 1 , ¨¨¨, y qi ; θ ˚q ´ρpqiq px 1 , x 2 , ¨¨¨, x qi ; θ ˚qρ pqiq py 1 , ¨¨¨, y qi ; θ ˚qdx 1 which is Op|W |q by Assumption (X2).The regularity conditions on e n pθq in Theorem A.2 are consequences of (F1), (X1). The stochastic behavior of e n pθ ˚q is easily deduced from the previous lemma. The result follows by showing that each component e i n pθ ˚q of e n pθ ˚q is stochastically bounded. By Chebyshev's inequality, we just need to bound |W n | ´1Varpe i n pθ ˚qq.

	All of its components are bounded by
	}g} 2 8 |W ||BpO, Rq| 2qi´2 pq i !q 2 ρ p2qiq px 1 Lemma A.4. The class of random vectors ż W sup diampxqăR sup diampyqăR sup y1PW
		#				+
			a	1 |W n |	e n pθ ˚q : n P N
	is stochastically bounded.
	Proof. Letting
		‰
	e i n pθq :"	ÿ x1,¨¨¨,xq i PXXWn	f i px 1 , ¨¨¨, x qi ; θq	n ´żW q i	f i px; θqρ pqiq px; θqdx,
	we know that Varpe i n pθ ˚qq is Op|W n |q by Lemma A.3 under Assumptions (X2) and (F2).

  θq∇ θ ρ pqiq px; θq T dx.[START_REF] Shirai | Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties[END_REF] Now, recall that f i , d dθ T f i , ρ pqiq and ∇ θ ρ pqiq are all continuously differentiable with respect to θ by Assumption (F1) and (X1). Moreover, the first and second derivatives of f i and ρ pqiq with respect to θ are bounded with respect to x and θ by the same assumptions. Therefore, since M α n pθ ˚q is a decreasing sequence of compact sets, there exist constants C 1 , C 2 ą 0 not depending on n, x and θ such that by a Taylor expansion, pqiq px; θ ˚q1 diampxqďR dx " Op|W n |q since ρ pqiq is bounded by Assumption (X2). This shows that Ersup θPM α n pθ ˚q }J i n pθq Ji

	By definition, for all i
	J i n pθq " ´qi !	ÿ xĂXXWn	d dθ T f i px; θq	`żW q i n	d dθ T f i px; θqρ pqiq px; θqdx
			|x|"qi
	`żW q i n f i px; sup θPM α n pθ ˚q }J i n pθq ´Ji n pθ ˚q} ď α a |W n | ¨C1 ÿ xĂXXWn 1 diampxqďR `C2	ż W	n 1 diampxqďR dudv ‹ ‹ q i '
						|x|"qi
	where the indicator functions arise as a consequence of Assumption (F2). Moreover,
		»				fi
						ż
	E	---xĂXXWn ÿ 1 diampxqďR	ffi ffi fl "	W	q i n
			|x|"qi
				n pθq "	¨J1 n pθq . . .	‹ ':"	´¨d dθ T e 1 n px; θq . . .	‹ '
						J l n pθq	d dθ T e l n px; θq
				imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018

ρ n pθ ˚q}s is Op a |W n |q.

  |W n |q by Lemma A.3. Hence it vanishes in probability when divided by |W n |. Denote by a n the first two terms in φ T J n pθ ˚qφ{|W n | and by b n the last term which is φ T H n pθ ˚qφ. Then Consider n to be large enough and θ to be in a neighbourhood of θ such that H n pθq is invertible and H n pθq ´1 is uniformly bounded with respect to n and θ. Let ẽn pθq " H n pθq ´1e n pθq and let us show that we can apply Theorem A.2 to ẽn . Obviously, ẽn has the same roots as e n , is continuously differentiable since θ Þ Ñ H n pθq and θ Þ Ñ H n pθq ´1 are continuously differentiable from Assumptions (F1) and (F2), and the family tẽ n pθ ˚q{ a |W n | : n P Nu is stochastically bounded. Let Jn pθq " ´d dθ T ẽn pθq. It remains to show the following lemma. Therefore, we only need to look at the behaviour of H n pθq ´1J n pθq.

