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Anil Kumar Bhattacharyya

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Anil kumar Bhattacharyya (Bengali: অনিল কুমার ভট্টাচার্য্য) (1 April 1915 – 17 July 1996) was an Indian statistician who worked at the Indian Statistical Institute in the 1930s and early 40s. He made fundamental contributions to multivariate statistics, particularly for his measure of similarity between two multinomial distributions, known as the Bhattacharyya coefficient, based on which he defined a metric, the Bhattacharyya distance.[1] This measure is widely used in comparing statistical samples in biology, genetics,[2] physics, computer science, etc.

Anil Kumar Bhattacharyya
Born(1915-04-01)1 April 1915
Bhatpara, Kolkata Bengal Presidency, British India (Now West Bengal, India)
Died17 July 1996(1996-07-17) (aged 81)
CitizenshipIndia
Alma materRajabazar Science College
(University of Calcutta)
Known forBhattacharyya distance, Bhattacharyya bound
Scientific career
FieldsStatistician
InstitutionsUniversity of Calcutta, Indian Statistical Institute

Life edit

Bhattacharyya was born to Bhavanath and Lilavati, sometime in March–April 1915 (in the month Chaitra Bengali: চৈত্র of the year 1321, the exact date is not known)[3] at Bhatpara in the district of 24 Parganas of West Bengal. His birth name was Anilkumar Bhattacharyya (does not have any middle name, his entire first name was Anilkumar, thus in all of his works his name appeared as A. Bhattacharyya). He passed the Matriculation Examination of Calcutta University in 1932 and I. Sc. Examination in 1934 from Hooghly Mohsin College. In 1936 he ranked first in the First Class at the B.A./B.Sc. examination from the same college and went over to the renowned Science College, Calcutta University for an M.Sc. in Mathematics. Here he had F. W. Levi and Raj Chandra Bose as his teachers and passed the M.A. Examination in 1938 with the first rank in the First Class.

In 1939, at Levy's suggestion, Bhattacharyya met P. C. Mahalanobis together with Bose and joined the Indian Statistical Institute as an honorary worker.[4][5] In 1941, he was made a part-time lecturer in the newly formed Statistics Department of Calcutta University, headed by Mahalanobis. Here he had C. R. Rao, H. K. Nandi, and T. P. Choudhury, as his students.[6] He went to Patna to take up the job of Statistical Officer of Bihar Government, in December 1943 and, in 1946, he returned to Calcutta to join Indian Statistical Institute as Superintending Statistician (in charge of training). Mahalanobis requested him to concurrently take classes in the Statistics Department of Presidency College. After the post was created, Bhattacharyya was made a full-time Senior Professor and Head of the Department in 1949. He occupied the post of Senior Professor until his retirement in March 1974, but in 1967 he stepped down from the leadership, apparently piqued by certain moves of the West Bengal Government's Education Department.[3] Almost since his retirement from Government service, he had been associated with the Narendrapur Ramakrishna Mission Residential College as a guest teacher. In 1994, on the occasion of its golden jubilee celebration Department of Statistics, Presidency University (then Presidency College), released a Festschrift in honour of Professor Bhattacharyya.[3]

Contributions to Statistics edit

Bhattacharyya has made contributions to the statistics in four directions. They are: (a) Measuring the divergence between two statistical populations (b) Characterisation of bivariate normal distributions through normal conditional distribution (c) setting up information bounds for the mean square error of estimators (not necessarily unbiased) that may not attain Fréchet-Cramér–Rao lower bound. (d) Unbiased statistics with minimum variance. Bhattacharyya also worked towards finding the distributional representations of dependent chi-square random variables.

Bhattacharyya distance edit

Main article: Bhattacharyya distance

Distance between statistical distributions had been addressed in 1936 by Mahalanobis, who proposed the D2 metric, now known as Mahalanobis distance. Subsequently, Bhattacharyya defined a cosine metric for the distance between multinomial distributions, this work despite being submitted for publication in 1941, appeared almost five years later in Sankhya.[7][4] Progress toward more general results, which defines the distance metric between two probability distributions which are absolutely continuous with respect to the Lebesgue measure, has been done by Bhattacharyya, which has come in 1942, at Proceedings of the Indian Science Congress.[8] The final work towards this direction appeared in 1943, in the Bulletin of the Calcutta Mathematical Society.[9]

