Shrinkage estimation under multivariate elliptic models

Article

Arashi, M., Khan, Shahjahan, Tabatabaey, S. M. M. and Soleimani, H.. 2013. "Shrinkage estimation under multivariate elliptic models." Communications in Statistics: Theory and Methods. 42 (11), pp. 2084-2103. https://doi.org/10.1080/03610926.2011.602492
Article Title

Shrinkage estimation under multivariate elliptic models

ERA Journal ID791
Article CategoryArticle
AuthorsArashi, M. (Author), Khan, Shahjahan (Author), Tabatabaey, S. M. M. (Author) and Soleimani, H. (Author)
Journal TitleCommunications in Statistics: Theory and Methods
Journal Citation42 (11), pp. 2084-2103
Number of Pages20
Year2013
PublisherTaylor & Francis
Place of PublicationPhiladelphia, PA. United States
ISSN0361-0926
1532-415X
Digital Object Identifier (DOI)https://doi.org/10.1080/03610926.2011.602492
Web Address (URL)http://www.tandfonline.com/loi/lsta20
Abstract

The estimation of the location vector of a p-variate elliptically contoured distribution (ECD) is considered using independent random samples from two multivariate
elliptically contoured populations when it is apriori suspected that the location vectors of the two populations are equal. For the setting where the covariance structure of the populations is the same, we define the maximum likelihood, Stein-type shrinkage and positive-rule shrinkage estimators. The exact expressions for the bias and quadratic risk functions of the estimators are derived. The comparison of the quadratic risk functions reveals the dominance of the Stein-type estimators if p ≥ 3. A graphical illustration of the risk functions under a 'typical' member of the elliptically contoured family of distributions is provided to confirm the analytical results.

Keywordsbias; risk functions; elliptically contoured distributions; Hotteling's T2 statistic; quadratic loss; Stein-type and positive-rule shrinkage estimators.
ANZSRC Field of Research 2020490509. Statistical theory
380204. Panel data analysis
490103. Calculus of variations, mathematical aspects of systems theory and control theory
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Byline AffiliationsShahrood University of Technology, Iran
Australian Centre for Sustainable Catchments
Ferdowsi University of Mashhad, Iran
Institution of OriginUniversity of Southern Queensland
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James-Stein estimators for the mean vector of a multivariate normal population based on independent samples from two normal populations with common covariance structure
Khan, Shahjahan and Hoque, Zahirul. 2002. "James-Stein estimators for the mean vector of a multivariate normal population based on independent samples from two normal populations with common covariance structure." Pakistan Journal of Statistics. 18 (3), pp. 359-381.
Shrinkage pre-test estimator of the intercept parameter for a regression model with multivariate student-t errors
Khan, Shahjahan and Saleh, A. K. Md. Ehsanes. 1997. "Shrinkage pre-test estimator of the intercept parameter for a regression model with multivariate student-t errors." Biometrical Journal: journal of mathematical methods in biosciences. 39 (2), pp. 131-147. https://doi.org/10.1002/bimj.4710390202
Estimation of the slope parameter for linear regression model with uncertain prior information
Khan, Shahjahan, Hoque, Zahirul and Saleh, A. K. Md. Ehsanes. 2002. "Estimation of the slope parameter for linear regression model with uncertain prior information." Journal of Statistical Research. 36, pp. 55- 73.
On the comparison of the pre-test and shrinkage estimators for the univariate normal mean
Khan, Shahjahan and Saleh, A. K. Md. E.. 2001. "On the comparison of the pre-test and shrinkage estimators for the univariate normal mean." Statistical Papers. 42 (4), pp. 451-473.