	is equal to g f f f f f e Var ¨ÿ xĂXXWn |x|"qi inf }φ}"1 φ T J n pθ ˚qφ ă l ˙ď lim d dθ T f i px; θ nÑ8 Pp inf ˚q‹ ‹ ' }φ}"1 b n ´ε{2 ă l, |a n | ă ε{2q ď lim nÑ8 Pp inf }φ}"1 b n ă εq " 0 which concludes the proof. which is Op a lim nÑ8 P ˆ1 |W n | To apply Theorem A.2 under the alternative Assumption (F3') instead of (F3), we proceed as follows. Lemma A.6. Under Assumptions (F1), (F2), (F3'), (X1) and (X2) we have sup θPM α n pθ ˚q › › › › 1 |W n | p Jn pθq ´J n pθ ˚qq › › › › P ÝÑ 0. (17) and P ˆlim nÑ8 1 |W n | inf }φ}"1 φ k"1 B Bθ j H n pθq ´1 i,k e n pθq k . (19) For any θ P M α n pθ ˚q, since all terms in (16) are bounded by Assumptions (F1) and (X1), using Assumption (F2) we get Er|e n pθq ´en pθ ˚q|s ď α a |W n | E ˜sup θPM α n pθ By (F1), (X1) and (F3'), B Bθj H n pθq ´1 " p B Bθj H n pθqqH n pθq ´2 is bounded on M α n pθ ˚q for a large enough n. It follows for all i, j, 1 |W n | sup θPM α n pθ ˚q › › › › › p ÿ k"1 B Bθ j H n pθq ´1 i,k e n pθ ˚qk ´Tn pθq i,j › › › › › P ÝÑ 0. Moreover, sup θPM α n pθ ˚q p ÿ k"1 B Bθ j H n pθq ´1 i,k e n pθ ˚qk |W n | P ÝÑ 0 as a consequence of Lemma A.4. Hence |W n | ´1 sup θPM α n pθ ˚q }T n pθq} P ÝÑ 0 and thus |W n | ´1}T n pθ ˚q} P sup θPM α n pθ ˚q › › › › 1 |W n | H n pθ ˚q´1 pJ n pθq ´Jn pθ ˚qq › › › › P ÝÑ 0. Finally, we observe that E ˜sup θPM α n pθ ˚q }pH n pθq ´1 ´Hn pθ ˚q´1 qJ n pθq} ḑ ÝÑ 0. From Lemma A.5 we know that α a |W n | E ˜sup θPM α n pθ ˚q }J n pθq} ¸sup θPM α n pθ ˚q sup 1ďjďp
	imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018

T Jn pθ ˚qφ ă 1{2 ˙" 0. (18) Proof. We have Jn pθq " H n pθq ´1J n pθq ´Tn pθq where T n pθq i,j " p ÿ ˚q }J n pθq} ¸" Op a W n q. imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018

  4, A.5 and A.6, we can apply Theorem A.2 and the first part in the statements of Theorem 2.1 is deduced. Now, for each n P N, we define θn as the closest root of e n to θ ˚, if e n has any, otherwise let θn " 0. Theorem A.2 tells us that Ppe n p θn q " 0q Ñ 1 and a |W n |p θn ´θ˚q is bounded in probability.To prove the asymptotic normality, we use the Taylor expansion e n p θn q " e n pθ ˚q is closer to θ ˚than θn with probability tending to 1. Moreover, we saw at the end of the proof of Theorem A.2 that the variance of the first two terms of |W n | ´1J n pθ ˚q vanishes when n Ñ 8 and the last term is equal to H n pθ ˚q. Finally, by Assumption (X3) and since |W n | ´1Varpe n pθ ˚qq is stochastically bounded (Lemma A.4), it follows by Slutsky's lemma that

							Jn
	pθ 0 n qp θn	´θ˚q where }θ 0 n	´θ˚} ď } θn	´θ˚} , which implies
	e n p θn q a |W n |	"	e n pθ a |W n | ˚q	`Jn pθ |W n | ˚q	a |W n |p θn	´θ˚q
							`1 |W n |	pJ n pθ 0 n q ´Jn pθ ˚qq	a |W n |p θn	´θ˚q .
	We know that e n p θn q{ a |W n | converges in distribution towards 0 and by Theorem A.2
	we also know that		› › › ›	1 |W n |	› › n q ´Jn pθ ˚qq pJ n pθ 0 › ›	P ÝÑ 0
	because θ 0 n |W n |Σ ´1{2 n	H n pθ ˚qp θn	´θ˚q L ÝÑ N p0, I p q.

