Estimate by the L 2 Norm of a Parameter Poisson Intensity Discontinuous
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1 Research Journal of Mathematics and Statistics 6: -5, 24 ISSN: , e-issn: Maxwell Scientific Organization, 24 Submied: September 8, 23 Accepted: November 23, 23 Published: February 25, 24 Estimate by the L 2 Norm of a Parameter Poisson Intensity Discontinuous Demba Bocar Ba UFR SET-Université de Thiès BP 967, Senegal Abstract: The model considered has two types of discontinuities. The first parameter is a parameter of scale; the second is a parameter of translation. The aim of study is to show that the minimum distance estimator is consistent and asymptotically normal. The problem of estimation studied in this work is the same time singular and regular. It is singular because the intensity is a discontinuous function and we obtains the consistency with the discontinuity and the study of asymptotic normality is based on the regularity of the function Λ(θθ, ). Keywords: Asymptotic properties, minimum distance estimator, parameter estimation, Poisson process, singular estimation INTRODUCTION In many areas of everyday life we use the inhomogeneous Poisson processes; there exists a large literature describing the different point processes and particularly Poisson with special aention to parameter estimation problems. The properties of the estimators (Maximum Likelihood (MLE), Bayesian (BE), Minimum Distance (MDE) are usually studied in the case of smooth w.r.t., parameter situation i.e., the intensity function has one or two derivative, w.r.t. unknown parameter. According to general theory all these estimators in regular (smooth) case are consistent and asymptotically normal see, Kutoyants (998). Moreover, the first two estimators are asymptotically efficient in usual sense. In non regular case, when, say intensity function is discontinuous, the properties of estimators are quite different. Remember that the MDE is a particular case of the minimum contrast estimator if we consider the function X() Λ() as a contrast see Le (972) for details. We have several possibilities of the choice of the space H. In the case H = L 2 (µ) the measure µ can also be chosen in different ways continuous, discrete, etc. Others definitions of the MDE can also be realized. Note that the asymptotic behavior of the estimator depends strongly of the chosen metric. We are mainly interested in the properties of the MDE in the case H = L 2 (µ). Particularly we show that under regularity conditions the MDE are consistent, asymptotically normal. For another metric this estimator is also consistent but its limit distribution is not Gaussian (Kutoyants and Liese, 992). We did a study on Regular properties of the MDE for Non Smooth Model of Poisson Process (Ba and Dabye, 29). We also studied the asymptotic behaviour of the maximum likelihood estimator and the Bayesian estimator (Ba et al., 28). The present study is devoted to the dimensional case i.e., we suppose that the intensity function SS(νν, ) depends to 2 dimensional parameter νν = (γγ, γγ 2 ). Our objective is to study the Minimum Distance Estimator (MDE) built from the norm L2. We show that the MDE is consistent and asymptotically normal. We studied the proprieties of the estimators using other norms and others models. Moreover in the case not hilbertien, it was shown in the studies of Kutoyants and Liese (992) that we caot have the asymptotic normality. STATISTICAL MODEL We observe trajectories XX jj = (XX jj (), [, ]) jj =,..., of the Poisson process with intensity function SS(νν oo, ) = ff(θθ + θθ 2 ) + λλ. Here νν oo = (θθ, θθ 2 ) in the unknown parameter, which we have to estimate, the function ff and the constant λλ are known and positive: EE θθoo XX jj () = Λ(θθ oo, ) = SS(θθ oo, ss)dddd with SS(θθ oo, ) = ff(θθ + θθ 2 ) + λλ Hypothesis A: The function ff( ) and the constant λλ known and positive. The function ff( ) is continuously differentiable on [aa, ττ ) (ττ, ττ 2 ) (ττ 2, ββ + ββ 2 )], we suppose that ββ 2 < αα + αα 2. The function ff( ) have two jumps at ττ, ττ 2 (ββ 2, αα + αα 2 ) with ττ < ττ 2 and ff(ττ ii +) ff(ττ ii ) = rr ii ii =,2.