Estimating moments of a selected Pareto population under asymmetric scale invariant loss function
Al-Mosawi, Riyadh Rustom and Khan, Shahjahan. 2018. "Estimating moments of a selected Pareto population under asymmetric scale invariant loss function." Statistical Papers. 57, pp. 183-198. https://doi.org/10.1007/s00362-016-0758-7
Changes in non-diabetic comorbid disease status following Laparoscopic Vertical Sleeve Gastrectomy (LVSG) versus Laparoscopic Roux-En-Y Gastric Bypass (LRYGB) procedures: a systematic review of randomized controlled trials
Osland, Emma, Yunus, Rossita Mohamad, Khan, Shahjahan, Memon, Muhammed Ashraf and Memon, Breda. 2017. "Changes in non-diabetic comorbid disease status following Laparoscopic Vertical Sleeve Gastrectomy (LVSG) versus Laparoscopic Roux-En-Y Gastric Bypass (LRYGB) procedures: a systematic review of randomized controlled trials." Obesity Surgery. 27 (5), pp. 1208-12221. https://doi.org/10.1007/s11695-016-2469-5
Statistics on spotlight: reflections on World Statistics Day
Khan, Shahjahan. 2016. "Statistics on spotlight: reflections on World Statistics Day." Journal of Islamic Countries Society of Statistical Sciences. 2 (2), pp. 175-184.
The correlated bivariate noncentral F distribution and its application
Khan, Shahjahan, Pratikno, Budi, Ibrahim, Adriana I. N. and Yunus, Rossita M.. 2016. "The correlated bivariate noncentral F distribution and its application." Communications in Statistics: Simulation and Computation. 45 (10), pp. 3491-3507. https://doi.org/10.1080/03610918.2014.999085
Mathematical reflection approach to instrumental variable estimation method for simple regression model
Saqr, Anwar and Khan, Shahjahan. 2016. "Mathematical reflection approach to instrumental variable estimation method for simple regression model." Pakistan Journal of Statistics. 32 (1), pp. 37-48.
Evidence based decision and meta-analysis with applications in cancer research studies
Khan, Shahjahan, Doi, Suhail A. R. and Memon, M. Ashraf. 2016. "Evidence based decision and meta-analysis with applications in cancer research studies." Applied Mathematics and Information Sciences. 10 (3), pp. 1-8. https://doi.org/10.12785/amis/paper-SK_print
A direct analysis of flood interval probability using approximately 100-year stream flow datasets
Brodie, Ian M. and Khan, Shahjahan. 2016. "A direct analysis of flood interval probability using approximately 100-year stream flow datasets." Hydrological Sciences Journal. 61 (12), pp. 2213-2225. https://doi.org/10.1080/02626667.2015.1099790
Improving statistical inference with uncertain non-sample prior information
Khan, Shahjahan, Memon, Muhammed Ashraf, Pratikno, Budi and Yunus, Rossita M.. 2015. "Improving statistical inference with uncertain non-sample prior information." Journal of Islamic Countries Society of Statistical Sciences. 1 (1), pp. 1-11.
Testing equality of two intercepts for the parallel regression model with non-sample prior information
Pratikno, Budi and Khan, Shahjahan. 2017. "Testing equality of two intercepts for the parallel regression model with non-sample prior information." Communications in Statistics: Theory and Methods. 46 (6), pp. 3099-3110. https://doi.org/10.1080/03610926.2015.1054941
Reply to comment to Meta-analysis and systematic review of laparoscopic versus open mesh repair for elective incisional hernia. Jensen K, Jorgensen LN
Awaiz, A., Rahman, F., Hossain, M. B., Yunus, R. M., Khan, S., Memon, B. and Memon, M. A.. 2015. "Reply to comment to Meta-analysis and systematic review of laparoscopic versus open mesh repair for elective incisional hernia. Jensen K, Jorgensen LN." Hernia. 19 (6), pp. 1027-1029. https://doi.org/10.1007/s10029-015-1432-z
Meta-analysis and systematic review of laparoscopic versus open mesh repair for elective incisional hernia
Awaiz, A., Rahman, F., Hossain, M. B., Yunus, R. M., Khan, S., Memon, B. and Memon, M. A.. 2015. "Meta-analysis and systematic review of laparoscopic versus open mesh repair for elective incisional hernia." Hernia. 19 (3), pp. 449-463. https://doi.org/10.1007/s10029-015-1351-z
