# 1.1  Statement of Research

The major contribution of my research is the development of many foundational results ranging from the distribution theory to the estimation theory for elliptical distributions with emphasis on the multivariate t-distribution. A recent text " Multivariate t Distributions and Their Applications" by Samuel Kotz and Saralees Nadarajah published by Cambridge University Press, UK contains many of  results with 8 references. My research papers have also been cited in reputed research journals, books published by John Wiley and Sons, CRC Press, etc. My research publications can be categorized into the following:

(i) Multivariate Estimation Theory

(ii)   Distribution Theory, Moments and Characterizations

(iii) Survey Sampling

(iv) Descriptive Statistics

The classical theory of Multivariate Analysis is based on the assumption that underlying observation vectors arise from independent multivariate normal distributions. The multivariate normal distributions have indeed played a predominant role in the historical development of statistical theory, and found applications in physical, biological, engineering  and other branches of science and business.

Samuel Kotz (1975) presented a systematic classification of multivariate distributions based on various criteria such as type of dependence, analogy of mathematical form, model and characterizations. Kelker (1970) was the first statistician to develop statistical theory for multivariate elliptical distributions, a class of distributions which accommodates  multivariate t-distribution and multivariate normal distribution as special cases. Fang and Anderson (1990) edited a book on multivariate elliptical distributions. Fang and Zhang (1990) were the first to come up with a  book on multivariate analysis with elliptical distributions.

The multivariate t-distribution has fatter tails and can characterize many financial data especially stock return data.  Zellner (1976) laid the foundation for modeling financial data under the assumption that observations follow t-distribution. Since then many authors tried to develop statistical theory for a t-population.

Multivariate Estimation Theory (11 Papers)

There have been numerous papers on the estimation of the covariance matrix of the multivariate normal distribution.  Maximum likelihood method has been one of the most popular method of estimation for the covariance matrix. The most powerful property of the maximum likelihood estimator is the asymptotic normality which stems out, in fact,  from the independence of the sample observations. However the assumption of the independence of the observations for multivariate elliptical distributions is true only for the special case of multivariate normal distribution. So some authors have tried to estimate the covariance by the criteria of  loss functions.

In a series of papers estimation strategies for covariance matrix of the multivariate t-distribution, its trace, characteristics roots etc have been developed under squared error loss function [Joarder (1995 b), Joarder and Ahmed (1996), Joarder and Beg (1999)]. The covariance matrix of the multivariate t-distribution has also been estimated under an entropy loss function (Joarder and Ali, 1997). Trace and covariance matrix of the scale mixture of multivariate normal distributions have also been studied [Joarder and Hossain (1995), Joarder and Ahmed (1998)]. The covariance matrix of the multivariate t-distribution has been estimated by the use of a regression type of estimator( Joarder and Singh ,1997) and also by using a known information  (Joarder and Singh, 2001). The covariance matrix based on multivariate normal distribution has been estimated by Joarder (1995c). Eigenvalues of a Wishart matrix based on multivariate normal distribution has been estimated by Pretest Method (see Ahmed, Volodin and Joarder, 2001)

Distribution  Theory, Moments and Characterizations  (13  Papers)

The distribution of the correlation coefficient based on bivariate elliptical distributions has been derived  (Ali and Joarder, 1991). This proves the robustness of the distribution correlation coefficient in the wider class of bivariate elliptical distributions and t-test for the uncorrelatedness. The distribution theory has been extended to multivariate elliptical distributions to derive the distribution of correlation matrix (Joarder and Ali, 1992).  Some integral results with applications to correlation analysis have been developed (Joarder, 2006b).

Characteristic functions of several multivariate distributions say Multivariate t-distribution, Multivariate Pearson Type II Distribution, Uniform Distribution on or inside Unit Hyper-Sphere  etc have been derived in terms of  well known special functions  [Joarder (1995 a), Joarder and Alam (1995), Joarder and Ali (1996 a), Joarder (1997)]. Spherical Distributions, a special class of elliptical distributions, have been characterized geometrically and analytically (Joarder and Ali ,1996 b). Some applications of  Macdonald function in the multivariate t-distribution has been discussed (Joarder, 1994). Stirling numbers of the second kind has been derived by an inductive method and applied to find moments of  integer valued random variables (Joarder and Mahmood ,1997).

Identities involving Wishart matrix based on the multivariate t-distribution have been derived ( Joarder and Ali ,1992 b). Some useful expected values of  important functions of Wishart matrices based on the multivariate t-model has been derived (Joarder, 1998 b). These are important for the loss theoretic estimation of covariance matrix and its characteristics. Product moments of bivariate Wishart distribution have been derived (Joarder, 2006a).