2 Res. J. Math. Stat., 6: -5, 24 We define this below the minimum distance estimator of θθ oo and we are intereste to their properties when. Let LL 2 ([, ]) the Hilbert space with the h( ) = h 2 () /2, then we define the minimum distance estimator as solution of the equation: XX jj = jj Λ(θθ, ) = iiiiii XX jj = jj Λ(θθ, ) STUDY OF THE CONSISTENCY The following theorem ensures the consistency of the MDE. Theorem : If the Hypothesis AA is satisfied, then the estimator of the minimum distance θθ is consistent: for all δδ > : PP ( θθoo ){θθ θθ oo > δδ} 4 ΛΛ(θθ oo,)dddd ψψ(δδ,θθ oo ) 2 with ψψ(δδ, θθ oo ) = inf θθ θθoo >δδ Λ(θθ, ) Λ(θθ oo, ) For the demonstration of this theorem, we need the following lemme. Lemme: If the Hypothesis AA is satisfied, then for all δδ >, the function ψψ(δδ, θθ oo ) >. Demonstration: To show ψψ(δδ, θθ oo ) >, proceed by contradiction i.e., we suppose that for a δδ >, we have ψψ(δδ, θθ oo ) =. In this case, it exist θθ oo θθ oo such: iiiiii Λ(θθ, ) Λ(θθ oo ) = Λ(θθ oo) Λ(θθ oo ) = so for all [, ]: Λ(θθ oo, ) = Λ(θθ oo, ) since Λ(θθ, ) is continuous function in. We deduce that Λ(θθ oo, ) = Λ(θθ oo, ) for all [, ] i.e.: ff(θθ + θθ 2 ) + λλ = ff(θθ + θθ 2) + λλ ff(θθ + θθ 2 ) = ff(θθ + θθ 2) with means that: we have: θθ ss + θθ = rr θθ ss 2 + θθ 2 = rr 2 ss ss 2 θθ θθ 2 = rr rr2 2 This leads to θθ = θθ et θθ 2 = θθ 2 i.e., θθ oo = θθ oo which contradicts the fact that θθ oo θθ oo. So the lemma is proved. Demonstration of the theorem: We put YY () = XX jj = jj () and ZZ () = (YY () Λ(θθ oo, ))which allows to write for all δδ > : PP () θθoo θθ () θθ oo > δδ PP θθoo inf YY ( ) Λ(θθ, ) > θθ θθ oo δδ inf YY ( ) Λ(θθ, ) θθ θθ oo δδ PP () θθoo 2 ZZ ( ) > ψψ(δδ, θθ oo ) where, we have used the properties of the norm: YY ( ) Λ(θθ oo ) + Λ(θθ oo, ) Λ(θθ, ) YY ( ) Λ(θθ, ) YY ( ) Λ(θθ oo, ) Λ(θθ oo, ) YY ( ) Λ(θθ, ) Using the lemma ψψ(δδ, θθ oo ) >. so, applying the inequality of Chebychev it follows: Because, PP θθ () {2 ZZ ( ) ψψ(δδ, θθ oo ) 4EE θθ oo ZZ ( ) 2 (δδ,θθ oo ) 2 EE θθoo ZZ ( ) 2 EE θθoo ( (YY () Λ(θθ oo, ))) 2 dddd = EE θθoo ( YY jj () Λ(θθ oo, ) 2 dddd = ( Λ(θθ oo, )dddd The laer term being bounded, which demonstrated the theorem. Asymptotic normality: In this section we will to prove the Asymptotic Normality of MDE. We define the functions FF(θθ, ) et GG(θθ, ) for all [, ] as: FF(θθ, ) = gg (θθ θθ ss + θθ 2 )dddd gg(θθ + θθ2) et GG(θθ, ) = gg(θθ + θθ 2 ) gg(θθ 2 ) Let us notice that the functions RR(θθ, ) et GG(θθ, ) are respectively the partial derives formal of the function Λ(θθ, ). Let us put AA, (θθ) = FF(θθ, ) 2 dddd: Λ 2,2 (θθ) = GG(θθ, ) 2 dddd AA 2 = AA 2, = FF (θθ, ) GG(θθ, )dddd