Suture cruroplasty versus prosthetic hiatal herniorrhaphy for large hiatal hernia: a meta-analysis and systematic review of randomized controlled trials
Memon, Muhammed Ashraf, Memon, Breda, Yunus, Rossita Mohamad and Khan, Shahjahan. 2016. "Suture cruroplasty versus prosthetic hiatal herniorrhaphy for large hiatal hernia: a meta-analysis and systematic review of randomized controlled trials." Annals of Surgery. 263 (2), pp. 258- 266. https://doi.org/10.1097/SLA.0000000000001267
Advances in the meta-analysis of heterogeneous clinical trials II: the quality effects model
Doi, Suhail A. R., Barendregt, Jan J., Khan, Shahjahan, Thalib, Lukman and Williams, Gail M.. 2015. "Advances in the meta-analysis of heterogeneous clinical trials II: the quality effects model." Contemporary Clinical Trials. 45 (A), pp. 123-129. https://doi.org/10.1016/j.cct.2015.05.010
Advances in the meta-analysis of heterogeneous clinical trials I: the inverse variance heterogeneity model
Doi, Suhail A. R., Barendregt, Jan J., Khan, Shahjahan, Thalib, Lukman and Williams, Gail M.. 2015. "Advances in the meta-analysis of heterogeneous clinical trials I: the inverse variance heterogeneity model." Contemporary Clinical Trials. 45 (A), pp. 130-138. https://doi.org/10.1016/j.cct.2015.05.009
Simulation comparison of the quality effects and random effects methods of metaanalysis
Doi, Suhail A. R., Barendregt, Jan J., Khan, Shahjahan, Thalib, Lukman and Williams, Gail M.. 2015. "Simulation comparison of the quality effects and random effects methods of metaanalysis." Epidemiology. 26 (4), pp. 42-44. https://doi.org/10.1097/EDE.0000000000000289
Contribution of Muslims to the regional social harmony, interreligious dialogue and culture
Khan, Shahjahan. 2014. "Contribution of Muslims to the regional social harmony, interreligious dialogue and culture." Harmes, Marcus K. and Batorowicz, Krzysztof (ed.) Peace, Interreligious Dialogue and Culture Seminar 2014. Toowoomba, Australia 13 - 14 Feb 2014 Toowoomba, Australia.
Classical and Bayesian prediction for multivariate simple regression model
Khan, Shahjahan. 2014. "Classical and Bayesian prediction for multivariate simple regression model." Ahmed, Munir (ed.) 13th Islamic Countries Conference on Statistical Sciences . Bogor, Indonesia 18 - 21 Dec 2014 Lahore, Pakistan.
Score test in robust M-procedure
Khan, Shahjahan and Yunus, Rossita M.. 2014. "Score test in robust M-procedure." Journal of Applied Probability and Statistics. 9 (2), pp. 55-75.
Effect of timing of pharmaconutrition (immunonutrition) administration on outcomes of elective surgery for gastrointestinal malignancies: a systematic review and meta-analysis
Osland, Emma, Hossain, Md Belal, Khan, Shahjahan and Memon, Muhammed Ashraf. 2014. "Effect of timing of pharmaconutrition (immunonutrition) administration on outcomes of elective surgery for gastrointestinal malignancies: a systematic review and meta-analysis." Journal of Parenteral and Enteral Nutrition. 38 (1), pp. 53-69. https://doi.org/10.1177/0148607112474825
Laparoscopic anterior versus posterior fundoplication for gastro-esophageal reflux disease: a meta-analysis and systematic review
Memon, Muhammed Ashraf, Subramanya, Manjunath S., Hossain, Md Belal, Yunus, Rossita Mohamad, Khan, Shahjahan and Memon, Breda. 2015. "Laparoscopic anterior versus posterior fundoplication for gastro-esophageal reflux disease: a meta-analysis and systematic review." World Journal of Surgery. 39 (4), pp. 981-996. https://doi.org/10.1007/s00268-014-2889-0
Statistics in planning and development
Khan, Shahjahan. 2013. "Statistics in planning and development." Pakistan Journal of Statistics. 29 (4), pp. 513-524.
Empirical link between economic growth, energy consumption and CO2 emission in Australia
Salahuddin, Mohammad and Khan, Shahjahan. 2013. "Empirical link between economic growth, energy consumption and CO2 emission in Australia." Journal of Developing Areas. 47 (2), pp. 81-92.
Reflection method of estimation for measurement error models
Saqr, Anwar and Khan, Shahjahan. 2012. "Reflection method of estimation for measurement error models." Journal of Applied Probability and Statistics. 7 (2), pp. 71-88.
