Geodesic
Contiguity and LAN-property of sequences of Poisson processes
Kybernetika, Tome 35 (1999) no. 3, pp. 281-308
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Using the concept of Hellinger integrals, necessary and sufficient conditions are established for the contiguity of two sequences of distributions of Poisson point processes with an arbitrary state space. The distribution of logarithm of the likelihood ratio is shown to be infinitely divisible. The canonical measure is expressed in terms of the intensity measures. Necessary and sufficient conditions for the LAN-property are formulated in terms of the corresponding intensity measures.
Using the concept of Hellinger integrals, necessary and sufficient conditions are established for the contiguity of two sequences of distributions of Poisson point processes with an arbitrary state space. The distribution of logarithm of the likelihood ratio is shown to be infinitely divisible. The canonical measure is expressed in terms of the intensity measures. Necessary and sufficient conditions for the LAN-property are formulated in terms of the corresponding intensity measures.
Classification : 60G55, 62B10, 62G20, 62M07
Keywords: Poisson point process; local asymptotic normality; Hellinger integral; likelihood ratio 
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Liese, Friedrich; Lorz, Udo. Contiguity and LAN-property of sequences of Poisson processes. Kybernetika, Tome 35 (1999) no. 3, pp. 281-308. http://geodesic.mathdoc.fr/item/KYB_1999_35_3_a1/

[1] Brown M.: Discrimination of Poisson processes. Ann. Math. Statist. 42 (1971), 773–776 | DOI | MR | Zbl

[2] Csiszár I.: Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität Markoffscher Ketten. Publ. Math. Inst. Hungar. Acad. Sci., Ser. A 8 (1963), 85–108 | MR

[3] Jacod J., Shiryaev A. N.: Limit Theorems for Stochastic Processes. Springer–Verlag, Berlin 1987 | MR | Zbl

[4] Karr A. F.: Point Processes and their Statistical Inference. Marcel Dekker, New York 1986 | MR | Zbl

[5] Kutoyants, Yu. A.: Parameter Estimation for Stochastic Processes. Helderman, Berlin 1984 | MR | Zbl

[6] Kutoyants, Yu. A.: Statistical inference for spatial Poisson processes. Lab. de Stat. et Proc. Univ. du Maine, Le Mans, manuscript of forthcoming monography (1996) | MR

[7] LeCam L.: Locally asymptotically normal families of distributions. Univ. Calif. Publ. Statist. 3 (1960), 37–98 | MR

[8] Liese F.: Eine informationstheoretische Bedingung für die Äquivalenz unbegrenzt teilbarer Punktprozesse. Math. Nachr. 70 (1975), 183–196 | DOI | MR | Zbl

[9] Liese F.: Hellinger integrals of diffusion processes. Statistics 17 (1986), 63–78 | DOI | MR | Zbl

[10] Liese F., Vajda I.: Convex Statistical Distances. Teubner, Leipzig 1987 | MR | Zbl

[11] Lorz U.: Sekundärgröen Poissonscher Punktprozesse – Grenzwertsätze und Abschätzungen der Konvergenzgeschwindigkeit. Rostock. Math. Kolloq. 29 (1986), 99–111 | MR

[12] Lorz U.: Beiträge zur Statistik unbegrenzt teilbarer Felder mit unabhängigen Zuwächsen. Dissertation, Univ. Rostock 1987 | Zbl

[13] Lorz U., Heinrich L.: Normal and Poisson approximation of infinitely divisible distribution function. Statistics 22 (1991), 627– 649 | DOI | MR

[14] Mecke J.: Stationäre zufällige Maße auf lokal–kompakten Abelschen Gruppen. Z. Wahrsch. verw. Geb. 9 (1967), 36–58 | DOI | MR

[15] Petrov V. V.: Sums of Independent Random Variables. Akademie–Verlag, Berlin 1975 | MR | Zbl

[16] Strasser H.: Mathematical Theory of Statistics. de Gruyter, Berlin 1985 | MR | Zbl

[17] Vajda I.: Theory of Statistical Inference and Information. Kluwer, Dordrecht 1989 | Zbl