Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-06T06:45:21.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for sums of general functions of m-spacings

Published online by Cambridge University Press:  24 October 2008

Peter Hall
Peter Hall
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Extract

Laws of large numbers and central limit theorems are proved for sums of general functions of m-spacings from general distributions. Explicit formulae are given for the norming constants. The results enable us to describe asymptotic properties of distributional tests under fixed alternatives. A generalization of Kimball's spacings test is considered in detail.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 517 - 532
Copyright
Copyright © Cambridge Philosophical Society 1984

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]Le Cam, L.. Un théorème sur la division d'un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris 7 (1958), 7–16.Google Scholar
[2]Cressie, N.. On the logarithms of high-order spacings. Biometrika 63 (1976), 343–355.CrossRefGoogle Scholar
[3]Cressie, N.. The minimum of higher order gaps. Austral. J. Statist 19 (1977), 132–143.CrossRefGoogle Scholar
[4]Cressie, N.. Power results for tests based on high-order gaps. Biometrika 65 (1978), 214–218.CrossRefGoogle Scholar
[5]Darling, D. A.. On a class of problems relating to the random division of an interval. Ann. Math. Statist. 24 (1953), 239–253.CrossRefGoogle Scholar
[6]Gnedenko, B. V. and Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, 1954).Google Scholar
[7]Hall, P.. Limit theorems for estimates based on inverses of spacings of order statistics. Ann. Probab. 10 (1982), 992–1003.CrossRefGoogle Scholar
[8]Holst, L.. Asymptotic normality of sum-functions of spacings. Ann. Probab. 7 (1979), 1066–1072.CrossRefGoogle Scholar
[9]Holst, L.. Some conditional limit theorems in exponential families. Ann. Probab. 9 (1981), 818–830.CrossRefGoogle Scholar
[10]Kimball, B. F.. On the asymptotic distribution of the sum of powers of unit frequency differences. Ann. Math. Statist 21 (1951), 263–271.CrossRefGoogle Scholar
[11]Koziol, J. A.. A note on limiting distributions for spacings statistics. Z. Wahrsch. Verw. Gebiete. 51 (1980), 55–62.CrossRefGoogle Scholar
[12]Delpino, G. E.. On the asymptotic distribution of fc-spacings with applications to goodness-of-fit tests. Ann. Statist., 7 (1979), 1058–1065.Google Scholar
[13]Pbescott, P.. On a test for normality based on sample entropy. J. Roy. Statist. Soc. Ser. B 38 (1976), 254–256.Google Scholar
[14]Pyke, R.. Spacings (with Discussion). J. Roy. Statist. Soc. Ser. B 7 (1965), 395–449.Google Scholar
[15]Pyke, R.. Spacings revisited. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 1 (1972), 417–427.Google Scholar
[16]Rao, J. S.. Some tests based on arc-lengths for the circle. Sankhya Ser. B 38 (1976), 329–338.Google Scholar
[17]Rao, J. S. and Sethtjraman, J.. Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors. Ann. Statis. 3 (1975), 299–313.CrossRefGoogle Scholar
[18]Sethubaman, J. and Rao, J. S.. Pitman efficiencies of tests based on spacings. In Non-parametric Techniques in Statistical Inference, ed. Puri, M. L. (Cambridge University Press, 1970), 267–275.Google Scholar
[19]Vasicek, O.. A test for normality based on sample entropy. J. Roy. Statist. Soc. Ser. B 38 (1976), 54–59.Google Scholar
[20]Weiss, L.. On asymptotic sampling theory for distributions approaching the uniform distribution. Z. Wahrsch. Verw. Gebiete 4 (1965), 217–221.CrossRefGoogle Scholar
[21]Weiss, L.. Limiting distributions of homogeneous functions of sample spacings for distributions approaching the uniform distribution. Z. Wahrsch. Verw. Gebiete 10 (1968), 193–197.CrossRefGoogle Scholar