Deriving the PDF of Normal Conditional Distribution edit

One of Bhattacharyya's research concerns towards mathematical probability was the characterization of classical bivariate normal distribution through normal conditional distributions. Normal conditional distributions are bivariate continuous probability distributions whose both conditional distributions are normal. In 1943 Bhattacharyya introduced the family of normal conditional distributions further he derived the probability density function of the normal conditional distribution.[10][11] In the same work, Bhattacharyya has given nine sets of sufficient conditions under which normal conditional distribution becomes classical bivariate normal distributions.[10] Barry C. Arnold noted Bhattacharyya's remarkable contribution way back in 1943 by introducing the family of normal conditional densities and has called the normal conditional distribution through various terms such as "Bhattacharyya's normal conditionals distribution", "Bhattacharyya distribution", and "Bhattacharyya's density" etc.[11][12]

Bhattacharyya Bound edit

Bhattacharyya's other research concern was the setting of lower bounds to the variance of an unbiased estimator.[13][14][15] His information lower bound is popularly known in the Statistical literature as the Bhattacharyya Bound.[1] Bhattacharyya's bound was extended for sequential samples as well.[16] The convergence property of Bhattacharyya Bound was well studied by other researchers.[17][18][19] P. K. Sen has studied the effectiveness of the Bhattacharyya Bound over the Fréchet-Cramér–Rao lower bound in the censoring scheme.[20]

Works edit

  • "A note on Ramamurti's problem of maximal sets", Sankhya, 6 (1942) 189 - 192.[21]
  • "On discrimination and divergence", Proc. Ind. Sc. Cong., 29th Session (1942).[8][22]
  • "On a measure of divergence between two statistical populations defined by their probability distributions", Bull. Cal. Math. Soc, 35 (1943) 99 - 109.[9]
  • "On some sets of sufficient conditions leading to the normal bivariate distribution", Sankhya, 6 (1943) 399 - 406.[10]
  • "On a measure of divergence between two multinomial populations", Sankhya, 7 (1946), 401 - 406.[7]
  • "On some analogues of the amount of information and their uses in statistical estimation" I, Sankhya, 8 (1946) 1 - 14.[13]
  • "On some analogues of the amount of information and their uses in statistical estimation" II, Sankhya, 8 (1947) 201 - 218.[14]
  • "On some analogues of the amount of information and their uses in statistical estimation" III, Sankhya, 8 (1948) 315 - 328.[15]
  • "Unbiased statistics with minimum variance", Proc. Roy. Soc. Edin., A, 63 (1950), 69 - 77.[24]
  • "The problem of regression in statistical population admitting local parameters", Bull. Int. Stat. Inst., 33, Part II (1951), 29 - 54.[25]
  • "Some uses of the t-statistic in multivariate analysis", Proc. Ind. Sc. Cong., 38th Session (1951).[26]
  • "On some uses of the t-distribution in multivariate analysis", Sankhya, 12 (1952), 89 - 104.[27]
  • "Notes on the use of unbiased and biased statistics in the binomial population", Cal. Stat. Assoc. Bull., 5 (1954), 149 - 164.[28]
  • "Some uses of the 'amount of information' in the statistical inference", (address of the Sectional President), Proc. Ind. Sc. Cong., 46th Session (1959).[29] An abridged form of the aforementioned presidential address (statistical section) is also available and appeared in the Calcutta Statistical Association Bulletin.[30]
  • "On a geometrical representation of probability distribution and its use in statistical inference", Cal. Stat. Assoc. Bull., 40 (1990–91), 23 - 49.[31]

