The Annals of Statistics

On the Behrens–Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM tests

Alexandre Belloni and Gustavo Didier

Full-text: Open access

Abstract

In this paper we provide a provably convergent algorithm for the multivariate Gaussian Maximum Likelihood version of the Behrens–Fisher Problem. Our work builds upon a formulation of the log-likelihood function proposed by Buot and Richards [5]. Instead of focusing on the first order optimality conditions, the algorithm aims directly for the maximization of the log-likelihood function itself to achieve a global solution. Convergence proof and complexity estimates are provided for the algorithm. Computational experiments illustrate the applicability of such methods to high-dimensional data. We also discuss how to extend the proposed methodology to a broader class of problems.

We establish a systematic algebraic relation between the Wald, Likelihood Ratio and Lagrangian Multiplier Test (WLRLM) in the context of the Behrens–Fisher Problem. Moreover, we use our algorithm to computationally investigate the finite-sample size and power of the Wald, Likelihood Ratio and Lagrange Multiplier Tests, which previously were only available through asymptotic results. The methods developed here are applicable to much higher dimensional settings than the ones available in the literature. This allows us to better capture the role of high dimensionality on the actual size and power of the tests for finite samples.

Article information

Source
Ann. Statist. Volume 36, Number 5 (2008), 2377-2408.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
http://projecteuclid.org/euclid.aos/1223908096

Digital Object Identifier
doi:10.1214/07-AOS528

Mathematical Reviews number (MathSciNet)
MR2458191

Zentralblatt MATH identifier
05368495

Subjects
Primary: 62H15: Hypothesis testing

Keywords
Behrens–Fisher Problem high-dimensional data hypothesis testing algorithm Wald Test Likelihood Ratio Test Lagrange Multiplier Test size power

Citation

Belloni, Alexandre; Didier, Gustavo. On the Behrens–Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM tests. Ann. Statist. 36 (2008), no. 5, 2377--2408. doi:10.1214/07-AOS528. http://projecteuclid.org/euclid.aos/1223908096.


Export citation

References

  • [1] Behrens, W. (1929). Ein beitrag zur fehlerberechnung bei wenigen beobachtungen [a contribution to error estimation with few observations]. Landwirtschaftliche Jahrbücher 68 807–837.
  • [2] Berndt, E. and Savin, N. E. (1977). Conflict among criteria for testing hypothesis in the multivariate linear regression model. Econometrica 45 1263–1278.
  • [3] Bonnans, J. F., Gilbert, J. C., Lemaréchal, C. and Sagastizábal, C. A. (2003). Numerical Optimization. Springer, Berlin.
  • [4] Breusch, T. S. (1979). Conflict among criteria for testing hypothesis: Extensions and comments. Econometrica 47 203–208.
  • [5] Buot, M.-L. G. and Richards, D. S. P. (2007). Counting and locating the solutions of polynomial systems of maximum likelihood equations. II. The Behrens–Fisher Problem. Statist. Sinica 17 1343–1354.
  • [6] Chien, J.-T. (2005). Decision tree state tying using cluster validity criteria. IEEE Trans. Speech Trans. Audio Processing 13 182–193.
  • [7] Conn, A. R., Gould, N. I. M. and Toint, P. L. (2000). Trust-Region Methods. SIAM, Philadelphia, PA.
  • [8] Cox, D. R. (1984). Effective degrees of freedom and the likelihood ratio test. Biometrika 71 487–493.
  • [9] Fisher, R. A. (1935). The fiducial argument in statistical inference. Ann. Eugenics 6 391–398.
  • [10] Fisher, R. A. (1939). The comparison of samples with possibly unequal variances. Ann. Eugenics 9 174–180.
  • [11] Godfrey, L. G. (1988). Misspecification Tests in Econometrics. Cambridge Univ. Press.
  • [12] Greene, W. H. (1997). Econometric Analysis, 3rd ed. Prentice-Hall Inc., Upper Saddle River, NJ.
  • [13] Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms. Springer, Berlin.
  • [14] Kelley, J. E. (1960). The cutting plane method for solving convex programs. J. Soc. Indust. Appl. Math. 8 703–712.
  • [15] Kim, S.-H. and Cohen, A. (1998). On the Behrens–Fisher Problem: A review. J. Educational and Behavioral Statist. 23 356–377.
  • [16] Kim, S.-J. (1992). A practical solution to the multivariate Behrens–Fisher Problem. Biometrika 79 171–176.
  • [17] Kirkpatrick, S. Gelatt Jr. C. D. and Vecchi, M. P. (1983). Optimization by simulated annealing. Science 220 671–680.
  • [18] Krishnamoorthy, K. and Yu, J. (2004). Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher Problem. Statist. Probab. Lett. 66 161–169.
  • [19] Laarhoven, P. J. M. and Aarts, E. H. L. (1987). Simulated Annealing: Theory and Applications. Kluwer Academic Publishers, Norwell, MA.
  • [20] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, New York.
  • [21] Murphy, B. J. (1987). Selecting out of control variables with the T2 multivariate quality control procedure. The Statistician 36 571–581.
  • [22] Nesterov, Y. and Nemirovskii, A. (1994). Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, PA.
  • [23] Robbins, H., Simons, G. and Starr, N. (1967). A sequential analogue of the Behrens–Fisher Problem. Ann. Math. Statist. 38 1384–1391.
  • [24] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • [25] Savin, N. E. (1976). Conflict among testing procedures in a linear regression model with autoregressive disturbances. Econometrica 44 1303–1315.
  • [26] Scheffé, H. (1943). On solutions of the Behrens–Fisher Problem, based on the t-distribution. Ann. Math. Statist. 14 35–44.
  • [27] Scheffé, H. (1970). Practical solutions of the Behrens–Fisher Problem. J. Amer. Statist. Assoc. 65 1501–1508.
  • [28] Schramm, C., Renn, S. and Biles, B. (1986). Controlling hospital cost inflation: New perspectives on state rate setting. Health Affairs Fall 22–33.
  • [29] Smale, S. (1986). Newton’s method estimates from data at one point. In The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Laramie, Wyo., 1985) 185–196. Springer, New York.
  • [30] Srivastava, M. S. (2007). Multivariate analysis with fewer observations than the dimension: A review. Preprint. Available at http://www.utstat.utoronto.ca/%7Esrivasta/.
  • [31] Srivastava, M. S. (2007). Multivariate theory for analyzing high-dimensional data. J. Japan Statist. Soc. To appear.
  • [32] Stuart, A. and Ord, J. K. (1994). Kendalls Advanced Theory of Statistics, 6th ed. Edward Arnold, London.
  • [33] Subrahmaniam, K. and Subrahmaniam, K. (1973). On the multivariate Behrens–Fisher problem. Biometrika 60 107–111.
  • [34] Welch, B. L. (1938). The significance of the difference between two means when the population variances are unequal. Biometrika 29 350–362.
  • [35] Welch, B. L. (1947). The generalization of ‘Student’s’ problem when several different population variances are involved. Biometrika 34 28–35.
  • [36] Yanagihara, Y. and Yuan, K.-H. (2005). Three approximate solutions to the multivariate Behrens–Fisher Problem. Comm. Statist.—Simul. Comput. 34 975–988.
  • [37] Yao, Y. (1965). An approximate degrees of freedom solution to the multivariate Behrens–Fisher Problem. Biometrika 52 139–147.
  • [38] Ye, Y. (1992). A new complexity result on minimization of a quadratic function with a sphere constraint. In Recent Advances in Global Optimization (C. Floudas and P. Pardalos, eds.) 19–31. Princeton Univ. Press.