The correct asymptotic variance for the sample mean of a homogeneous Poisson marked point process.(English)Zbl 1274.60064

Summary: The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general $$d$$-dimensional case is also established.

MSC:

 60F05 Central limit and other weak theorems 60K99 Special processes
Full Text:

References:

 [1] Ballani, F., Kabluchko, Z. and Schlather, M. (2012). Random marked sets. Adv. Appl. Prob. 44 , 603-616. · Zbl 1266.60091 [2] Brillinger, D. R. (1973). Estimation of the mean of a stationary time series by sampling. J. Appl. Prob. 10 , 419-431. · Zbl 0258.62050 [3] Ivanov, A. V. and Leonenko, N. N. (1986). Statistical Analysis of Random Fields . Kluwer Publishers, Dordrecht. · Zbl 0721.62097 [4] Karr, A. F. (1986). Inference for stationary random fields given Poisson samples. Adv. Appl. Prob. 18 , 406-422. · Zbl 0598.60052 [5] Karr, A. F. (1991). Point Processes and Their Statistical Inference , 2nd edn. Marcel Dekker, New York. · Zbl 0733.62088 [6] Kutoyants, Y. A. (1984a). On nonparametric estimation of intensity function of inhomogeneous Poisson process. Problems Control Inf. Theory 13 , 253-258. · Zbl 0555.62068 [7] Kutoyants, Y. A. (1984b). Parameter Estimation for Stochastic Processes (Res. Exposition Math. 6 ). Heldermann, Berlin. [8] Masry, E. (1983). Nonparametric covariance estimation from irregularly-spaced data. Adv. Appl. Prob. 15 , 113-132. · Zbl 0508.62035 [9] Politis, D. N., Paparoditis, E. and Romano, J. P. (1999). Resampling marked point processes. In Multivariate Analysis, Design of Experiments, and Survey Sampling (Statist. Textbooks Monogr. 159 ), ed. Subir Ghosh, Marcel Dekker, New York, pp. 163-185. \endharvreferences · Zbl 0946.62087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.