TRANSTORNOS EMOCIONALES Y DE CONDUCTA POR EL CONSUMO DE DROGAS Y ALCOHOL

As in the reviewed literature, only methods based on the ET divergence and the EL divergence will be considered. Even for the one sample situation there are no bootstrap methods in the literature that use both fixed side information constraints and profiling con- straints. Biased bootstrap tilting methods, a combination of the biased bootstrap from Hall and Presnell (1999) and the bootstrap tilting methods from Efron (1981), will be obtained by first replacing the role of Fn with Fwe, where the weights we are obtained by minimizing

an ET divergence of the form

n

X

i=1

nwilog (nwi)

or an EL divergence of the form

−

n

X

i=1

log (nwi)

using the fixed side information constraint, e.g.

n

X

i=1

wixi =µx0

corresponding to the side informationE(X) =µx0, as in the biased bootstrap methods from

Hall and Presnell (1999).

In the spirit of Efron (1981)Fw_e is then embedded in a resampling family of distributions,

to the fixed constraint and the profiling constraint n X i=1 wixi =µx0, n X i=1 wiyi =µy

The adjusted estimator for E(Y) is the calibration estimator

e µy = n X i=1 e wiyi

and w_e =w(µ_ey) shows the embedding.

Alternatively Fw_e may be embedded in another resampling family of distributions, each

member Fw with weightsw=w(µy) that minimize an ET divergence of the form n X i=1 wi e wi log wi e wi

or an EL divergence of the form

− n X i=1 log wi e wi

subject only to the profiling constraint, e.g.

n

X

i=1

wiyi =µy

Confidence intervals forµy may be constructed by extending the methods from Efron (1981).

Instead of resampling from several members of the resampling families, resampling fromFw_e

or even fromFnmay be possible by using importance sampling reweighting as in Hesterberg

(1999). The method may be extended to the two independent samples situation to generalize the methods from Jing and Robinson (1997).

REFERENCES

Adimari, G. (1995). Empirical likelihood confidence intervals for the difference between means. Statistica 55, 87–94.

Baggerly, K. A. (1998). Empirical likelihood as a goodness-of-fit measure. Biometrika 85, 535–547.

Barbe, P. and Bertail, P. (1995). The Weighted Bootstrap. Springer-Verlag, New York, U.S.A.

Bera, A. K. and Bilias, Y. (2002). The mm, me, ml, el, ef and gmm approaches to estimation: a synthesis. Journal of Econometrics 107, 51–86.

Bhapkar, V. P. (1966). A note on the equivalence of two test criteria for hypotheses in categorical data. Journal of the American Statistical Association 61, 228–235.

Brown, B. W. and Newey, W. K. (2002). Generalized method of moments, efficient boot- strapping, and improved inference. Journal of Business and Economic Statistics 20, 507–517.

Chen, J. and Qin, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80, 107–116.

Chen, J. and Sitter, R. R. (1999). A pseudo empirical likelihood approach to the effective use of auxiliary information in complex surveys. Statistica Sinica 9, 385–406.

Chen, S. X., Leung, D. H. Y. and Qin, J. (2003). Information recovery in a study with surrogate endpoints. Journal of the American Statistical Association 98, 1052–1062. Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on

the sum of observations. Ann. Math. Statist 23, 493–507.

Chuang, C. S. and Lai, T. L. (2000). Hybrid resampling methods for confidence intervals (with comments). Statistica Sinica 10, 1–50.

Cochran, W. G. (1963). Sampling Techniques (Second edition). Wiley, New York.

Corcoran, S. A. (2000). Empirical exponential family likelihood using several moment con- ditions. Statistica Sinica 10, 545–557.

Csisz´ar, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar.2, 299–318.

Csisz´ar, I. (1975). I-divergence geometry of probability distributions and minimization prob- lems. The Annals of Probability 3, 146–158.

Deming, W. E. and Stephan, F. F. (1940). On a least squares adjustment of a sampled fre- quency table when the expected marginal totals are known. The Annals of Mathematical Statistics 11, 427–444.

Deville, J. C. and S¨arndal, C. E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association 87, 376–382.

DiCiccio, T. J., Hall, P. and Romano, J. P. (1991). Empirical likelihood is bartlett- correctable. The Annals of Statistics 19, 1053–1061.

DiCiccio, T. J. and Romano, J. P. (1990). Nonparametric confidence limits by resampling methods and least favorable families. International Statistical Review 58, 59–76.

