Hostname: page-component-788cddb947-tr9hg Total loading time: 0 Render date: 2024-10-14T14:50:38.146Z Has data issue: false hasContentIssue false

Asymptotic distributions of weighted pontograms under contiguous alternatives

Published online by Cambridge University Press:  24 October 2008

Barbara Szyszkowicz
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6

Extract

Let {N(x), x ≥ 0} be a non-homogeneous, also called non-stationary, Poisson process with density function λ(x), x ≥ 0. We consider the problem of testing the null hypothesis of λ(x) having a constant value against the alternative that λ(x) is a function of x.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Billingsley, P.. Convergence of Probability Measures (Wiley, 1968).Google Scholar
[2]Chibisov, D.. Some theorems on the limiting behaviour of empirical distribution functions. Selected Transl. Math. Statist. Prob. 6 (1964), 147156.Google Scholar
[3]Csörgő, M.. Quantile Processes with Statistical Applications (SIAM, 1983).Google Scholar
[4]Csörgő, M., Csörgő, S., Horváth, L. and Mason, D.. Weighted empirical and quantile processes. Ann. Probab. 14 (1986), 3185.CrossRefGoogle Scholar
[5]Csörgő, M. and Horváth, L.. Asymptotic distributions of pontograms. Math. Proc. Cambridge Philos. Soc. 101 (1987), 131139.CrossRefGoogle Scholar
[6]Csörgő, M. and Horváth, L.. Nonparametric methods for changepoint problems. In Handbook of Statistics, vol. 7 (Elsevier Science Publishers B.V., 1988), pp. 403425.Google Scholar
[7]Csörgő, M. and Horváth, L.. Invariance principles for changepoint problems. J. Multivariate Anal. 27 (1988), 151168.Google Scholar
[8]Csörgő, M. and Horváth, L.. On the distributions of L p norms of weighted uniform empirical and quantile processes. Ann. Probab. 16 (1988), 142161.Google Scholar
[9]Csörgő, M. and Horváth, L.. Weighted Approximations in Probability and Statistics (Book in preparation).Google Scholar
[10]Csörgő, M., Horváth, L. and Shao, Q. M.. Random integrals and summability of partial sums. Technical Report Series of the Laboratory for Research in Statistics and Probability, Carleton University, 168 (1991).Google Scholar
[11]Csörgő, M. and Révész, P.. Strong Approximations in Probability and Statistics (Academic Press, 1981).Google Scholar
[12]Csörgő, M., Shao, Q. M. and Szyszkowicz, B.. A note on local and global functions of a Wiener process and some Rényi-type statistics. Studia Sci. Math. Hungar. 26 (1991), in press.Google Scholar
[13]Eastwood, V. R.. Some recent developments concerning asymptotic distributions of pontograms. Math. Proc. Cambridge Philos. Soc. 108 (1990), 559567.CrossRefGoogle Scholar
[14]Greenwood, P. E. and Shiryayev, A. N.. Contiguity and the Statistical Invariance Principle (Gordon and Breach, 1985).Google Scholar
[15]Hájek, J. and Šidák, Z.. Theory of Rank Tests (Academic Press, 1967).Google Scholar
[16]Itô, K. and McKean, H. P. Jr,. Diffusion Processes and their Sample Paths (Springer-Verlag, 1965).Google Scholar
[17]Kendall, D. G. and Kendall, W. S.. Alignments in two-dimensional random sets of points. Adv. in Appl. Prob. 12 (1980), 380424.CrossRefGoogle Scholar
[18]Cam, L. Le. Locally asymptotically normal families of distributions. Univ. California Publ. Statist. 3 (1960), 3798.Google Scholar
[19]Cam, L. Le. Asymptotic Methods in Statistical Decision Theory (Springer-Verlag, 1986).CrossRefGoogle Scholar
[20]O'Reilly, N.. On the weak convergence of empirical processes in sup-norm metrics. Ann. Probab. 2 (1974), 642651.CrossRefGoogle Scholar
[21]Roussas, G. G.. Contiguity of Probability Measures: Some Applications in Statistics (Cambridge University Press, 1972).Google Scholar
[22]Shorack, G. R. and Wellner, J. A.. Empirical Processes with Applications to Statistics. (Wiley, 1986).Google Scholar
[23]Szyszkowick, B.. Changepoint problem and contiguous alternatives. Statist. Probab. Lett. 11 (1991), 299308.Google Scholar