Behrens–Fisher problem
Is an approximation analogous to Fisher's argument necessary to solve the Behrens–Fisher problem?
In
Specification
One difficulty with discussing the Behrens–Fisher problem and proposed solutions, is that there are many different interpretations of what is meant by "the Behrens–Fisher problem". These differences involve not only what is counted as being a relevant solution, but even the basic statement of the context being considered.
Context
Let X1, ..., Xn and Y1, ..., Ym be
Requirements of solutions
Solutions to the Behrens–Fisher problem have been presented that make use of either a
The task of specifying interval estimates for this problem is one where a frequentist approach fails to provide an exact solution, although some approximations are available. Standard Bayesian approaches also fail to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found.[citation needed] Thus study of the problem can be used to elucidate the differences between the frequentist and Bayesian approaches to interval estimation.
Outline of different approaches
Behrens and Fisher approach
Ronald Fisher in 1935 introduced fiducial inference[3][4] in order to apply it to this problem. He referred to an earlier paper by Walter-Ulrich Behrens from 1929. Behrens and Fisher proposed to find the probability distribution of
where and are the two
Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be
Welch's approximate t solution
A widely used method is that of B. L. Welch,[6] who, like Fisher, was at University College London. The variance of the mean difference
results in
Welch (1938) approximated the distribution of by the Type III Pearson distribution (a scaled chi-squared distribution) whose first two moments agree with that of . This applies to the following number of degrees of freedom (d.f.), which is generally non-integer:
Under the null hypothesis of equal expectations, μ1 = μ2, the distribution of the Behrens–Fisher statistic T, which also depends on the variance ratio σ12/σ22, could now be approximated by
This is a random variable. A t distribution with a random number of degrees of freedom does not exist. Nevertheless, the Behrens–Fisher T can be compared with a corresponding quantile of
This method also does not give exactly the nominal rate, but is generally not too far off.[citation needed] However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it is more accurate to use Student's t-test.[citation needed]
Other approaches
A number of different approaches to the general problem have been proposed, some of which claim to "solve" some version of the problem. Among these are,[7]
In Dudewicz’s comparison of selected methods,[7] it was found that the Dudewicz–Ahmed procedure is recommended for practical use.
Exact solutions to the common and generalized Behrens–Fisher problems
For several decades, it was commonly believed that no exact solution to the common Behrens–Fisher problem existed.[citation needed] However, it was proved in 1966 that it has an exact solution.[12] In 2018 the probability density function of a generalized Behrens–Fisher distribution of m means and m distinct standard errors from m samples of distinct sizes from independent normal distributions with distinct means and variances was proved and the paper also examined its asymptotic approximations.[13] A follow-up paper showed that the classic paired t-test is a central Behrens–Fisher problem with a non-zero population correlation coefficient and derived its corresponding probability density function by solving its associated non-central Behrens–Fisher problem with a nonzero population correlation coefficient.[14] It also solved a more general non-central Behrens–Fisher problem with a non-zero population correlation coefficient in the appendix.[14]
Variants
A minor variant of the Behrens–Fisher problem has been studied.[15] In this instance the problem is, assuming that the two population-means are in fact the same, to make inferences about the common mean: for example, one could require a confidence interval for the common mean.
Generalisations
One generalisation of the problem involves multivariate normal distributions with unknown covariance matrices, and is known as the multivariate Behrens–Fisher problem.[16]
The nonparametric Behrens–Fisher problem does not assume that the distributions are normal.[17][18] Tests include the Cucconi test of 1968 and the Lepage test of 1971.
Notes
- ^ Lehmann (1975) p.95
- ^ Lehmann (1975) Section 7
- hdl:2440/15222.
- ^ "R. A. Fisher's Fiducial Argument and Bayes' Theorem by Teddy Seidenfeld" (PDF).
- ^ "Sezer, A. et al. Comparison of confidence intervals for the Behrens–Fisher Problem Comm. Stats. 2015".