imsart-bj ver. 2014/10/16 file: eeGeneral.tex date: June 15, 2018 Appendix B: Proof of Lemma A.1

  u ´u1 ; θq k |}∇ θ ρ pqiq p.; θq} 8 dudu 1 Wn f i p0, u ´u1 ; θq∇ θ ρ pqiq p0, u ´u1 ; θq T dudu 1

	"	|W n zW aR n | |W n | }∇ 1 ż ż
		|W n |	W aR n
	"	|W aR n | |W n |	ż }t}ďR	f i p0, t; θq∇ θ ρ pqiq p0, t; θq T dt
		ż
	Ñ

θ ρ pqiq p.; θq} 8 ż }t}ďR |f i p0, t; θq k |dt ď |BW n ' R| |W n | }∇ θ ρ pqiq p.; θq} 8 ż }t}ďR |f i p0, t; θq k |dt Ñ 0, where ε n,kl denotes the klth entry of the matrix ε n and f i p¨q k the kth component of the vector f i p¨q. Moreover, }t}ďR f i p0, t; θq∇ θ ρ pqiq p0, t; θq T dt,

Table S1 .

 S1 Percentage of times the estimation methods have converged for the models and estimators considered in Section 4.1 of the main manuscript.

		ρ	α	R " 0.05 R " 0.1 R " 0.25 Adaptive
		50	0.02	0.75	0.72	0.72	0.72
			0.04	0.97	0.85	0.80	0.85
			0.07	0.92	1.00	0.99	0.98
		100	0.01	0.66	0.69	0.72	0.64
			0.03	0.99	0.96	0.89	0.93
			0.05	1.00	1.00	1.00	1.00
		1000 0.005	1.00	0.95	0.95	0.95
			0.01	1.00	1.00	1.00	1.00
			0.015	1.00	1.00	1.00	1.00
	Inhom 0.005	0.93	0.92	0.95	0.93
			0.01	0.98	0.98	0.98	0.96
			0.015	1.00	1.00	1.00	1.00
	ρ	α	R " 0.05 R " 0.1 R " 0.25 ε " 0.01 ε " 0.05
	50	0.02		5.49	5.45	5.95	5.53	7.13
		0.04		14.92	8.81	8.79	8.87	8.71
		0.07		13.08	8.10	8.07	8.04	8.82
	100	0.01		2.30	2.27	2.45	2.49	2.77
		0.03		5.05	4.99	5.16	5.10	5.27
		0.05		5.75	4.40	4.47	4.50	5.10
	1000 0.005		0.68	0.87	0.83	0.73	0.73
		0.01		0.57	0.59	0.61	0.56	0.59
		0.015		0.47	0.46	0.52	0.47	0.51
	Inhom 0.005		1.58	1.65	1.66	1.61	1.57
		0.01		1.34	1.36	1.36	1.32	1.37
		0.015		1.43	1.47	1.48	1.40	1.46

). imsart-bj ver. 2014/10/16 file: eeGeneral-supplementary.tex date: June 15, 2018

Table S2 .

 S2 RMSE (ˆ10 3 ) for the same simulations as in Table

  -bj ver. 2014/10/16 file: eeGeneral-supplementary.tex date: June 15, 2018 σ R " 0.05 R " 0.1 R " 0.25 ε " 0.01 ε " 0.05

								K	clik	Palm
	0.02	κ	17	21	21	21	20	23	28	21
			(0.40)	(0.48)	(0.47)	(0.48)	(0.47)	(0.54)	(0.70) (0.49)
		σ	1.04	1.84	1.94	1.79	1.51	2.60	1.54	1.92
			(0.02)	(0.06)	(0.08)	(0.07)	(0.04)	(0.09)	(0.03) (0.09)
	0.035	κ	35	31	40	38	35	33	121	35
			(0.79)	(0.75)	(0.95)	(0.89)	(0.85)	(0.81)	(4.20) (0.90)
		σ	4.80	5.54	7.92	6.24	4.59	5.76	8.04	5.50
			(0.09)	(0.14)	(0.32)	(0.21)	(0.10)	(0.14)	(0.07) (0.12)
	0.05	κ	54	49	53	47	53	35	554	39
			(1.24)	(0.92)	(2.05)	(1.74)	(1.89)	(1.55) (13.64) (1.02)
		σ	18.30	36.47	12.17	11.22	8.94	8.12	23.69	19.47
			(1.28)	(1.47)	(0.41)	(0.53)	(0.61)	(0.25)	(0.13) (0.74)

Table

imsart-bj ver. 2014/10/16 file: eeGeneral.tex date:[START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF]

Acknowledgements

Rasmus Waagepetersen is supported by The Danish Council for Independent Research -Natural Sciences, grant DFF -7014-00074 "Statistics for point processes in space and beyond", and by the "Centre for Stochastic Geometry and Advanced Bioimaging", funded by grant 8721 from the Villum Foundation.