Meta-analyses of lightweight versus conventional (heavy weight) mesh in inguinal hernia surgery
Memon, M. Ashraf, Khan, S. and Osland, E.. 2012. "Meta-analyses of lightweight versus conventional (heavy weight) mesh in inguinal hernia surgery." Hernia. 16 (5), pp. 497-502. https://doi.org/10.1007/s10029-012-0987-1
Testing of intercept when slope is under suspicion
Pratikno, Budi and Khan, Shahjahan. 2012. "Testing of intercept when slope is under suspicion." Ahmad, Munir (ed.) 11th Islamic Countries Conference on Statistical Sciences. Lahore, Pakistan 19 - 22 Dec 2011 Lahore, Pakistan.
Testing base load with non-sample prior information on process load
Khan, Shahjahan and Pratikno, Budi. 2013. "Testing base load with non-sample prior information on process load." Statistical Papers. 54 (3), pp. 605-617. https://doi.org/10.1007/s00362-012-0451-4
Early versus traditional postoperative feeding in patients undergoing resectional gastrointestinal surgery: a meta-analysis
Osland, Emma, Yunus, Rossita Mohamad, Khan, Shahjahan and Memon, Muhammed Ashraf. 2011. "Early versus traditional postoperative feeding in patients undergoing resectional gastrointestinal surgery: a meta-analysis." Journal of Parenteral and Enteral Nutrition. 35 (4), pp. 473-487. https://doi.org/10.1177/0148607110385698
Meta-analysis of D1 versus D2 gastrectomy for gastric adenocarcinoma
Memon, Muhammed Ashraf, Subramanya, Manjunath S., Khan, Shahjahan, Hossain, Md. Belal, Osland, Emma and Memon, Breda. 2011. "Meta-analysis of D1 versus D2 gastrectomy for gastric adenocarcinoma." Annals of Surgery. 253 (5), pp. 900-911. https://doi.org/10.1097/SLA.0b013e318212bff6
The bivariate noncentral chi-square distribution – a compound distribution approach
Yunus, Rossita M. and Khan, Shahjahan. 2011. "The bivariate noncentral chi-square distribution – a compound distribution approach." Applied Mathematics and Computation. 217 (13), pp. 6237-6247. https://doi.org/10.1016/j.amc.2010.12.112
Early enteral nutrition within 24 h of intestinal surgery versus later commencement of feeding: a systematic review and meta-analysis [Letter to the Editor]
Osland, Emma, Yunus, Rossita M., Khan, Shahjahan and Memon, Muhammed Ashraf. 2009. "Early enteral nutrition within 24 h of intestinal surgery versus later commencement of feeding: a systematic review and meta-analysis [Letter to the Editor]." Journal of Gastrointestinal Surgery. 13 (6), pp. 1163-1165. https://doi.org/10.1007/s11605-009-0846-2
Meta-analysis of laparoscopic and open distal gastrectomy for gastric carcinoma
Memon, Muhammed Ashraf, Khan, Shahjahan, Yunus, Rossita Mohamad, Barr, Richard and Memon, Breda. 2008. "Meta-analysis of laparoscopic and open distal gastrectomy for gastric carcinoma." Surgical Endoscopy: surgical and interventional techniques. 22 (8), pp. 1781-1789. https://doi.org/10.1007/s00464-008-9925-9
Estimation of univariate normal mean using p-value
Kabir, Md Enamul and Khan, Shahjahan. 2007. "Estimation of univariate normal mean using p-value." 9th Islamic Countries Conference on Statistical Sciences. Shah Alam, Malaysia 12 - 14 Dec 2007 Kuala Lumpur, Malaysia.
Test for intercept after pre-testing on slope - a robust method
Yunus, Rossita M. and Khan, Shahjahan. 2007. "Test for intercept after pre-testing on slope - a robust method." Ahmad, Munir, Mohamed, Ibrahim, Osman, Jamil and Ibrahim, Noor Akma (ed.) 9th Islamic Countries Conference on Statistical Sciences. Shah Alam, Malaysia 12 - 14 Dec 2007 Kuala Lumpur, Malaysia.
Observed data based estimator and both observed data and prior information based estimator under asymmetric losses
Hoque, Zahirul, Wesolowski, Jacek and Khan, Shahjahan. 2007. "Observed data based estimator and both observed data and prior information based estimator under asymmetric losses." American Journal of Mathematical and Management Sciences. 27 (1-2), pp. 93-110. https://doi.org/10.1080/01966324.2007.10737690
Prediction distribution of future regression and residual sum of squares matrices for multivariate simple regression model with correlated normal responses
Khan, Shahjahan. 2006. "Prediction distribution of future regression and residual sum of squares matrices for multivariate simple regression model with correlated normal responses." Journal of Applied Probability and Statistics. 1 (1), pp. 15-30.
