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An Optimal Confidence Region for for the Best and the Worst Populations

An Optimal Confidence Region for for the Best and the Worst Populations. Speaker: Prof. Hubert J. Chen , Institute of Finance National Cheng Kung U., Taiwan. Jointly with: Prof. Shu-Fei Wu , Tamkang U., Taiwan. 2012.11.21 Invited Talk at Dept. of Statistics, Ping-Tung Edu . U.

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An Optimal Confidence Region for for the Best and the Worst Populations

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  1. An Optimal Confidence Region for for the Best and the Worst Populations Speaker: Prof. Hubert J. Chen, Institute of FinanceNational Cheng Kung U., Taiwan Jointly with: Prof. Shu-Fei Wu, Tamkang U., Taiwan 2012.11.21 Invited Talk at Dept. of Statistics, Ping-Tung Edu. U.

  2. Annual Returns of 8 Mutual Funds in U.S. from 1974 to 2010 They are correlated & all depend on the market Washington Mutual (AWSHX) Fidelity Contra (FCNTX) American Income (AMECX) New Perspective (ANWPX) Growth Fund of American (AGTHX) Fidelity Puritan (FPURX) Vanguard Windsor (VWNDX) Janus Family-Janus (JANSX)

  3. Table 1 Annual Returns of 8 Mutual Funds in U.S. from 1974 to 2010

  4. Correlation Matrix

  5. Table 2 Summary of Statistics

  6. Contents Literature Review 2. Optimal Confidence Region 3. Application - Mutual Funds 4. Comparisons 5. Conclusions

  7. Literature Review Estimation of the Largest MeanSaxena, Krishna and Tong (1969)Dudewicz (1972) Optimal Interval for the Largest MeanDudewicz and Tong (1971)Chen and Dudewicz (1976)Chen and Chen (2004) Interval for the Largest Mean of Correlated PopulationsChen, Li and Wen (2008) Conf. Region for the Largest & Smallest MeansChen and Wu (2012)

  8. Confidence Region k correlated populations (Treatments) Means : Common Vari. : ( > 0 ) Common Corr. : ( > 0 ) Random Vectors Distrib. of X

  9. Correlation Matrix Sample Mean Vector Sample Covariance Matrix

  10. Best Variance Estimate of

  11. Individual intervals:

  12. Goal:

  13. Bonferroni Inequality

  14. Application 8 U.S. diversified mutual funds: Washington Mutual (AWSHX) Fidelity Contra (FCNTX) American Income (AMECX) New Perspective (ANWPX) Growth Fund of American (AGTHX) Fidelity Puritan (FPURX) Vanguard Windsor (VWNDX) Janus Family-Janus (JANSX)

  15. Annual Returns (1974-2010) Table 2. Summary of Statistics Passed Normality test. Passed Independence test within each sample. Passed Equal variance test.

  16. Correlation Matrix • All coefficients are positive. • DidNOT pass equal correlation test.( Lawley (1963) ) • Use minimum correlation, 0.59.

  17. Critical values at P*=0.9 ( by Fortran ) :

  18. 90%Confidence Region for Best and Worst • and

  19. Interpretation The mutual fund AGTHX has the highest mean return. The mutual fund AWSHX has the lowest mean return. With a 90% confidence, the true highest mean return falls into the interval (9.89% to 19.72% ) and the lowest mean return falls into the interval (7.48% to 17.31%). 4. Both lower interval limits are much larger than the long-term bond rate of 3%. Wide range of overlapping intervals, the mean returns among funds are not significantly different. 6. All of these funds are considered equivalent. 7. Investor’s choice : the ones with lower transaction and management fees.

  20. Appendix: Table 1 Annual Returns of 8 Mutual Funds in U.S. from 1974 to 2010

  21. Sources: finance.yahoo.com/funds, mutual online performance 2011.11.30

  22. A Class of Individual Intervals

  23. Comparing to Intercepting Regions

  24. Comparing to Simultaneous Intervals A set of simultaneous confidence intervals by Bonferronifor the true mean returns is given by

  25. Comparing to Bonferroni Intervals

  26. Comparing to Bonferroni Intervals Optimal intervals are shorter. AGTHX in (10.06, 18.86) Length=8.80 AWSHX in ( 7.90, 16.70) Length=8.80 Intercepting intervals are medium shorter. AGTHX in (9.08, 20.18) Length=11.1 AWSHX in (7.02, 18.12) Length=11.1 Bonferroni intervals are wider. AGTHX in (7.23, 23.84 ) Length=16.61 AWSHX in (5.24, 18.09) Length=12.85

  27. Conclusions Optimal intervals are shortest. Intercepting intervals are medium shorter. Bonferroni intervals are widest.

  28. References • Bechhofer, R.E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics, 25, 16-39. • Chen, H.J. (1975). Strong consistency and asymptotic unbiasedness of a natural estimator for a ranked parameterSankhya Ser. B, Vol. 38, 92-94. • Chen, H.J. and Chen, S.Y. (1999). A nearly optimal confidence interval for the largest normal mean. Communications in Statistics: Simulation and Computation, 28(1), 131-146. • Chen, H. J. and Chen, S. Y. (2004). Optimal confidence interval for the largest normal mean with unknown variance. Computational Statistics and Data Analysis, 47, 845-866. • Chen, H.J. and Dudewicz, E.J. (1976). Procedures for fixed-width interval estimation of the largest normal mean. Journal of the American Statistical Association, 71, 752-756. • Chen, H.J. and Dudewicz, E.J. (1973). Estimation of ordered parameters and subset selection. Technical Report, Ohio State University, Columbus, Ohio. • Chen, H.J., Li, H.L. and Wen, M.J. (2008). Optimal Confidence Interval for the Largest Mean of Correlated Normal Populations and Its Application to Stock Fund Evaluation. Computational Statistics and Data Analysis, 52, 4801-4813. • Chen, H.J., and Wu, S.F. (2012). A confidence region for the largest and the smallest means under heteroscedasticity. Computational Statistics and Data Analysis, 56, 1692-1702.

  29. References • Dudewicz, E.J. (1972). Two-sided confidence intervals for ranked means. Journal of the American Statistical Association, 67, 462-464. • Dudewicz, E.J. and Tong, Y.L. (1971). Optimal confidence interval for the largest location parameter. Statistical Decision Theory and Related Topics. (S.S. Gupta and J. Yackel, Eds.) New York: Academic Press, Inc., 363-375. • Johnson, N.L. and Kotz, S. (1972). Distribution in Statistics: Continuous Multivariate Distributions. New York: Wiley. • Johnson, R.A. and Wichern, D.W. (2002). Applied Multivariate Statistical Analysis (4th Ed.), Upper Saddle River, New Jersey: Prentice Hall. • Lawley, D. N. (1963). On testing a set of correlation coefficients for equality. The Annals of Mathematical Statistics, 34, 149-151. • Saxena, K.M.L. (1976). A single-sample procedure for the estimation of the largest mean. Journal of the American Statistical Association, 71, 147-148. • Saxena, K.M.L., Krishna, M.L, Tong, Y.L. (1969). Interval estimation of the largest mean of k normal populations with known variance. Journal of the American Statistical Association, 64, 296-299. • Tong, Y.L. (1973). An asymptotically optimal sequential procedure for the estimation of the largest mean. The Annals of Statistics, 1, 175-179.

  30. for your attendance Thank you

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