Efron, B. (1981). Nonparametric standard errors and confidence intervals. The Canadian Journal of Statistics 9, 139–172.

Fienberg, S. E. and Tanur, J. M. (1987). Experimental and sampling structures: parallels diverging and meeting. International Statistical Review55, 75–96.

Haberman, S. J. (1984). Adjustment by minimum discriminant information. Annals of Statistics 12, 971–988.

Hall, P. and Horowitz, J. (1996). Bootstrap critical values for tests based on generalized method of moments. Econometrica 64, 891–916.

Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood.

International Statistical Review 58, 109–127.

Hall, P. and Presnell, B. (1999). Intentionally biased bootstrap methods. J. R. Statist. Soc. B 61, 143–158.

Hansen, L. (1982). Large sample properties of generalized method of moments estimators.

Econometrica 50, 1029–1054.

Hartley, H. O. and Rao, J. N. K. (1968). A new estimation theory for sample surveys.

Biometrika 55, 547–557.

Hesterberg, T. (1999). Bootstrap tilting confidence intervals. Technical Report Research Report No. 84, MathSoft, Inc., Seattle, U.S.A.

Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without re- placement from a finite population. Journal of the American Statistical Association 47, 663–685.

models. Journal of Econometrics 107, 87–98.

Imbens, G. W., Spady, R. H. and Johnson, P. (1998). Information theoretic approaches to inference in moment condition models. Econometrica 66, 333–357.

James, G. (1951). The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika38, 324–329.

Jing, B. Y. (1995). Two-sample empirical likelihood method. Statistics & Probability Letters

24, 315–319.

Jing, B. Y. and Robinson, J. (1997). Two-sample nonparametric tilting method. Australian Journal of Statistics 39, 25–34.

Kitamura, Y. and Stutzer, M. (1997). An information-theoretic alternative to generalized method of moments estimation. Econometrica 65, 861–874.

Koch, G. G., Amara, I. A., Davis, G. W. and Gillings, D. B. (1982). A review of some statistical methods for covariance analysis of categorical data. Biometrics38, 563–595. Koch, G. G., Imrey, P. B., Singer, J. M., Atkinson, S. S. and Stokes, M. E. (1985). Analysis

of Categorical Data. Les Presses de l’Universite de Montreal, Montreal, Canada.

Koch, G. G., Tangen, C. M., Jung, J. W. and Amara, I. A. (1998). Issues for covariance analysis of dichotomous and ordered categorical data from randomized clinical trials and non-parametric strategies for addressing them. Statistics in Medicine 17, 1863–1892. Kullback, S. (1959). Information Theory and Statistics. John Wiley, New York, U.S.A. Kullback, S. and Leibler, R. A. (1951). On information and sufficiency. The Annals of

Mathematical Statistics 22, 79–86.

Lee, S. M. S. and Young, G. A. (1999). Nonparametric likelihood ratio confidence intervals.

Biometrika 86, 107–118.

Lesaffre, E., Bogaerts, K., Li, X. and Bluhmki, E. (2002). On the variability of covari- ate adjustment: experience with koch’s method for evaluating the absolute difference in proportions in randomized clinical trials. Controlled Clinical Trials 23, 127–142.

Lesaffre, E. and Senn, S. (2003). A note on non-parametric ancova for covariate adjustment in randomized clinical trials. Statistics in Medicine22, 3583–3596.

Mann, H. B. and Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics 18, 50–60. Namba, A. (2004). Simulation studies on bootstrap empirical likelihood tests. Communica-

Newey, W. K. and Smith, R. J. (2004). Higher order properties of gmm and generalized empirical likelihood estimators. Econometrica 72, 219–255.

Neyman, J. (1949). Contributions to the theory of the χ2 _{test. In Proceedings of the First} Berkeley Symposium on Mathematical Statistics and Probability, pages 239–273. Univer- sity of California Press.

Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional.

Biometrika 75, 237–249.

Owen, A. (1990a). Empirical likelihood for linear models.Annals of Statistics19, 1725–1747. Owen, A. (1990b). Empirical likelihood ratio confidence regions. Annals of Statistics 18,

90–120.