Survey Sampling (10  Papers)

Scrambled randomized response method has been applied to  linear regression model (Singh, Joarder and King , 1996).  Optional randomized response technique has been developed for sensitive qualitative variable (Singh and Joarder, 1997 b).

Finite population variance has been estimated by the use of random non-response (Singh and Joarder 1998). Unknown repeated trials in randomized response sampling has been considered by Singh and Joarder (1997 a). Regression type estimators in the presence of non-response has been discussed by Singh, Joarder and Tracy (2000). Some regression type estimators have been studied for random

non-response in different situations under the assumption that the number of sampling units on which information cannot be obtained due to random non-response follows some probability distribution (see Singh, Joarder and Tracy (2001).

Distribution function and median in two phase sampling have been estimated by Singh and Joarder (2002). General class of estimators in multicharacter surveys is

considered in Singh, Grewal and Joarder (2004). Singh, Chandra , Joarder and Singh (2006)  have developed family of estimators of mean, ratio and product of a finite population under random nonresponse.

Descriptive Statistics and Miscellaneous (12 Papers)

It has been proved that statistical independence and linear dependence of a square contingency matrix are equivalent (Joarder, 1998 a). The dependence structure of conditional probabilities in a square contingency table has been considered by Joarder and Al-Sabah ( 2002). A halving method  (Joarder 2003) has been proposed for the quartiles. A remainder method (Firozzaman  and Joarder, 2001) has been proposed for quartiles and deciles . Linear interpolation has been viewed by Joarder (2002) from six different perspectives. Sample variance has been calculated by Joarder (2002) without the use of sample mean and by Joarder

(2003) by the first order differences of observations. A comparison and contrast for sample quartiles has been considered by Joarder and Latif (2004). Some inequalities among some measures of location have been developed by Laradji and Joarder (2006). Algebraic inequalities have been developed by Joarder and Laradji (2005) for measures of dispersion.

#### References

Kelker, D. (1970). Distribution theory of spherical distributions and a location scale parameter generalization. Sankhya, Ser. A, 32, 419—430.

## Kotz, S. (1975). Multivariate distributions at a cross-road. In Statistical Distributions in Scientific Work, (eds. G.P. Patil, S. Kotz and J.K. Ord, D. Reidel Publ. Co.), 1,  247—270.

Fang, K.T. and Anderson, T.W. (1990). Statistical Inference in Elliptically Contoured and Related Distributions. Allerton Press, New York.

Fang, K.T. and Zhang, Y. (1990). Generalized Multivariate Analysis. Springer Verlag.

Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error term.  Journal of American Statistical Association, 71, 400—405 (correction, 71, 1000).

For other references see the following list.

A Chronological List of Research Publications

01. Ali, M.M. and Joarder, A.H. (1991). Distribution of the correlation coefficient for the

class of  bivariate elliptical models. Canadian Journal of Statistics, 19(4),  447--452.

(ISI)

02. Joarder, A.H. and Ali, M.M. (1992 b). On some generalized Wishart expectations.

Communications in Statistics – Theory and Methods, 21(1), 283--294. (ISI)

03. Joarder, A.H. and Ali, M.M. (1992 a). Distribution of the correlation matrix for a

class of elliptical models, Communications in Statistics – Theory and  Methods,

21(7), 1953--1964. (ISI)  [PDF]

04. Joarder, A.H. (1994). Some applications of  Macdonald function in the multivariate t-

distribution. Journal of  Statistical Studies, 14, 115--124. [PDF]

05. Joarder, A.H. (1995 a). The characteristic function of the univariate t-distribution.

Dhaka University Journal of Science, 43(1), 117--125. [PDF]

06. Joarder, A.H. and Alam, A.U. (1995). The characteristic function of the elliptical  t

distribution using a conditional expectation approach. Journal of Information and

Optimization Sciences, 16(2), 307--310. [PDF]

07. Joarder, A.H. (1995 b). Estimation of the scale matrix of a multivariate t-model.

Journal of Statistical Research, 29(1), 55-- 66.   [PDF]

08. Joarder, A.H. (1995 c). Estimation of the covariance matrix of the multivariate

normal distribution. Pakistan Journal of Statistics, 11(3), 159--165.  [PDF]

09. Joarder, A.H. and Hossain, M.A. (1995). Estimation of the trace of the scale matrix

of   scale mixture of multivariate normal distributions. Journal of Information and

optimization  Sciences, 16(3), 565--572.  [PDF]

10. Joarder, A.H. and Ali, M.M. (1996 a). On the characteristic function of the   multivariate

t-distribution. Pakistan Journal of Statistics, 12(1), 55—62.  [PDF]

11. Joarder, A.H. and Ali, M.M. (1996 b). On the characterization of spherical

distributions. Journal of Information and Optimization Sciences, 17(1), 177-- 184.

