D.D. Kosambi pp 101-107 | Cite as

# A Test of Significance for Multiple Observations

## Abstract

In the years at Fergusson College, Kosambi did extensive statistical measurements and was devising new ways of analysing the data. This work, a precursor to “Statistics in Function Space”, was reviewed by Abraham Wald who says “For the purpose of discriminating multivariate normal populations with respect to their mean values, test functions have been introduced and studied by H. Hotelling, R.A. Fisher, P.C. Mahalanobis, R.C. Bose, S.N. Roy and others. If the number of variates as well as the number of populations is greater than two, the application of the test would require the knowledge of certain probability distributions which have not yet been tabulated”. DDK proposes a method in this paper to overcome this lack, but falls short of convincing his reviewer Wald, who adds “The author states that \(F^*\) has the ordinary *F*-distribution tabulated by R.A. Fisher and others. It seems to the reviewer that this statement of the author would be correct only if the coefficients \(\lambda _1, \ldots , \lambda _p\) were chosen independently of the sample. Since \(\lambda _1, \ldots , \lambda _p\) are functions of the sample values, the sampling distribution of \(F^*\) will arise partly from the sampling variation of \(\lambda _1, \ldots , \lambda _p\) and consequently the distribution of \(F^*\) need not be the same as that of *F*.

## Keywords

Characteristic Root Multivariate Normal Distribution Degenerate Kernel Cranial Shape Orthonormal Function## References

- 1.H. Hoteling, The generalization of student’s ratio. Ann. Math. Stat.
**2**, 360–378 (1931).Google Scholar - 2.R.A. Fisher,
*Statistical Methods for Research Workers*, 7th edn. 294—298 (1938).Google Scholar - 3.P.C. Mahalanobis,
*Proc. Natl. Inst. Sci. India***2**, 49–55 (1936); R.C. Bose,*Sankhyā***2**, 143–154, 379–384 (1936); S.N. Roy,*Ibid.*, 385–396.Google Scholar - 4.S.S. Wilks, Certain generalizations in the analysis of variance. Biometrika
**24**, 471–494 (1932).Google Scholar - 5.D.D. Kosambi, A bivariate extension of Fisher’s \(z\)-test. Curr. Sci.
**10**, 191–492 (1941).Google Scholar - 6.S. Banach,
*Thèorie de l’Intègrale*, ed. by S. Saks (1933), pp. 204–272.Google Scholar - 7.P.L. Hsu, Biometrika
**31**, 221–237; Ann. Math. Stat. 9, 231–243 1939. J. London Math. Soc.**16**(1941), 183–194 (1940).Google Scholar - 8.F. Hausdorff, Math. Annalen
**79**, 157–179 (1919).Google Scholar