Owen, A. B. (2001). Empirical Likelihood. Chapman & Hall/CRC, Boca Raton, U.S.A. Pincus, T., Koch, G. G., Sokka, T., Lefkowith, J., Wolfe, F., Jordan, J. M., Luta, G.,

Callahan, L. F., Wang, X., Schwartz, T., Abramson, S. B., Caldwell, J. R., Harrell, R. A., Kremer, J. M., Lautzenheiser, R. L., Markenson, J. A., Schnitzer, T. J., Weaver, A., Cummins, P., Wilson, A. and Morant, S. and, F. J. (2001). A randomized, double-blind, crossover clinical trial of diclofenac plus misoprostol versus acetaminophen in patients with osteoarthritis of the hip or knee. Arthritis & Rheumatism 44, 1587–1598.

Pincus, T., Swearingen, C. and Wolfe, F. (1999). Toward a multidimensional health assess- ment questionnaire (mdhaq): assessment of advanced activities of daily living and psy- chological status in the patient-friendly health assessment questionnaire format. Arthritis & Rheumatism 42, 2220–2230.

Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. Wiley, New York, U.S.A.

Qin, G. and Y., J. B. (2001). Empirical likelihood for censored linear regression.Scandinavian Journal of Statistics 28, 661–673.

Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations.Annals of Statistics 22, 300–325.

Qin, J. and Lawless, J. (1995). Estimating equations, empirical likelihood and constraints on parameters. The Canadian Journal of Statistics 23, 145–159.

Quade, D. (1967). Rank analysis of covariance. Journal of the American Statistical Associ- ation 62, 1187–1200.

R Development Core Team (2006). R: A Language and Environment for Statistical Com- puting. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.

Read, T. R. C. (1984). Small-sample comparisons for the power divergence goodness-of-fit statistics. Journal of the American Statistical Association 79, 929–935.

Read, T. R. C. and Cressie, N. A. C. (1988). Goodness-of-Fit Statistics for Discrete Multi- variate Data. Springer-Verlag, New York, U.S.A.

Reid, N. (1988). Saddlepoint methods and statistical inference. Statistical Science 3, 213– 238.

Renssen, R. H. and Nieuwenbroek, N. J. (1997). Aligning estimates for common variables in two or more survey samples. Journal of the American Statistical Association92, 368–374. Sheehy, A. (1987). Kullback-Leibler estimation of probability measures with an application

to clustering. PhD thesis, University of Washington, Seattle.

Sheehy, A. (1988). Kullback-leibler constrained estimation of probability measures. Technical Report Technical report no. 137, Department of Statistics, University of Washington, Seattle, U.S.A.

Snedecor, G. W. and Cochran, W. G. (1980). Statistical Methods, 7th ed. Iowa State University Press, Ames, U.S.A.

Stein, C. (1956). Efficient nonparametric testing and estimation. InProceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pages 187–195. University of California Press.

Tangen, C. M. and Koch, G. G. (2001). Non-parametric analysis of covariance for confir- matory randomized clinical trials to evaluate dose-response relationships. Statistics in Medicine 20, 2585–2607.

Wu, C. (2002). Empirical likelihood method for finite populations. In Chaubey, Y. P., editor,

Recent Advances in Statistical Methods, pages 339–351. Imperial College Press, London. Wu, C. (2004). Combining information from multiple surveys through the empirical likeli-

hood method. Canadian Journal of Statistics 32, 15–26.

Wu, C. (2005). Algorithms and r codes for the pseudo empirical likelihood method in survey sampling. Survey Methodology 31, 239–243.

Yang, L. and Tsiatis, A. A. (2001). Efficiency study of estimators for a treatment effect in a pretest-posttest trial. The American Statistician 55, 314–321.

Zhang, B. (1996). Confidence intervals for a distribution function in the presence of auxiliary information. Computational Statistics & Data Analysis 21, 327–342.

Zhang, B. (1997). Empirical likelihood confidence intervals for m-functionals in the presence of auxiliary information. Statistics & Probability Letters 32, 87–97.

Zhang, B. (1999). Bootstrapping with auxiliary information. The Canadian Journal of Statistics 27, 237–249.

Zhong, B., Chen, J. and Rao, J. N. K. (2000). Empirical likelihood inference in the presence of measurement error. The Canadian Journal of Statistics 28, 841–852.

Zhong, B. and Rao, J. N. K. (2000). Empirical likelihood inference under stratified random sampling using auxiliary population information. Biometrika87, 929–938.

Zhou, M. (2005). emplik: Empirical likelihood ratio for censored/truncated data. R package version 0.9-2.

Zieschang, K. D. (1990). Sample weighting methods and estimation of totals in the consumer expenditure survey. Journal of the American Statistical Association 85, 986–1001.