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The Ghosh distribution is a univariate probability distribution defined with the probability density function (PDF) [1]:
where [k] is the largest integer ≤ k.
The formula provides a counterexample to the theorem on similar regions in hypothesis testing (also called “regions similar to the sample space with respect to composite parameters”). Ghosh’s work revealed that all similar regions didn’t always have detailed similarity. In other words, it is what we would call Neyman Structure today; The Neyman Structure helps ensure that all variables are taken into account when determining a statistic and that its results are reliable.
Ghosh Distribution History
The distribution is named after Manindra Nath Ghosh (1918-1965), a foundational member of the Calcutta Statistical association who died “in the prime of his activities” from coronary heart failure in 1965 [2].
Ghosh’s distribution originally appeared in a 1948 article titled On the Problem of Similar Regions [3] which was published in volume 8 of Sankhyā: The Indian Journal of statistics. Outside of its entry in Index to the Distributions of Mathematical Statistics, it is seldom referred to in the literature — and never in modern texts. As such, the distributions seems lost to history.
References
[1] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
[2] Calcutta Statistical Association Bulletin. Vol. 14, September & December, 1965. Nos, 55&56. Online: https://journals.sagepub.com/doi/abs/10.1177/0008068319650301?journalCode=csaa
[3] Ghosh, M. N. On the problem of similar regions. Sankhyā: The Indian Journal of Statistics (1933-1960) Vol. 8, No. 4 (Jun., 1948), pp. 329-338 (10 pages).