Edgeworth size corrected W, LR and LM test in the formation of the preliminary test estimator
Khan, Shahjahan, Hoque, Zahirul and Billah, Baki. 2006. "Edgeworth size corrected W, LR and LM test in the formation of the preliminary test estimator." Journal of Statistical Research. 40 (2), pp. 55-64.
Optimal tolerance regions for some functions of multiple regression model with Student-t errors
Khan, Shahjahan. 2006. "Optimal tolerance regions for some functions of multiple regression model with Student-t errors." Journal of Statistics and Management Systems. 9 (3), pp. 699-715.
On the Size Corrected Tests in Improved Estimation
Hoque, Z., Billah, B. and Khan, S.. 2005. "On the Size Corrected Tests in Improved Estimation." Calcutta Statistical Association Bulletin. 57 (3-4), pp. 143-160. https://doi.org/10.1177/0008068320050301
Estimation of parameters of the simple multivariate linear model with student t-errors
Khan, Shahjahan. 2005. "Estimation of parameters of the simple multivariate linear model with student t-errors ." Journal of Statistical Research. 39 (2), pp. 79-94.
Estimation of the intercept parameter for linear regression model with uncertain non-sample prior information
Khan, Shahjahan, Hoque, Zahirul and Saleh, A. K. Md. E.. 2005. "Estimation of the intercept parameter for linear regression model with uncertain non-sample prior information." Statistical Papers. 46 (3), pp. 379-395. https://doi.org/10.1007/bf02762840
Bridging the divide by screencasting in an introductory statistics class at an Australian university
Khan, Shahjahan, Loch, Birgit and McDonald, Christine. 2010. "Bridging the divide by screencasting in an introductory statistics class at an Australian university." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Meta-analysis of laparoscopic posterior and anterior fundoplication for gastro-oesophageal reflux disease
Subramanya, M. S., Hossain, M. B., Khan, S., Memon, B. and Memon, M. A.. 2010. "Meta-analysis of laparoscopic posterior and anterior fundoplication for gastro-oesophageal reflux disease." Li, Yan, Yang, Jiajia, Wen, Peng and Wu, Jinglong (ed.) 2010 IEEE/ICME International Conference on Complex Medical Engineering (ICME 2010). Gold Coast, Australia 13 - 15 Jul 2010 Piscataway, NJ. United States. https://doi.org/10.1109/ICCME.2010.5558829
M-tests for multivariate regression model
Yunus, Rossita M. and Khan, Shahjahan. 2011. "M-tests for multivariate regression model." Journal of Nonparametric Statistics. 23 (1), pp. 201-218. https://doi.org/10.1080/10485252.2010.503896
Preliminary results of a meta-analysis evaluating the effect of immunonutrition on outcomes on outcomes of elective gastrointestinal surgery
Osland, Emma, Hossain, Md. Belal, Khan, Shahjahan and Memon, Muhammed A.. 2010. "Preliminary results of a meta-analysis evaluating the effect of immunonutrition on outcomes on outcomes of elective gastrointestinal surgery." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Benefits of early feeding versus traditional nil-by-mouth nutritional postoperative management in gastrointestinal resectional surgery patients: a meta-analysis
Osland, Emma, Yunus, Rossita Mohamad, Khan, Shahjahan and Memon, Muhammed A.. 2010. "Benefits of early feeding versus traditional nil-by-mouth nutritional postoperative management in gastrointestinal resectional surgery patients: a meta-analysis." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Preliminary results of meta-analysis of endoscopic retrograde cholangiopancreatography (ERCP) versus conservative treatment for gall stone pancreatitis
Burstow, Matthew J., Yunus, Rossita M., Khan, Shahjahan, Memon, Breda and Memon, Muhammed A.. 2010. "Preliminary results of meta-analysis of endoscopic retrograde cholangiopancreatography (ERCP) versus conservative treatment for gall stone pancreatitis." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Preliminary results of meta-analysis of laparoscopic and open inguinal hernia repair
Burstow, Matthew John, Yunus, Rossita Mohamad, Khan, Shahjahan, Memon, Breda and Memon, Muhammed Ashraf. 2010. "Preliminary results of meta-analysis of laparoscopic and open inguinal hernia repair." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
M-test of two parallel regression lines under uncertain prior information
Khan, Shahjahan and Yunus, Rossita M.. 2010. "M-test of two parallel regression lines under uncertain prior information." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Meta-analysis of D1 versus D2 gastrectomy for gastric adenocarcinoma
Subramanya, Manjunath S., Hossain, Md. Belal, Khan, Shahjahan, Memon, Breda and Memon, Muhammed A.. 2010. "Meta-analysis of D1 versus D2 gastrectomy for gastric adenocarcinoma." Amin, Zeinab and Hadi, Ali S. (ed.) 10th Islamic Countries Conference on Statistical Sciences. Cairo, Egypt 20 - 23 Dec 2009 Lahore, Pakistan.
