There are several topics to be considered beyond the works completed in this dissertation. We now discuss some possible future research work.
The circular block bootstrap (Politis and Romano, 1992), alternatives to mov-ing block bootstrap, which have an advantage of reducmov-ing the bias of the bootstrap variance, can be extended in longitudinal data. The tapered block bootstrap method (Paparoditis and Politis, 2001; Paparoditis and Politis, 2002) in which each block end points are shrunk toward a target value before being concatenated to form a bootstrap pseudo-series, which indeed leads a more accurate variance estimator, can be used in a longitudinal setting. We need to explore the optimal block length using the other block bootstrap methods in longitudinal data. The correlation structure of different time measurements can be extended to long range dependence and nonstationary dependence.
While most of our work has focused on linear constraints in our repeated measure-ment model, developmeasure-ments with nonlinear constraints might also be possible. Other possible extensions include binary or polytomous data in repeated measurements.
The possible nonparametric or semiparametric estimation methods using block bootstrap technique in the analysis of longitudinal data pose interesting problems for future research. Furthermore, it may be desirable to develop a methodology for irregularly spaced repeated measurement data.
REFERENCES
Arnold, S. (1981). The Theory of Linear Models and Multivriate Analysis. New York:
Wiley.
Babu, G. J. (1986). “Bootstrapping statistics with linear combinations of chi-squares as weak limit.” Sankhya Ser A., 56, 85–93.
Bhattacharya, R. N. and Qumsiyeh, M. (1988). “Second order and Lp comparisons between the bootstrap and empirical edgeworth expansion methodologies.” The Annals of Statistics, 17, 160–169.
Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximations and Asymp-totic Expansions. Malabar, FL: Krieger.
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. 2nd ed.
New York: Springer-Verlag.
B¨uhlmann, P. (1994). “Blockwise bootstrapped empirical process for stationary se-quences.” The Annals of Statistics, 22, 995–1012.
— (1997). “Sieve bootstrap for time series.” Bernoulli, 3, 128–148.
B¨uhlmann, P. and Kunsch, H. R. (1995). “The blockwise bootstrap for general pa-rameters of a stationary time series.” Scan. Journal of Statistics, 22, 35–54.
Carlstein, E. (1986). “The use of subseries methods for estimating the variance of a general statistic from a stationary time series.” The Annals of Statistics, 14, 1171–1179.
Davis, C. S. (2002). Statistical Methods for the Analysis of Repeated Measurements.
New York: Springer-Verlag.
Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application.
Cambridge, UK: Cambridge University Press.
Diggle, P. J., Liang, K. Y., and Zeger, S. L. (1994). Analysis of Longitudinal Data.
New York: Oxford University Press Inc.
Dobson, A. J. (2002). An Introduction to Generalized Linear Models. 2nd ed. New York: Chapman & Hall/CRC.
Efron, B. (1979). “Bootstrap methods: Another look at the jackknife.” The Annals of Statistics, 7, 1–16.
Feng, Z., McLerran, D., and Grizzle, J. (1996). “A comparison of statistical methods for clustered data analysis with Gaussian error.” Statistics in Medicine, 15, 1793–
1806.
Goncalves, S. and White, H. (2002). “The bootstrap of the mean for dependent heterogeneous arrays.” Econometric Theory, 18, 1367–84.
Granger, C. W. and Joyuex, R. (1980). “An introduction to long-memory time series models and fractional differencing.” J. Time Series Anal., 1, 15–29.
Graybill, F. A. (1976). Theory and Application of Linear Model . MA: Duxbury Press.
Hall, P. (1992). The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.
Hall, P., Horowitz, J., and Jing, B. Y. (1995). “On blocking rules for the bootstrap withe dependent data.” Biometrika, 82, 561–574.
Hand, D. J. and Crowder, M. J. (1996). Practical Longitudinal Data Analysis. Lon-don, UK: Chapman & Hall.
Hartley, H. O. and Rao, J. N. K. (1967). “Maximum likelihood estimation for the mixed analysis of variance model.” Biometrika, 54, 93–108.
Hocking, R. R. (1985). The Analysis of Linear Models. Pacific Grove, CA: Brooks Cole.
Hosking, J. R. M. (1981). “Fractional differencing.” Biometrika, 68, 165–176.