[PDF]

12. Joarder, A.H. and Ahmed, S.E. (1996). Estimation of  characteristic roots of scale

matrix. Metrika, 44, 259--267. (ISI)   [PDF]

13. Joarder, A.H. and Hossain, M.A. (1996). Estimation of the eigenvalues of the scale

matrix of a class of elliptical distributions. Statistica, 56(3), 314--319.  [PDF]

14. Singh, S; Joarder, A.H. and King, M.L. (1996). Regression analysis using

scrambled  responses. Australian Journal of Statistics, 38(2), 201--211. (ISI) [PDF]

15. Singh, S. and Joarder, A.H. (1997 a). Unknown repeated trials in randomized

response sampling. Indian Society of Agricultural Statistics, 50(1), 103-105.

16. Singh, S. and Joarder, A.H. (1997 b). Optional randomized response

technique for  sensitive quantitative variable. Metron, 55(1-2), 151--157.

17. Joarder, A.H. and Ali, M.M. (1997). Estimation of the scale matrix of a

multivariate t-model under entropy loss. Metrika, 46, 21--32.  (ISI)

18. Joarder, A.H. and Singh, S. (1997). Estimation of the trace of the scale matrix of a

multivariate t-model using regression type estimator. Statistics, 29, 161--168.  [PDF]

19. Joarder, A.H. (1997). On the characteristic function of the multivariate Pearson

Type II  distribution. Journal of Information and Optimization Sciences, 18(1), 177—

182.  [PDF]

20. Joarder, A.H. and Mahmood, M. (1997). An inductive derivation of Stirling

numbers of  the  second kind and their applications in statistics. Journal of Applied

Mathematics and Decision Sciences, 1(2), 151--157.  [PDF]

21. Joarder, A.H. (1998 a). On the statistical independence in a contingency

table. International Journal of Mathematical Education in Science and

Technology29(5), 780--782.   [PDF]

22. Joarder, A.H. (1998 b).  Some useful Wishart expectations based on the

multivariate  t-model. Statistical Papers, 39, 223--229. (ISI) [PDF]

23. Singh, S. and Joarder, A.H. (1998). Estimation of finite population variance using

random non-response  in survey sampling. Metrika, 47, 241--249. (ISI)

24.  Joarder, A.H. and Ahmed, S.E. (1998). Estimation of the scale matrix of a class of

elliptical distributions. Metrika, 48, 149—160. (ISI) [PDF]

25.  Joarder, A.H. and Beg, G.K. (1999). Estimation of the trace of the scale matrix of

the multivariate t-model under a squared error loss. Statistica, 59(2), 181--191.  [PDF]

26. Singh, S.,  Joarder, A.H. and Tracy, D.S. (2000). Regression type estimators for

random non-response in survey sampling. Statistica, 60(1), 39-- 44.  [PDF]

27. Singh, S.,  Joarder, A.H. and Tracy, D.S. (2001). Median estimation using double

sampling. Australian and New Zealand Journal of Statistics, 43(1), 33-46. (ISI)  [PDF]

28. Joarder, A.H. and Firozzaman, M. (2001). Quartiles for discrete data.

Teaching Statistics, 23(3), 86-89.

29. Ahmed, S.E. ; Volodin, A.I. and Joarder, A. H. (2002). Pretest estimation of

eigenvalues of a Wishart matrix. Inernational Mathematical Journal, 1 (3), 259--272.  [PDF]

30. Firozzaman, M. and Joarder, A.H. (2001). A refinement over the usual formulae for

deciles. International Journal of Mathematical Education in Science and

Technology, 32 (5), 761-765.  [PDF]

31. Joarder, A.H. and Singh, S. (2001). Estimation of the trace of the scaled

covariance matrix of a multivariate t-model using a known information. Metrika,

54 (1), 53-58. (ISI) [PDF]

32. Joarder, A.H. (2002). Six ways to look at linear interpolation, International

Journal of Mathematical Education in Science and  Technology, 32 (6), 932-

937. [PDF]

33. Joarder, A.H. and Al-Sabah, W.S. (2002). The dependence structure of

conditional probabilities in a contingency table. International Journal of  Mathematical

Education in Science and Technology, 33(3), 475-480. [PDF]

34. Joarder, A.H. (2002). On some representations of sample variance. International

Journal of Mathematical Education in Science and Technology, 33(5), 772-784.  [PDF]

35. Singh, S.  and Joarder, A.H. (2002).Estimation of  the distribution function

and median in two phase sampling. Pakistan Journal of Statistics (S. E Ahmed

special issue edited by Serge B. Provost, The University of Western Ontario, Canada),

18(2), 301-319. [PDF]

36. Joarder, A.H. (2003). The halving method for sample quartilesInternational Journal of

Mathematical Education in Science and Technology. 34(4), 629-633.