Statistical inference
Khan, Shahjahan. 2010. "Statistical inference." Gaber, Mohamed Medhat (ed.) Scientific data mining and knowledge discovery: principles and foundations. Berlin, Germany. Springer. pp. 53-76
Increasing the power of the test through pre-test: a robust method
Yunus, Rossita M. and Khan, Shahjahan. 2011. "Increasing the power of the test through pre-test: a robust method." Communications in Statistics: Theory and Methods. 40 (4), pp. 581-597. https://doi.org/10.1080/03610920903350572
M-test for intercept after pre-testing on slope
Yunus, Rossita M. and Khan, Shahjahan. 2010. "M-test for intercept after pre-testing on slope." Journal of Statistical Modeling and Analytics. 1 (1), pp. 45-57.
Increasing power of robust test through pre-testing in multiple regression model
Yunus, Rossita M. and Khan, Shahjahan. 2010. "Increasing power of robust test through pre-testing in multiple regression model." Pakistan Journal of Statistics. 26 (1), pp. 151-170.
Scientific and scholarly contributions of Distinguished Research and Emeritus Professor A K Md Ehsanes Saleh
Khan, Shahjahan. 2009. "Scientific and scholarly contributions of Distinguished Research and Emeritus Professor A K Md Ehsanes Saleh." International Journal of Statistical Sciences. 9, pp. 7-31.
Meta-analysis of randomised clinical trials comparing open and laparoscopic anti-reflux surgery
Peters, Matthew James, Mukhtar, Athar, Yunus, Rossita Mohamad, Khan, Shahjahan, Pappalardo, Juanita, Memon, Breda and Memon, Muhammed Ashraf. 2009. "Meta-analysis of randomised clinical trials comparing open and laparoscopic anti-reflux surgery." The American Journal of Gastroenterology. 104 (6), pp. 1548-1561. https://doi.org/10.1038/ajg.2009.176
Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors
Khan, Shahjahan. 2009. "Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors." Statistical Papers. 50 (3), pp. 511-525. https://doi.org/10.1007/s00362-007-0095-y
Prediction distribution of generalized geometric series distribution and its different forms
Hassan, Anwar and Khan, Shahjahan. 2009. "Prediction distribution of generalized geometric series distribution and its different forms." Pakistan Journal of Statistics. 25 (1), pp. 47-57.
Improved meta analysis using predicted relative risk
Khan, Shahjahan and Hasan, Md Masud. 2008. "Improved meta analysis using predicted relative risk." Chowdhury, Abdul Hannan, Khan, Shahjahan, Ahmed, Nasar U., Rahman, M. Saydur and Kibria, B. M. Golam (ed.) 2008 International Conference on Recent Development in Statistical Sciences. Dhaka, Bangladesh 26 - 27 Dec 2008 Dhaka, Bangladesh.
Estimation of slope for linear regression model with uncertain prior information and student-t error
Khan, Shahjahan and Saleh, A. K. Md. E.. 2008. "Estimation of slope for linear regression model with uncertain prior information and student-t error." Communications in Statistics: Theory and Methods. 37 (16), pp. 2564-2581. https://doi.org/10.1080/03610920802040399
Sine square distribution: a new statistical model based on the sine function
Al-Faris, Rand Q. and Khan, Shahjahan. 2008. "Sine square distribution: a new statistical model based on the sine function." Journal of Applied Probability and Statistics. 3 (1), pp. 163-173.
