Exact Laws for Sums of Logarithms of Uniform Spacings
DOI:
https://doi.org/10.17713/ajs.v32i1&2.448Abstract
In this paper we provide some simple expressions for the exact law of sums of logarithms of uniform spacings. We also obtain closed form expressions for the corresponding cumulants of arbitrary order, in terms of the Riemann zeta function. Our results follow from two representations of this statistic through sums of independent random variables, the latter being of independent interest. We conclude by showing that the distribution of sums of logarithms of uniform spacings are very closely approximated by a Gamma distribution even for very small values of the sample size. This renders easy the use of this statistic for goodness-of-fit tests of the uniformity assumption.
References
M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover, New York, 9th edition, 1970.
E.-E.A.A. Aly. Normalized spacings and goodness-of-fit tests. J. Statist. Comput. Simul., 36:117–127, 1990.
E.-E.A.A. Aly, J. Beirlant, and L. Horváth. Strong and weak approximations of k-spacings processes. Z. Wahrscheinlichkeit. verw. Gebiete, 66:461–484, 1984.
J. Beirlant, P. Deheuvels, J.H.J. Einmahl, and D.P. Mason. Bahadur-Kiefer theorems for uniform spacings processes. Teoria Verojantost. Primen., 36:724–743, 1991a.
J. Beirlant, L. Győrfi, and G. Lugosi. On the asymptotic normality of the L1-errors and L2-errors in histogram density estimation. Can. J. Statist., 22:308–318, 1994.
J. Beirlant and L. Horváth. Approximations of m-overlapping spacings processes. Scand. J. Statist., 11:225–245, 1984.
J. Beirlant, P. Janssen, and N. Veraverbeke. On the asymptotic normality of functions of uniform spacings. Can. J. Statist., 19:93–101, 1991b.
J. Beirlant, E.C. van der Meulen, and F.H. Ruymgaart. On functions bounding the empirical distribution of uniform spacings. Z.Wahrscheinlichkeit. verw. Gebiete, 61:417–430,
S. Blumenthal. Contribution to sample spacings theory, I: Limit distributions of sums of ratios of spacings. Ann. Math. Statist., 37:904–924, 1966a.
S. Blumenthal. Contribution to sample spacings theory, II: Test of the parametric goodness of fit and two-sample problems. Ann. Math. Statist., 37:925–939, 1966b.
S. Blumenthal. Logarithms of sample spacings. SIAM J. Appl. Math., 16:1184–1191, 1968.
V.P. Borovikov. Limit theorems for statistics that are partial-sums of functions of spacings. Theor. Probab. Appl., 32:86–97, 1988.
D.R. Cox. Some statistical methods connected with series of events. J. R. Statist. Soc. B, 17:129–164, 1955.
N. Cressie. On the logarithms of high-order spacings. Biometrika, 63:343–355, 1976.
N. Cressie. Power results for tests based on high-order gaps. Biometrika, 65:214–218, 1978.
F. Czekala. Asymptotic distribution of statistics based on logarithm of spacings. Zastos. Mat., 21:511–519, 1993.
D.A. Darling. On a class of problems related to the random division of an interval. Ann. Math. Statist., 24:239–253, 1953.
H.A. David. Order Statistics. J. Wiley, New York, 2nd edition, 1981.
P. Deheuvels. Strong bounds for multidimensional spacings. Z. Wahrscheinlichkeit. verw. Gebiete, 64:411–424, 1983.
G. del Pino. On the asymptotic distribution of some goodness of fit tests based on spacings (abstract). Bull. Inst. Math. Statist., 4:137, 1975.
J.H.J. Einmahl and M.C.A. van Zuijlen. Strong bounds for weighted empirical distribution functions based on uniform spacings. Ann. Probab., 16:108–125, 1988.
M. Ekstrom. Strong limit theorems for sums of logarithms of high order spacings. Statistics, 33:153–169, 1991.
I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series and Products. Academic Press, New York, 1982.