References edit

  1. ^ a b Dodge, Yadolah (2003). The Oxford Dictionary of Statistical Terms. Oxford University Press. ISBN 978-0-19-920613-1.
  2. ^ Chattopadhyay, Aparna; Chattopadhyay, Asis Kumar; B-Rao, Chandrika (2004). "Bhattacharyya's distance measure as a precursor of genetic distance measures". Journal of Biosciences. 29 (2): 135–138. doi:10.1007/BF02703410. ISSN 0250-5991. PMID 15295209. S2CID 18485884.
  3. ^ a b c Mukherjee, S. P.; Chaudhuri, Arijit; Basu, Sujit K., eds. (1994). Essays on probability and statistics: Festschrift in honour of Professor Anil Kumar Bhattacharyya. A.M. Gun on behalf of the Organising Committee, Golden Jubilee Celebrations, Dept. of Statistics, Presidency College, Calcutta, Calcutta, 1994. OCLC 230185222.
  4. ^ a b Sen, Pranab Kumar (1996). "Anil Kumar Bhattacharyya (1915-1996): A Reverent Remembrance". Calcutta Statistical Association Bulletin. 46 (3–4): 151–158. doi:10.1177/0008068319960301. ISSN 0008-0683. S2CID 164326977.
  5. ^ Gun, A. M. (1994). "Truly a teacher in the Indian tradition". Essays on Probability and Statistics: Festschrift in Honour of Professor Anil Kumar Bhattacharyya: vi–xxii. OCLC 230185222.
  6. ^ Rao, C. R. (1973). "Prasantha Chandra Mahalanobis, 1893-1972". Biographical Memoirs of Fellows of the Royal Society. 19: 455–492. doi:10.1098/rsbm.1973.0017. ISSN 0080-4606.
  7. ^ a b Bhattacharyya, A. (1946). "On a Measure of Divergence between Two Multinomial Populations". Sankhyā. 7 (4): 401–406. JSTOR 25047882. MR 0018387.
  8. ^ a b Bhattacharyya, A (1942). "On discrimination and divergence". Proceedings of the Indian Science Congress. Abstract portion of the work
  9. ^ a b Bhattacharyya, A (1943). "On a measure of divergence between two statistical populations defined by their probability distributions". Bulletin of the Calcutta Mathematical Society. 35: 99–109. MR 0010358. S2CID 235941388. The article: "On a measure of divergence between two statistical populations defined by their probability distributions" is on Pages: 99-109
  10. ^ a b c Bhattacharyya, A. (1943). "On Some Sets of Sufficient Conditions Leading to the Normal Bivariate Distribution". Sankhyā. 6 (4): 399–406. JSTOR 25047806. MR 0010360.
  11. ^ a b Arnold, Barry C. (1999). "Variations on a Bhattacharyya Theme". Calcutta Statistical Association Bulletin. 49 (1–2): 23–30. doi:10.1177/0008068319990102. ISSN 0008-0683. MR 1744884. S2CID 125755221.
  12. ^ Arnold, Barry C. (1994). "Bhattacharyya's normal conditionals distribution". Essays on Probability and Statistics: Festschrift in Honour of Professor Anil Kumar Bhattacharyya: 1–13. OCLC 230185222.
  13. ^ a b Bhattacharyya, A. (1946). "On Some Analogues of the Amount of Information and Their Use in Statistical Estimation". Sankhyā. 8 (1): 1–14. JSTOR 25047921. MR 0020242.
  14. ^ a b Bhattacharyya, A. (1947). "On Some Analogues of the Amount of Information and Their Use in Statistical Estimation (Contd.)". Sankhyā. 8 (3): 201–218. JSTOR 25047948. MR 0023503.
  15. ^ a b Bhattacharyya, A. (1948). "On Some Analogues of the Amount of Information and Their Use in Statistical Estimation (Concluded)". Sankhyā. 8 (4): 315–328. JSTOR 25047897. MR 0026302.
  16. ^ Seth, G. R. (1949). "On the Variance of Estimates". The Annals of Mathematical Statistics. 20 (1): 1–27. doi:10.1214/aoms/1177730088. ISSN 0003-4851. JSTOR 2236801.
  17. ^ BLIGHT, B. J. N.; RAO, P. V. (1974). "The convergence of Bhattacharyya bounds". Biometrika. 61 (1): 137–142. doi:10.1093/biomet/61.1.137. ISSN 0006-3444.
  18. ^ Alharbi, Abdulghani (1994). "On the convergence of the Bhattacharyya bounds in the multiparametric case". Applicationes Mathematicae. 22 (3): 339–349. doi:10.4064/am-22-3-339-349. ISSN 1233-7234.
  19. ^ Shanbhag, D. N. (1972). "Some characterizations based on the Bhattacharya matrix". Journal of Applied Probability. 9 (3): 580–587. doi:10.2307/3212327. ISSN 0021-9002. JSTOR 3212327. S2CID 123725201.
  20. ^ Sen, P. K. (1994). "A Bhattacharyya detour of MSE bounds for some censoring schemes". Essays on Probability and Statistics: Festschrift in Honour of Professor Anil Kumar Bhattacharyya: 153–172. OCLC 230185222.
  21. ^ Bhattacharyya, A. (1942). "A Note on Ramamurti's Problem of Maximal Sets". Sankhyā. 6 (2): 189–192. JSTOR 25047757. MR 0008326.
  22. ^ "Indian Statistical Institute: Eleventh Annual Report: 1942-43". Sankhyā: The Indian Journal of Statistics (1933-1960). 6 (4): 431–441. 1943. ISSN 0036-4452. JSTOR 25047812. Page 10 of this supporting document shows that a paper on divergence named 'On discrimination and divergence' was submitted to the proceedings of the Indian Science Congress.
  23. ^ Bhattacharyya, A. (1945). "A Note on the Distribution of the Sum of Chi-Squares". Sankhyā. 7 (1): 27–28. JSTOR 25047828. MR 0013271.
  24. ^ Bhattacharyya, A. (1950). "Unbiased Statistics with Minimum Variance". Proceedings of the Royal Society of Edinburgh. Section A: Mathematics. 63 (1): 69–77. doi:10.1017/S0080454100006993. MR 0035947. S2CID 124275255.
  25. ^ Bhattacharyya, A (1951). "The problem of regression in statistical population admitting local parameters". Bulletin of the International Statistical Institute. 33: 29–54. MR 0069464.
  26. ^ Bhattacharyya, A (1951). "Some uses of the t-statistic in multivariate analysis". Proceedings of the Indian Science Congress. Abstract portion of the work (on page 65)
  27. ^ Bhattacharyya, A. (1952). "On the Uses of the t-Distribution in Multivariate Analysis". Sankhyā. 12 (1/2): 89–104. JSTOR 25048117. MR 0058921.
  28. ^ Bhattacharyya, A (1954). "Notes on the use of unbiased and biased statistics in the binomial population". Calcutta Statistical Association Bulletin. 5: 149–164. MR 0067422.
  29. ^ Indian Science Congress (1959). INDIAN SCIENCE CONGRESS ASSOCIATION PART-2. INDIAN SCIENCE CONGRESS ASSOCIATION, CALCUTTA.
  30. ^ "Forty Sixth Indian Science Congress Proceedings of the Statistics Section". Calcutta Statistical Association Bulletin. 9 (1–2): 29–45. 1959. doi:10.1177/0008068319590103. S2CID 220749750. Summary of Prof. A Bhattacharyya's presidential address at the Statistics Section of the Indian Science Congress Association Meeting in Delhi, 1959
  31. ^ Bhattacharyya, A. (1990). "On a Geometrical Representation of Probability Distributions and its use in Statistical Inference". Calcutta Statistical Association Bulletin. 40 (1–4): 23–49. doi:10.1177/0008068319900504. MR 1172634. S2CID 125793582.