37. Joarder, A.H.(2003). The sample variance and first-order Differences of

observations. Mathematical Scientist, 28, 129-133. [PDF]

38. Joarder, A.H. and Latif, R.M. (2004). A  comparison and contrast of some methods

for sample quartiles. Journal of Probability and Statistical Science,  2(1), 95-109

39. Barone, L; Voulgaridis, G.Z and Joarder, A.H. (2004). On the dispersion of data in

nonsymmetric distributions. International Journal of Mathematical Education in Science

and Technology,  35(3), 419-424.  [PDF]

40. Singh, S., Grewal, I.S. and Joarder, A.H. (2004). General class of estimators in

multicharacter surveys. Statistical Papers, 45, 571-582. (ISI) [PDF]

41. Joarder, A.H. and Laradji, A. (2005). Algebraic inequalities for measures of

dispersion. Journal of Probability and Statistical Science, 3(2), 317-326.

42. Joarder, A.H. and Latif, R.M. (2006). Standard deviation for small samples,

Teaching Statistics, 28(2), 40-43. [PDF]

43. Joarder, A.H. (2006a). Product moments of a bivariate Wishart distribution.

Journal of Probability and Statistical Science, 4(2), 2006, 233-244. [PDF]

44. Kibria, B. M. G. and Joarder A. H. (2006). A short review of multivariate

t-distribution. Journal of Statistical Research, 40(1), 59-72. [PDF]

45. Laradji, A. and Joarder, A.H. (2007). Inequalities among some measures

of  location. Communications in Statistics – Theory and Methods. 35,

(11), 1963-1970. (ISI) [PDF]

46. Joarder, A.H.  and Omar, M.H. (2007). Evaluation of moment integrals

without integration. International Journal of Mathematical Education in

Science and Technology.  38(4), 538-543. [PDF]

47.    A.H. Joarder and M.R. Abujiya (2007). The remainder method for sample percentiles.

International Journal of Mathematical Education in Science and Technology, 38(5), 667-676.

48. Joarder, A.H.  and Al-Sabah, W.S. (2007). Probability issues in without replacement

sampling. International Journal of Mathematical Education in Science and Technology,

38(6), 823-831. [PDF]

49. Singh, H.P.; Chandra, P; Joarder A.H. and Singh, S. (2007). Family of estimators of

mean, ratio and product of a finite population using random nonresponse. Test: A Journal

of the Spanish Statistical Society, 16(3), 565-597.  (ISI)

50.  Joarder, A.H. (2007). On  some characteristics of the bivariate t-distribuion.

International Journal of Modern Mathematics. 2(2), 191-204 [October 2007] [PDF]

51. Joarder, A.H.  (2007). Some useful integrals and their applications in  correlation

analysis. Statistical Papers, 49(2), 211-224. (ISI) [PDF]

52. Joarder, A.H.; Al-Sabah, W.S. and Omar, M.H.  (2008). On the distribution of

the norms of spherical distributions.  Journal of Probability and Statistical Science,

6(1), 115-123. [PDF]

53. Joarder, A.H. (2008). Some characteristics of Mahalanobis distance for  bivariate

distributions. International Journal of Modern Mathematics, 3(3), 315 - 325.

54.  Joarder, A.H. and Abujiya, M.R. (2008). Standardized moments for bivariate

chi-square distribution. Journal of Applied Statistical Science, 16(4), 1-9. [PDF]

55. Joarder, A.H. (2008). Moments of product and ratio of two correlated chi-square variables.

Statistical Papers, DOI: 10.1007/s00362-007-0105-0. (ISI)  [PDF]

56. Sarr, A.; Gupta, A.K. and Joarder, A.H. (2008). Estimation of the precision matrix of multivariate

Pearson type II model. Metrika, DOI: 10.1007/s00184-008-0172-9. (ISI) [PDF]

57. Joarder, A.H. and Omar, M.H. (2008). A mass function based on correlation coefficient and its

application. Statistics and Probability Letters, 75(18), 3344-3349. (ISI) [PDF]

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