Huang, J. Z., Wu, W. O., and Zhou, L. (2002). “Varying-coefficient models and basis function approximations for the analysis of repeated measurements.” Biometrika, 89, 111–128.
Keselman, H. J., Algina, J., Kowalchuk, R. K., and Wolfinger, R. D. (1998). “A comparision of two approaches for selecting covariance structure in the analysis of repeated measurements.” Communication and Statistics., Series. B , 27, 591–604.
K¨unsch, H. R. (1989). “The jackknife and the bootstrap for general stationary ob-servations.” The Annals of Statistics, 17, 1217–1261.
Lahiri, S. N. (1992). “Bootstrapping m-estimators of a multiple linear regression parameter.” The Annals of Statistics, 20, 1548–1570.
— (1993). “On the moving block bootstrap under long range dependence.” Statistics and Probability Letters, 18, 405–413.
— (1994). “On two-term Edgeworth expansions and bootstrap approximations for studentized multivariate m-estimators.” Sankhya Ser A., 56, 201–226.
— (1995). “On the asymptotic behaviour of the moving block bootstrap for nor-malized sums of heavy-tailed random variables.” The Annals of Statistics, 23, 1331–1349.
— (1996). “On Edgeworth and moving block bootstrap for studentized m-estimators in multiple linear regression models.” Journal of Multivariate Analysis, 56, 42–59.
— (1999). “Theoretical comparisons of block bootstrap methods.” The Annals of Statistics, 27, 386–404.
— (2001). “Effects of block lengths on the validity of block resampling methods.”
Probability Theory and Related Fields, 121, 73–97.
Laird, N. M. and Ware, J. H. (1982). “Random-effects models for longitudinal data.”
Biometrics, 38, 963–974.
Liang, K. Y. and Zeger, S. L. (1986). “Longitudinal data analysis using generalized linear models.” Biometrika, 73, 13–22.
Lindsey, J. K. (1999). Models for Repeated Measurements. 2nd ed. New York: Oxford University Press Inc.
Liu, R. Y. and Singh, K. (1992). “Moving blocks jackknife and bootstrap capture weak dependence.” In Exploring the Limits of Bootstrap. New York: Wiley, pp.
225-248.
Paparoditis, E. and Politis, D. N. (2001). “Tapered block bootstrap.” Biometrika, 88, 1105–1119.
— (2002). “Tapered block bootstrap for general statistics from stationary sequences.”
Econometric Journal , 5, 131–148.
Politis, D. N. and Romano, J. P. (1992). “A circular block resampling procedure for stationary data.” In Exploring the Limits of Bootstrap. New York: Wiley, pp.
263-270.
Porell, F., Caro, F. G., Silva, A., and Monane, M. (1988). “A longitudinal analysis of the nursing home outcomes.” Health Services Research, 33, 835–865.
Rao, C. R. (1973). Linear Statistical Inference and Its Applications. New York: John Wiley & Sons, Inc.
Royall, R. M. (1986). “Model robust confidence intervals using maximum likelihood estimators.” International Statistical Review , 54, 221–226.
Searle, S. R. (1997). Linear Models. Classic ed. New York: Wiley.
Searle, S. R., Casella, G., and McCulloch, C. E. (1992). Variance Components. New York: Wiley.
Seber, G. A. F. (1977). Linear Regression Analysis. New York: Wiley.
Singh, K. (1981). “On the asymptotic accuracy of the Efron’s bootstrap.” The Annals of Statistics, 9, 1187–1195.
Xie, M. and Yang, Y. (2003). “Asymptotics for generalized estimating equations with large cluster sizes.” The Annals of Statistics, 31, 13–22.
VITA
Hyunsu Ju, son of Sung Moon Ju and Il Sun Yang, was born on October 7, 1970, in Junnam, Korea. He received a Bachelor of Science in statistics in February of 1996 from Chung-Ang University. In February 1998, he received a Master of Arts in statistics from Seoul National University. He went to America to pursue his Ph.D.
in August of 1999 in the Department of Statistics at Texas A&M University. He received his Ph.D. in statistics under the direction of Dr. Suojin Wang in December 2004. His permanent address is
530-62 Sadang 1 Dong SeoChoGU Seoul, Korea, 130-061