External links edit

  1. ^ Mukherjee, S. P.; Chaudhuri, Arijit; Basu, Sujit K., eds. (1994). Essays on probability and statistics: Festschrift in honour of Professor Anil Kumar Bhattacharyya. A.M. Gun on behalf of the Organising Committee, Golden Jubilee Celebrations, Dept. of Statistics, Presidency College, Calcutta, Calcutta, 1994. OCLC 230185222.
  2. ^ Gupta, S. Das (1995). "Review of Essays on Probability and Statistics: Festschrift in Honour of Professor Anil Kumar Bhattacharyya". Sankhyā: The Indian Journal of Statistics, Series A (1961-2002). 57 (1): 166. ISSN 0581-572X. JSTOR 25051041.
  3. ^ "Review of Essays on Probability and Statistics: Festschrift in Honour of Professor Anil Kumar Bhattacharyya". Biometrics. 52 (3): 1163. 1996. doi:10.2307/2533086. ISSN 0006-341X. JSTOR 2533086.
  4. ^ Senroy, Sugata (1995). "Book Review". Calcutta Statistical Association Bulletin. 45 (1–2): 131–136. doi:10.1177/0008068319950110. ISSN 0008-0683. S2CID 220748897.
  5. ^ Rbl (1997). "Review of Essays on Probability and Statistics. Festschrift in Honour of Professor Anil Kumar Bhattacharyya". Journal of the American Statistical Association. 92 (437): 390. doi:10.2307/2291500. ISSN 0162-1459. JSTOR 2291500.
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