3 Res. J. Math. Stat., 6: -5, 24 and CC (θθ) = FF (θθ, )FF(θθ, ss)λ(θθ, ss ) dddddddd CC 2,2 (θθ) = GG (θθ, )GG(θθ, ss)λ(θθ, ss ) dddddddd CC,2 (θθ) = CC 2, (θθ) = FF (θθ, )GG(θθ, ss)λ(θθ, ss ) dddddddd Let be carrees matrix of order 2: and AA(θθ) = AA (θθ) AA, (θθ) CC(θθ) = CC (θθ) CC 2 (θθ) AA,2 (θθ) AA 2, (θθ) CC 2 (θθ) CC 2,2 (θθ) the determinant of the matrix AA(θθ) is: dde(aa(θθ)) = FF FF (θθ, )GG(θθ, )dddd 2 (θθ, ) 2 dddd GG (θθ, ) 2 dddd To prove the asymptotic normality, we suppose that: Hypothesis B: AA (θθ) AA 2,2 (θθ) (AA,2 (θθ)) 2 Théorème 2: If the Hypothesis AA and BB are satisfied then the minimum distance estimator is asymptotically normal i.e.: (θθ θθ oo ) NN(, AA CC(θθ oo )AA (θθ oo )) Demonstration: We define the function HH (θθ) by: HH (θθ) = XX jj = jj () Λ(θθ, ) 2 dddd Let us remind that the minimum distance estimator θθ is a solution of the equation: θθ = aaaaaa inf θθ= Θ HH (θθ) Let us note that θθ = (θθ,, θθ 2, ) and θθ = (θθ, θθ 2 ) the function HH (θθ) is derivable. So the MDE θθ minimize the function HH (θθ) and is consistent in θθ oo element of Θ. It with a probability numeric. This estimator is a solution of system: HH (θθ) θθ=θθ = HH (θθ) θθ=θθ2 = The calculation of the partial derives by report at θθ and θθ 2 gives us: 3 HH θθ (θθ) = 2 θθ gg (θθ ss + θθ 2 )dddd (θθ + θθ 2 ) jj = θθ 2 HH (θθ) = XX jj () Λ(θθ, ) dddd 2 [ gg(θθ θθ + θθ 2 gg(θθ 2 )] Λ(heeaa, dd thus using (3) and (4), we obtain: gg (θθ, ss + θθ 2, )dddd (θθ, XX jj = jj () Λ(θθ, ) dddd = gg(θθ, + θθ 2, ) gg(θθ 2, ) XX jj = jj () Λ(θθ, ) dddd = We can write this system as: FF (θθ, ) Λ(θθ,)dd= GG (θθ, ) Λ(θθ,)dd= jj = XX jj jj = XX jj jj = XX jj () + θθ 2, ) () Λ(θθ oo, ) + Λ(θθ oo, ) () Λ(θθ oo, ) + Λ(θθ oo, ) We introduce the process us WW () define by: thus, Then, WW () = ( jj = XX jj () Λ(θθ oo, )) ()FF(θθ, ) dddd = FF(θθ, )[Λ(θθ, ) Λ(θθ oo, ) dddd ()GG(θθ, ) dddd = GG(θθ, )[Λ(θθ, ) Λ(θθ oo, ) dddd [ Λ(θθ + h, ) Λ(θθ, ) (h, Λ(θθ, ) (h, Λ (θθ, ) 2 dddd = θθ( h 2 ) Using the continuity of this functions and the consistency of the estimator, we can write the system as: ()FF(θθ oo, )dddd = FF(θθ oo, )(uu, Λ (θθ oo, )dddd + θθ
