Hostname: page-component-6bf8c574d5-n2sc8 Total loading time: 0 Render date: 2025-03-04T04:43:10.894Z Has data issue: false hasContentIssue false
  • English
  • Français

From Characteristic Function to Distribution Function: A Simple Framework for the Theory

Published online by Cambridge University Press:  11 February 2009

N.G. Shephard
Get access Rights & Permissions [Opens in a new window]

Abstract

A unified framework is established for the study of the computation of the distribution function from the characteristic function. A new approach to the proof of Gurland's and Gil-Pelaez's univariate inversion theorem is suggested. A multivariate inversion theorem is then derived using this technique.

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 4 , December 1991 , pp. 519 - 529
Copyright
Copyright © Cambridge University Press 1991

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.Bohmann, H. Approximate Fourier analysis of distribution functions. Arkiv für Matematik 4 (1961): 99–1157.10.1007/BF02592003CrossRefGoogle Scholar
2.Bohmann, H. A method to calculate the distribution function when the characteristic function is known. Nordisk Tidskr. Informationsbehandling (BIT) 10 (1970): 237–242.Google Scholar
3.Cramér, H. Mathematical methods of statistics Princeton: Princeton University Press, 1946.Google Scholar
4.Davies, R.B. Numerical inversion of a characteristic function. Biometrika 60 (1973): 415–417.10.1093/biomet/60.2.415CrossRefGoogle Scholar
5.Davies, R.B. AS 155: The distribution of a linear combination of x2 random variables. Applied Statistics 29 (1980): 323–333.10.2307/2346911CrossRefGoogle Scholar
6.Farebrother, R.W. Testing linear restrictions with unequal variances, a problem. Econometric Theory 4 (1988): 349.10.1017/S0266466600012111CrossRefGoogle Scholar
7.Farebrother, R.W. Testing linear restrictions with unequal variances, a solution. Econometric Theory 5 (1989): 324–326.10.1017/S0266466600012494Google Scholar
8.Feller, W. Introduction to Probability Theory and Its Applications, Volume Two 2nd ed. New York: Wiley, 1971.Google Scholar
9.Gil-Pelaez, J. Note on the inversion theorem. Biometrika 37 (1951): 481–482.10.1093/biomet/38.3-4.481CrossRefGoogle Scholar
10.Gurland, J. Inversion formulae for the distribution of ratios. Annals of Mathematical Statistics 19 (1948): 228–237.10.1214/aoms/1177730247CrossRefGoogle Scholar
11.Hewitt, E. & Stromberg, K.R.. Real and Abstract Analysis New York: Springer-Verlag, 1965.Google Scholar
12.Imhof, J.P. Computing the distribution of quadratic forms in normal variables. Biometrika 48 (1961): 419–426.10.1093/biomet/48.3-4.419Google Scholar
13.Kendall, M.G., Stuart, A. & Ord, J.K.. Advanced Theory of Statistics, Vol. I 5th ed. London: Griffin, 1987.Google Scholar
14.Knott, M. The distribution of the Cramér-von Mises statistic for small sample sizes. Journal of the Royal Statistical Society, Series B 36 (1974): 430–438.Google Scholar
15.Léy, P. Calcul des probabilités Paris: Gauthier-Villars, 1925.Google Scholar
16.Lukacs, E. Characteristic Functions London:Griffin, 1970.Google Scholar
17.Phillips, P.C.B. Exact small sample theory in the simultaneous equations model. In Handbook of Econometrics, Volume 1, Griliches, Z. & Intriligator, M.D. (eds.), Amsterdam: North-Holland Publishing Company, 1983.Google Scholar
18.Phillips, P.C.B. The distribution of matrix quotients. Journal of Multivariate Analysis 16 (1985): 157–161.10.1016/0047-259X(85)90056-9CrossRefGoogle Scholar
19.Rubin, W. Real and Complex Analysis New York: McGraw-Hill, 1970.Google Scholar
20.Shephard, N.G. Numerical integration rules for multivariate inversions. Journal of Statistical Computation and Simulation 39 (1991): 37–46.10.1080/00949659108811337CrossRefGoogle Scholar
21.Shephard, N.G. Evaluating the distribution function of the maximum likelihood estimator of a first order moving average process and a local level model. Working paper, London School of Economics, 1990.Google Scholar
22.Shephard, N.G. Tabulation of Farebrother's test for linear restrictions in linear regression models under heteroscedasticity. Working paper, London School of Economics, 1990.Google Scholar
23.Shively, T.S. Numerical inversion of a bivariate characteristic function. Working paper, University of Texas at Austin, 1988.Google Scholar
24.Shively, T.S. An analysis of tests for regression coefficient stability. Journal of Econometrics 39 (1988): 367–386.10.1016/0304-4076(88)90064-4CrossRefGoogle Scholar
25.Watson, M.W. & Engle, R.F.. Testing for regression coefficient stability with a stationary AR(1) alternative. Review of Economics and Statistics 67 (1985): 341–346.10.2307/1924737CrossRefGoogle Scholar