Supplementary material for "Adaptive estimating function inference for non-stationary determinantal point processes"

FR ÉD ÉRIC LAVANCIER 1,* ARNAUD POINAS 2,** RASMUS WAAGEPETERSEN 3, † 1 Laboratoire de Mathématiques Jean Leray -BP 92208 -2, Rue de la Houssinière -F-44322 Nantes Cedex 03 -France. Inria, Centre Rennes Bretagne Atlantique, France.

E-mail: * frederic.lavancier@univ-nantes.fr 2 IRMAR -Campus de Beaulieu -Bat. 22/23 -263 avenue du Général Leclerc -35042 Rennes -France E-mail: ** arnaud.poinas@univ-rennes1.fr 3

Department of Mathematical Sciences -Aalborg University -Fredrik Bajersvej 7G -DK-9220

Aalborg -Denmark E-mail: † rw@math.aau.dk

Supplementary tables for Section 4.1 of the main manuscript

For the simulation study carried out in Section 4.1 of the main manuscript, considering estimation for a DPP model with a Bessel-type kernel, we report in Figure S1 the boxplots representing the distribution of the estimators and in Table S1 the percentages of times each estimation method has converged in our simulation study. These percentages are similar for all estimation methods. Table S2 displays the root mean square errors of the estimators considered in Section 4.1 where, for comparison, we also include results for the adaptive estimator using ε " 0.05. Conclusions based on these tables are given in the main paper.

Two-step versus simultaneous

Referring to Section 4.2, Figure S2 shows how irregular the contrast function epψq for the simultaneous approach can be in comparison with the contrast function e 2 pψq for the two-step approach. The underlying point pattern is displayed on the left. This is a realisation on the unit square of a homogeneous DPP with a Gaussian kernel, with intensity ρ " 1000 and range α " 0.01. Figure S3 reports the distributions of estimates of ρ over 1000 realisations on the unit square of a DPP with a Bessel-type kernel with ρ " 1000 and α " 0.01. The two first estimators come from the simultaneous approach, see equation ( 10) of the main manuscript where ψ " α in this setting. For the first one, the numerical solution of epαq " 0 to get α was initialized at the true value 0.01 of α. For the second one, α was fixed to the true value, i.e. α " 0.01. The last estimator on the right of Figure S3 is simply ρ " N pX X W q{|W |, corresponding to the first step of the two-step procedure. The respective root mean square errors are 33.6, 31.4 and 26. See the main manuscript for further discussion.

Some simulations for the Thomas model

The adaptive estimating function is also useful for clustered point processes. Here we consider a Thomas model on r0, 1s 2 , with parent intensity κ " 100, offspring intensity µ " 10 and various values of the dispersal kernel standard deviation σ. The same three estimation methods as in Section 4.1 of the main manuscript have been evaluated, where for the adaptive version both ε " 0.01 and ε " 0.05 have been considered. A point pattern sample and the distribution of the estimators of κ and σ based on 1000 replications are shown in Figure 3 for σ " 0.02, σ " 0.035 and σ " 0.05 respectively. Estimators of the library spatstat [1] of R [2] with default settings have also been added. These are: minimum contrast estimation based on the K-function, Guan's composite likelihood, and Palm likelihood, see also Section 2.3 in the main manuscript. Table S3 summarises the estimated root mean square errors for each estimation method.

Also for the Thomas process, the adaptive method, both with ε " 0.01 and ε " 0.05, performs well compared with the three fixed R estimators. In fact for σ " 0.05, the imsart-bj ver. 2014/10/16 file: eeGeneral-supplementary.tex date: June 15, 2018 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -50 0 50 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 200 400 600 800 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 500 1000 1500 2000 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.005 0.000 0.005 0.010 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00