- ^ Welch (1938, 1947)
- ^ a b Dudewicz, Ma, Mai, and Su (2007)
- .
- ^ Prokof'yev, V. N.; Shishkin, A. D. (1974). "Successive classification of normal sets with unknown variances". Radio Engng. Electron. Phys. 19 (2): 141–143.
- ^ Dudewicz & Ahmed (1998, 1999)
- arXiv:2210.16473 [math.ST].
- S2CID 120965543.
- . Retrieved 21 May 2020.
- ^ S2CID 125245802. Retrieved 21 May 2020.
- ISBN 0-521-83971-8(page 204)
- ^ Belloni & Didier (2008)
- .
- . Retrieved 26 September 2016.
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2010) |
References
- Behrens, W. U. (1929). "Ein Beitrag zur Fehlerberechnung bei wenigen Beobachtungen" [A contribution to error estimation with few observations]. Landwirtschaftliche Jahrbücher. 68. Berlin: Wiegandt and Hempel: 807–37.
- Belloni, A.; Didier, G. (2008). "On the Behrens–Fisher Problem: A Globally Convergent Algorithm and a Finite-Sample Study of the Wald, LR and LM Tests". S2CID 15968707.
- Chang, CH; Pal, N (2008). "A revisit to the Behrens–Fisher problem: Comparison of five test methods". Communications in Statistics - Simulation and Computation. 37 (6): 1064–1085. S2CID 32811488.
- Dudewicz, E. J.; Ahmed, S. U. (1998). "New exact and asymptotically optimal solution to the Behrens–Fisher problem, with tables". American Journal of Mathematical and Management Sciences. 18 (3–4): 359–426. .
- Dudewicz, E. J.; Ahmed, S. U. (1999). "New exact and asymptotically optimal heteroscedastic statistical procedures and tables, II". American Journal of Mathematical and Management Sciences. 19 (1–2): 157–180. .
- Dudewicz, E. J.; Ma, Y.; Mai, S. E.; Su, H. (2007). "Exact solutions to the Behrens–Fisher problem: Asymptotically optimal and finite sample efficient choice among". Journal of Statistical Planning and Inference. 137 (5): 1584–1605. .
- Fisher, R. A. (1935). "The fiducial argument in statistical inference". Annals of Eugenics. 8 (4): 391–398. hdl:2440/15222.
- Fisher, R. A. (1941). "The Asymptotic Approach to Behrens' Integral with further Tables for the d Test of Significance". Annals of Eugenics. 11: 141–172. .
- Fraser, D. A. S.; .
- Lehmann, E. L. (1975) Nonparametrics: Statistical Methods Based on Ranks, Holden-Day ISBN 0-07-037073-7
- Ruben, H. (2002)"A simple conservative and robust solution of the Behrens–Fisher problem", Sankhyā:The Indian Journal of Statistics, Series A, 64 (1),139–155.
- Pardo, JA; Pardo, MD (2007). "A simulation study of a new family of test statistics for the Behrens–Fisher problem". Kybernetes. 36 (5–6): 806–816. .
- Sawilowsky, Shlomo S (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ1 ≠ σ2". Journal of Modern Applied Statistical Methods. 1 (2). .
- Welch, B. L. (1938). "The significance of the difference between two means when the population variances are unequal". JSTOR 2332010.
- Welch, B. L. (1947), "The generalization of "Student's" problem when several different population variances are involved", PMID 20287819
- Voinov, V.; Nikulin, M. (1995). "On the problem of means of weighted normal populations". Questiio. 19 (2): 7–20.
- Zheng, SR; Shi, NZ; Ma, WQ (2010). "Statistical inference on difference or ratio of means from heteroscedastic normal populations". Journal of Statistical Planning and Inference. 140 (5): 1236–1242. .
External links
- Dong, B.L. (2004) The Behrens–Fisher Problem: An Empirical Likelihood Approach Econometrics Working Paper EWP0404, University of Victoria