A partially observed dynamic germ-grain model with renewal dropping process is considered. The expected fraction of free area function is estimated by a product integral-type estimator. Uniform consistency and asymptotic Gaussianity of the estimator are proved. Confidence bands and simulation results are also provided.
P. K. Andersen, O. Borgan, R. D. Gill, and N. Keiding, Statistical Models Based on Counting Processes, Springer, New York, 1997.
View at: Google ScholarP. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, 2nd edition, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNetV. Capasso, M. De Giosa, and R. M. Mininni, “Asymptotic properties of the maximum likelihood estimators of parameters of a spatial counting process modelling crystallization of polymers,” Stochastic Analysis and Applications, vol. 13, no. 3, pp. 279–294, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNetV. Capasso, A. Micheletti, M. De Giosa, and R. M. Mininni, “Stochastic modelling and statistics of polymer crystallization processes,” Surveys on Mathematics for Industry, vol. 6, no. 2, pp. 109–132, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNetD. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Series in Statistics, Springer, New York, 1988.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. De Giosa and R. M. Mininni, “Free area estimation in a partially observed dynamic germ-grain model,” Journal of Applied Mathematics and Stochastic Analysis, vol. 15, no. 4, pp. 301–321, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNetR. D. Gill and S. Johansen, “A survey of product-integration with a view toward application in survival analysis,” The Annals of Statistics, vol. 18, no. 4, pp. 1501–1555, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press, New York, 2nd edition, 1975.
View at: Google Scholar | Zentralblatt MATH | MathSciNetR. D. Ripley, Spatial Statistics, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1981.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. Schumacher, “Two-sample tests of Cramér-von Mises- and Kolmogorov-Smirnov-type for randomly censored data,” International Statistical Review. Revue International de Statistique, vol. 52, no. 3, pp. 263–281, 1984.
View at: Google Scholar | Zentralblatt MATH | MathSciNetD. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and Its Applications, John Wiley & Sons, New York, 2nd edition, 1995.
View at: Google Scholar | Zentralblatt MATH