4 Res. J. Math. Stat., 6: -5, 24 ()GG(θθ oo, )dddd = GG(θθ oo, )(uu, Λ (θθ oo, )) dddd + θθ where we put uu = (uu θθ ) et uu 2, = ( (θθ 2, YY = (YY, YY (2) ) with,, uu 2), uu, = (θθ, θθ 2 ) we introduce the vector YY = FF (θθ oo, ) WW () dddd YY (2) = GG (θθ oo, ) WW () dddd we obtain YY = AA, (θθ oo )uu, ) + AA,2 (θθ oo )uu 2, + θθ YY (2) = AA 2, (θθ oo )uu, ) + AA 2,2 (θθ oo ) + θθ or under matrix shape: AA (θθ oo ) uu = YY + θθ From the hypothesis BB, the matrix of order 2 AA(θθ oo ) is invertible thus we can write BB(θθ oo ) = AA(θθ oo ), so: uu = BB(θθ oo )YY + θθ We show that EE θθoo (θθ θθ oo )(θθ θθ oo ) = BB(θθ oo )CC(θθ oo ) BB(θθ oo ) + θθ We have: FF (θθ oo, )WW ()dddd = ( jj = XX jj () Λ(θθ oo, ) FF(θθ oo, ) dddd ( XX jj = jj () Λ(θθ oo, )FF(θθ oo, ) dddd = jj = with ψψ jj ψψ jj = ( XX jj () Λ(θθ oo, ))FF(θθ oo, )dddd GG(θθ oo, ) WW () dddd = ) ( jj = XX jj () Λ(θθ oo, ) FF(θθ oo, ) dddd = ( XX jj = jj () Λ(θθ oo, )GG(θθ oo, )dddd = (2) jj = with ψψ jj ψψ jj = ( XX jj () Λ(θθ oo, ))GG(θθ oo, )dddd EE θθoo ψψ jj ψψ jj (2) = CC l,ii (θθ oo ) Let us note that the terms of the matrix are: YY = FF(θθ oo, ss) dddd dddd(λ(θθ oo, )) YY (2) = GG(θθ oo, ss)dddd dddd Λ(θθ oo, ) We have the following relation: 4 CC l,ii (θθ oo ) = EE θθoo (YY (l) YY (ii) ) We obtain: and, CC, (θθ oo ) CC 2,2 (θθ oo ) = FF(θθ oo, ss) dddd = GG(θθ oo, ss) dddd 2 2 SS(θθ oo, ) dddd SS(θθ oo, ) dddd CC,2 (θθ oo ) = CC 2, (θθ oo ) = FF(θθ oo, ss) dddd GG (θθ oo, rr) dddd SS(θθ oo, ) dddd ψψ = [ XX () Λ(θθ oo, )] FF(θθ oo, ) dddd = ss [ddxx (ss) SS(θθ oo, ) dddd] FF(θθ oo, ) dddd ss ss = FF(θθ oo, ) dddd[ddxx (ss) SS(θθ oo, ) dddd] = FF(θθ oo, ) dddddddd(ss) For ψψ (2) we have: ss ψψ (2) = GG(θθ oo, )dddd dddd(ss) On the other hand using: EE θθoo [XX () Λ(θθ oo, )][XX (ss) Λ(θθ oo, ss) = Λ(θθ oo, ss] we obtain for example: EE θθoo ψψ ψψ (2) = FF(θθ oo, )GG(θθ oo, ss)λ(θθ oo, ssddss dd Using the central theorem limit, we obtain: YY NN(, CC(θθ oo )) as (θθ θθ oo ) = BB(θθ oo ) YY + θθ we deduce: uu = (θθ θθ oo ) NN(, BB(θθ oo ) CC(θθ oo ) BB(θθ oo ) ) i.e.: (θθ θθ oo ) NN(, AA (θθ oo ) CC(θθ oo ) AA (θθ oo ) ) So the asymptotic normality is proved. REFERENCES Ba, D.B. and A.S. Dabye, 29. On regular properties of the MDE for non smooth model of poisson process, Indian J. Math., 5:
5 Res. J. Math. Stat., 6: -5, 24 Ba, D.B., A.S. Dabye, A. Diakhaby and F.N. Diop, 28. Comportment asymptotique des estimateurs de paramètre d un processus de Poisson pour un modèle non régulier. Aales de l Université Ndjamena, 3: Kutoyants, Y.A., 998. Statistical Inference for Spatial Poisson Processes, Lect. Notes Statist. 34, Springer, New York, pp: 276. Kutoyants, Y.A. and F. Liese, 992. On minimum distance estimation for spatial poisson processes. A. Academial. Scient. Feicae, ser. A. I., 7: Le, C., 972. Limit of experiments. Proceeding of the 6th Berkeley Symposium on Mathematical Statistics and Probability,:
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