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On fractal distribution function estimation and applications

Introduction

Theoretical background for affine IFS

2.1 Minimization approach

Theorem 1

Theorem 2 (Collage theorem)

Theorem 3 (Forte and Vrscay, 1995)

2.2 Direct approach

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Corollary 6

Theorem 7

2.3 The choice of affine maps

3 Fourier analysis results

Theorem 8 (Forte and Vrscay, 1998)

Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

4 Statistical applications

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 10

Theorem 11

Theorem 12 (Gill and Levit, 1995)

Theorem 13

4.2 Characteristic function and Fourier density estimation

5 Monte Carlo analysis

5.1 Applications to survival analysis

Final remarks

References

On fractal distribution function estimation and applications

References

[1] Barnsley, M.F., Demko, S., "Iterated function systems and the global construction of fractals", Proc. Roy. Soc. London, Ser A, 399, 243-275, 1985.

[2] Beran, R., "Estimating a distribution function", Ann. Statist., 5, 400-404, 1977.

[3] Dvoretsky, A., Kiefer, J. and Wolfowitz, J., "Asymptotic minimax character of the sample distribution function and of the classical multinomial estimators", Ann. Math. Statist., 27, 642-669, 1956.

[4] Efromovich, S., "Second order efficient estimating a smooth distribution function and its applications", Meth. Comp. App. Probab., 3, 179-198, 2001.

[5] Kaplan, E. and Meyer, P., "Nonparametric estimator from incomplete observations", J. Amer. Statist. Ass, 53, 457-81, 1958.

[6] Forte, B., Vrscay, E.R., "Solving the inverse problem for function/image approximation using iterated function systems, I. Theoretical basis", Fractal, 2, 3, 325-334, 1995.

[7] Forte, B., Vrscay, E.R., "Inverse problem methods for generalized fractal transforms", in Fractal Image Encoding and Analysis, NATO ASI Series F, Vol. 159, ed. Y. Fisher, Springer Verlag, Heidelberg, 1998.

[8] Fleming, T.R. and Harrington, D.P. , Counting processes and survival analysis, Wiley, New York, 1991.

[9] Kiefer, J. and Wolfowitz, J., "Asymptotic minimax character of the sample distribution function for vector chance variables", Ann. Math. Stat., 30, 463-489, 1959.

[10] Gill, R. D. and Levit, B. Y., "Applications of the van Trees inequality: A Bayesian Cram´er-Rao bound", Bernoulli, 1, 59-79, 1995.

[11] Golubev, G. K. and Levit, B. Y., "On the second order minimax estimation of distribution functions", Math. Methods. Statist., 5, 1-31, 1996a.

[12] Golubev, G. K. and Levit, B. Y., "Asymptotic efficient estimation for analytic distributions", Math. Methods. Statist., 5, 357-368, 1996b.

[13] Hutchinson, J., "Fractals and self-similarity", Indiana Univ. J. Math., 30, 5, 713-747, 1981.

[14] Iacus, S.M. and La Torre, D., "Approximating distribution functions by iterated function systems", submitted, available as Acrobat PDF file at http://159.149.74.117/~web/R/ifs/ifs.pdf, 2001.

[15] Ihaka, R. and Gentleman, R., "R: A Language for Data Analysis and Graphics", Journal of Computational and Graphical Statistics, 5, 299-314, 1996.

[16] Levit, B.Y., "Infinite-dimensional information inequalities", Theory Probab. Applic., 23, 371-377, 1978.

[17] Millar, P.W., "Asymptotic minimax theorems for sample distribution functions", Z. Warsch. Verb. Geb., 48, 233-252, 1979.

[18] Silverman, B. W., Density Estimation, London, Chapman and Hall, 1986.

[19] Tarter, M.E. and Lock, M.D, Model free curve estimation, Chapman & Hall, New York, 1993.

[20] Venables, W. N. and Ripley, B. D., Modern Applied Statistics with S-PLUS, New York, Springer, forthcoming, 2002. Springer, New York, 1998. [21] Winter, B.B., "Strong uniform consistency of integrals of density estimators", Can. J. Statist., 1, 247-253, 1973.

[22] Winter, B.B., "Convergence rate of perturbed empirical distribution functions", J. Appl. Prob., 16, 163-173, 1979.

[23] Yukish, J.E. , "A note on limit theorems for perturbed empirical processes". Stoch. Proc. Appl., 33, 163-173, 1989.

Stefano M. Iacus, Davide La Torre

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