Computing moments of ratios of quadratic forms in normal variables

Computational Statistics and Data Analysis

Volumn 42, Issue 3, 2003, Pages 313-331

  Paolella, Marc S ^a  

^a UNIVERSITY OF KIEL   (Germany)

Author keywords

Autocorrelation function; Saddlepoint approximation

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; DATA REDUCTION; FUNCTIONS; NUMERICAL METHODS;

AUTOCORRELATION FUNCTIONS;

STATISTICS;

NUMERICAL METHOD;

EID: 0037471381     PISSN: 01679473     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-9473(02)00213-X     Document Type: Article

References (34)

1. Adler, R.J., Feldman, R.E., Taqqu, M.S. (Eds.), 1998. A Practical Guide to Heavy Tails. Birkhäuser, Boston.
2. Ali, M.M., 1984. Distributions of the sample autocorrelations when observations are from a stationary autoregressive-moving-average process. J. Bus. Econom. Statist. 2(3), 271-278.
3. Anderson, O.D., 1993. Exact general-lag serial correlation moments and approximate low-lag partial correlation moments for gaussian white noise. J. Time Ser. Anal. 14 (6), 551-574.
4. Boshnakov, G.N., 1996. Bartlett's formula-closed forms and recurrent equations. Ann. Inst. Statist. Math. 48 (1), 49-59.
5. Butler, R.W., Paolella, M.S., 1998. Approximate distributions for the various serial correlograms. Bernoulli 4 (4), 497-518.
6. Butler, R.W., Paolella, M.S., 1999. Saddlepoint approximations to the density and distribution of ratios of quadratic forms in normal variables with application to the sample autocorrelation function. Working Paper, Department of Statistics, Colorado State University.
7. Daniels, H.E., 1954. Saddlepoint approximation in statistics. Ann. Math. Statist. 25, 631-650.
8. De Gooijer, J.G., 1980. Exact moments of the sample autocorrelations from series generated by general arima processes of order (p,d,q) d=0 or 1. J. Econometrics, 14 365-379.
9. Dufour, J.-M, Roy, R., 1985. Some robust exact results on sample autocorrelations and tests of randomness. J. Econometrics 29, 257-273.
10. Farebrother, R.W., 1990. The distribution of a quadratic form in normal variables. Appl. Statist. 39, 294-309.
11. Ghazal, G.A., 1994. Moments of the ratio of two dependent quadratic forms. Statist. Probab. Lett. 20, 313-319.
12. Graybill, F.A., 1983, Matrices with Applications in Statistics. Wadsworth and Brooks/Cole, Pacific Grove, CA.
13. Imhof, P., 1961. Computing the distribution of quadratic forms in normal variates. Biometrika 48, 419-426.
14. Kanto, A.J., 1988. Covariances between estimated autocorrelations of an arma process. Econom. Lett. 26, 253-258.
15. Krämer, W., Sonnberger, H., 1986. The Linear Regression Model Under Test. Physica-Verlag, Heidelberg
16. Lieberman, O., 1994. A laplace approximation to the moments of a ratio of quadratic forms. Biometrika 81 (4), 681-690.
17. Lieberman, O., 1994. Saddlepoint approximation for the distribution of a ratio of quadratic forms in normal variables. J. Amer. Statist. Assoc. 89 (427), 924-928.
18. Lugannani, R., Rice, S.O., 1980. Saddlepoint approximations for the distribution of sums of independent random variables. Adv. Appl. Probab. 12, 475-490.
19. Magnus, J.R., 1986. The exact moments of a ratio of quadratic forms in normal variables. Ann. Econom. Statist. 4, 95-109.
20. Magnus, J.R., Pesaran, B., 1993. Evaluation of moments of quadratic forms and ratios of quadratic forms in normal variables. Comput. Statist. 8 (1), 21-56.
21. Mathai, A. M., & Provost, S. B. (1992). Quadratic Forms in Normal Random Variables. Marcel Dekker, New York.
22. Mehta, J.S., Swamy, P.A., 1978. The existence of moments of some simple bayes estimators of coefficients in a simultaneous equation model. J. Econometrics 7, 1-13.
23. Moran, P.A., 1948. Some theorems on time series, ii: The significance of the serial correlation coefficient. Biometrika 35, 255-260.
24. Moran, P.A., 1967. Testing for serial correlation with exponentially distributed variates. Biometrika 54, 395-401.
25. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1989. Numerical Recipes: The Art of Scientific Computing (FORTRAN Version). Cambridge University Press, New York.
26. Provost, S.B., Rudiuk, E.M., 1996. The exact distribution of indefinite quadratic forms in noncentral normal vectors. Ann. Inst. Statist. Math. 48 (2), 381-394.
27. Roberts, L.A., 1995. On the existence of moments of ratios of quadratic forms. Econometric Theory 11, 750-774.
28. Sawa, T., 1972. Finite sample properties of the k-class estimator. Econometrica 40 (4), 653-680.
29. Sawa, T., 1978. The exact moments of the least squares estimator for the autoregressive model. J. Econometrics 8, 159-172.
30. Smith, M.D., 1989. On the expectations of a ratio of quadratic forms in normal variables. J. Multivariate Anal. 31, 244-257.
31. Smith, M.D., 1993. Expectations of ratios of quadratic forms in normal variables: Evaluating some top-order invariant polynomials. Austral. J. Statist. 35 (3), 271-282.
32. Smith, M.D., 1994. Discussion of: "a laplace approximation to the moments of a ratio of quadratic forms," by O. Lieberman. Mimeo.
33. Ullah, A., Srivastava, V.K., 1994. Moments of the ratio of quadratic forms in non-normal variables with econometric examples. J. Econometrics 62, 129-141.
34. Ullah, A., Srivastava, V.K., Roy, N., 1995. Moments of the function of non-normal random vector with applications to econometric estimators and test statistics. Econometric Rev., 14 (4), 459-471.