Estimation in multiple regression model with elliptically contoured errors under MLINEX loss
Arashi, M., Tabatabaey, S. M. M. and Khan, Shahjahan. 2008. "Estimation in multiple regression model with elliptically contoured errors under MLINEX loss." Journal of Applied Probability and Statistics. 3 (1), pp. 23-35.
Prediction distribution for linear regression model with multivariate student-t errors
Rahman, Azizur and Khan, Shahjahan. 2008. "Prediction distribution for linear regression model with multivariate student-t errors." Far East Journal of Theoretical Statistics. 24 (1), pp. 35-48.
Shrinkage estimators of intercept parameters of two simple regression models with suspected equal slopes
Khan, Shahjahan. 2008. "Shrinkage estimators of intercept parameters of two simple regression models with suspected equal slopes." Communications in Statistics: Theory and Methods. 37 (2), pp. 247-260. https://doi.org/10.1080/03610920701648961
Comparison of estimators of means based on p-samples from multivariate Student-t population
Khan, Shahjahan and Saleh, A. K. Md. Ehsanes. 1998. "Comparison of estimators of means based on p-samples from multivariate Student-t population." Communications in Statistics: Theory and Methods. 27 (1), pp. 193-210. https://doi.org/10.1080/03610929808832660
Prediction distribution for linear regression model with multivariate Student-t errors under the Bayesian approach
Rahman, Azizur and Khan, Shahjahan. 2007. "Prediction distribution for linear regression model with multivariate Student-t errors under the Bayesian approach." Ismail, Fudziah, Tarmizi, Rohani, Midi, Habsha, Said, Mohamad and Abbas, Zulkifly (ed.) 3rd International Conference on Research and Education in Mathematics (ICREM3). Kuala Lumpur, Malaysia 10 - 12 Apr 2007 Kuala Lumpur, Malaysia.
Shrinkage estimation of the slope parameters of two parallel regression lines under uncertain prior information
Khan, Shahjahan. 2006. "Shrinkage estimation of the slope parameters of two parallel regression lines under uncertain prior information." Model Assisted Statistics and Applications: an international journal. 1 (3), pp. 193-205.
Estimation of the parameters of two parallel regression lines under uncertain prior information
Khan, Shahjahan. 2003. "Estimation of the parameters of two parallel regression lines under uncertain prior information." Biometrical Journal: journal of mathematical methods in biosciences. 45 (1), pp. 73-90. https://doi.org/10.1002/bimj.200290017
The role of the shape parameter in the estimation of the mean vector in multivariate Student-t population
Khan, Shahjahan. 2004. "The role of the shape parameter in the estimation of the mean vector in multivariate Student-t population." International Journal of Statistical Sciences. 3 (Special issue), pp. 69-89.
Improved estimation of the mean vector for Student-t model.
Khan, Shahjahan. 2000. "Improved estimation of the mean vector for Student-t model." Communications in Statistics: Theory and Methods. 29 (3), pp. 507-527.
Distribution of sum of squares and products matrices for the generalized multilinear matrix-T model
Khan, Shahjahan. 2002. "Distribution of sum of squares and products matrices for the generalized multilinear matrix-T model." Journal of Multivariate Analysis. 83, pp. 124-140. https://doi.org/10.1006/jmva.2001.2040
Distribution of future location vector and residual sum of squares for multivariate location-scale model with spherically contoured errors
Khan, Shahjahan and Kabir, Md Enamul. 2006. "Distribution of future location vector and residual sum of squares for multivariate location-scale model with spherically contoured errors." Journal of Statistical Theory and Applications. 5 (2), pp. 91-104.
Predictive distribution of regression vector and residual sum of squares for normal multiple regression model
Khan, Shahjahan. 2004. "Predictive distribution of regression vector and residual sum of squares for normal multiple regression model." Communications in Statistics: Theory and Methods. 33 (10), pp. 2423-2443. https://doi.org/10.1081/STA-200031471
Preliminary test estimators for the multivariate normal mean based on the modified W, LR and LM tests
Khan, S. and Hoque, Z.. 2003. "Preliminary test estimators for the multivariate normal mean based on the modified W, LR and LM tests." Journal of Statistical Research. 37 (1), pp. 43-55.