P. Guttorp and R.A. Lockhart. On the asymptotic distributions of high-order spacing statistics. Can. J. Statist., 19:419–426, 1989.
P. Hall. Limit theorems for estimators based on inverses of spacings of order statistics. Ann. Probab., 10:992–1003, 1982.
P. Hall. Limit theorems for general functions of m-spacings. Math. Proc. Cambridge Philos. Soc., 96:517–532, 1984.
L. Holst. Asymptotic normality of sum-functions of spacings. Ann. Probab., 7:1066–1072, 1979.
L. Holst. On convergence of the coverage by random arcs on a circle and the largest spacing. Ann. Probab., 9:648–655, 1981.
J. Koziol. A note on limiting distributions for spacings statistics. Z. Wahrscheinlichkeit verw. Gebiete, 51:55–62, 1980.
L. Le Cam. Un théorème sur la division d’un intervalle par des points pris au hasard. Publ. Inst. Statist. Univ. Paris, 7:7–16, 1958.
P. L’Ecuyer. Tests based on sum-functions of spacings for uniform random numbers. J. Stat. Comput. Sim., 59:251–269, 1997.
P. Lévy. Sur la division d’un segment par des points choisis au hasard. C. R. Acad. Sci. Paris, 208:147–149, 1939.
S. Malmquist. On a property of order statistics from a rectangular distribution. Skand. Aktuarietidskr., 33:214–222, 1950.
J. Naus. A power comparison of two tests of non random clustering. Technometrics, 8: 493–517, 1966.
V.V. Petrov. Sums of Independent Random Variables. Springer, New York, 1st edition, 1975.
R. Pyke. Spacings. J. R. Statist. Soc. B, 27:395–436 and discussion 437–449, 1965.
R. Pyke. Spacings revisited. Proc. 6th Berkeley Symp., 1:417–427, 1972.
B. Ranneby. The maximum spacing method: An estimation method related to the maximum likelihood method. Scand. J. Statist., 11:93–112, 1984.
C.R. Rao. Some tests based on arc-lengths for the circle. Sankhy¯a Ser. B, 33:1–10, 1976.
J. Sethuraman and J.S. Rao. Pitman efficiencies of tests based on spacings. In M.L. Puri, editor, Nonparametric Techniques in Statistical Inference, pages 405–415. Cambridge University Press, 1969.
Y. Shao and M.G. Hahn. Limit theorems for the logarithms of sample spacings. Statist. Prob. Lett., 24:121–132, 1995.
Y. Shao and R. Jiménez. Entropy for random partitions and its applications. J. Theor. Probab., 11:417–433, 1998.
G.R. Shorack. Convergence of quantile and spacings processes with applications. Ann. Math. Statist., 43:1400–1411, 1972.
G.R. Shorack and J.A. Wellner. Empirical Processes with Applications to Statistics. J. Wiley, New York, 1st edition, 1986.
A.F. Siegel. Random arcs on the circle. J. Appl. Probab., 15:774–779, 1978.
A.F. Siegel. Asymptotic coverage distributions on the circle. Ann. Probab., 7:651–661, 1979.
E. Slud. Entropy and maximal spacings for random partitions. Z. Wahrscheinlichkeit. verw. Gebiete, 41:341–352, 1978.
J. Spanier and K.B. Oldham. An Atlas of Functions. Hemisphere Publ. Co, Washington, 1987.
P.V. Sukhatme. Tests of significance for samples of the Â2 population with two degrees of freedom. Ann. Eugen., 8:52–56, 1937.
T. Swartz. Goodness of fit tests using Kullback-Leibler information. Commun. Stat. Simulat., 21:711–729, 1992.
L. Weiss. A certain class of tests of fit. Ann. Math. Statist., 27:1165–1170, 1956.
L. Weiss. The asymptotic power of certain tests of fit based on sample spacings. Ann. Math. Statist., 28:783–786, 1957.
Downloads
Published
Issue
Section
License
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.
How to Cite
