# Ing. Jitka Hrabáková

## Publikace

### Existence, Consistency and Computer Simulation for Selected Variants of Minimum Distance Estimators

Autoři
Kůs, V.; Morales, D.; Hrabáková, J.; Frýdlová, I.
Rok
2018
Publikováno
Kybernetika. 2018, 54(2), 336-350. ISSN 0023-5954.
Typ
Článek
Pracoviště
Anotace
The paper deals with sufficient conditions for the existence of general approximate minimum distance estimator (AMDE) of a probability density function $f_0$ on the real line. It shows that the AMDE always exists when the bounded $\phi$-divergence, Kolmogorov, L\'evy, Cram\'er, or discrepancy distance is used. Consequently, $n^{-1/2}$ consistency rate in any bounded $\phi$-divergence is established for Kolmogorov, L\'evy, and discrepancy estimators under the condition that the degree of variations of the corresponding family of densities is finite. A simulation experiment empirically studies the performance of the approximate minimum Kolmogorov estimator (AMKE) and some histogram-based variants of approximate minimum divergence estimators, like power type and Le\,Cam, under six distributions (Uniform, Normal, Logistic, Laplace, Cauchy, Weibull). A comparison with the standard estimators (moment/maximum likelihood/median) is provided for sample sizes $n=10,20,50,120,250$. The simulation analyzes the behaviour of estimators through different families of distributions. It is shown that the performance of AMKE differs from the other estimators with respect to family type and that the AMKE estimators cope more easily with the Cauchy distribution than standard or divergence based estimators, especially for small sample sizes.

### Notes on consistency of some minimum distance estimators with simulation results

Autoři
Hrabáková, J.; Kůs, V.
Rok
2017
Publikováno
Metrika. 2017, 80(2), 243-257. ISSN 0026-1335.
Typ
Článek
Pracoviště
Anotace
We focus on the minimum distance density estimators f_n of the true probability density f_0 on the real line. The consistency of the order of n^−1/2 in the (expected) L_1-norm of Kolmogorov estimator (MKE) is known if the degree of variations of the nonparametric family D is finite. Using this result for MKE we prove that minimum Lévy and minimum discrepancy distance estimators are consistent of the order of n^−1/2 in the (expected) L_1-norm under the same assumptions. Computer simulation for these minimum distance estimators, accompanied by Cramér estimator, is performed and the function s(n)=a_0+a_1√n is fitted to the L_1-errors of f_n leading to the proportionality constant a1 determination. Further, (expected) L_1-consistency rate of Kolmogorov estimator under generalized assumptions based on asymptotic domination relation is studied. No usual continuity or differentiability conditions are needed.

### Minimum Distance Density Estimates of a Probability Density on the Real Line

Autoři
Hrabáková, J.; Kůs, V.
Rok
2013
Publikováno
Essays on Mathematics and Statistics: Volume 3. Athens: Athens Institute for Education and Research, 2013. p. 97-107. ISBN 978-960-9549-34-9.
Typ
Kapitola v knize
Pracoviště
Anotace
This paper focuses on the minimum distance density estimate f_n of probability density f on a real line. The rate of consistency of Kolmogorov density estimate for n tending to infinity is known if the degree of variations is finite. The rate of consistency is studied under more general conditions. For this purpose, the generalization of degree of variation - the partial degree of variation is defined for density g of nonparametric family D containing the unknown density f. If the partial degree of variation is finite and some additional, but not as restrictive as a finiteness of degree of variation, assumptions are fulfilled then the Kolmogorov density estimate is consistent with the order n to the -1/2 in L1-norm and also in the expected L1-norm. A small generalization of previous theory is made. Furthermore, some other minimum distance density estimates are explored. (Namely Lévy, discrepanci, and Cramer-von Mises distance.) And with the aid of inequalities between statistical distances, Levy and discrepancy minimum distance estimates are proved to be consistent with the order n to the -1/2 in the L1-norm and in the expected L1-norm as well. Further, numerical simulations of minimum Cramer-von Mises distance estimate and of the others estimates, which are proved to be consistent, are performed and graphs of consistency are presented and discussed.

### The Consistency and Robustness of Modified Cramer-Von Mises and Kolmogorov-Cramer Estimators

Autoři
Hrabáková, J.; Kůs, V.
Rok
2013
Publikováno
Communication in Statistics-Theory and Method. 2013, 42(20), 3665-3677. ISSN 0361-0926.
Typ
Článek
Pracoviště
Anotace
This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér - von Mises distance. The generalized power form of Cramér - von Mises distance is defined together with the so-called Kolmogorov - Cramér distance which includes both standard Kolmogorov and Cramér - von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L1 - norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L1 - norm on contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov - Cramér, generalized Cramér -von Mises, standard Kolmogorov, and Cramér -von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the ontaminated and uncontaminated pseudo-random samples.

### Kolmogorov-Cramér Type Estimators

Autoři
Rok
2012
Publikováno
Doktorandské dny \'12. Praha: Matfyzpress, 2012. pp. 83-85. ISBN 978-80-7378-217-7.
Typ
Stať ve sborníku
Pracoviště
Anotace
We study consistency of MD estimates and their robustness, as well as its impact to the physical data processing coming from an accelerated physics. We introduce a wide class of modifications of the Kolmogorov-Cramér distance by implementing data-based weight functions, random selecting of differences to be summed up, and using various coefficient modifications. All these new estimates are defined to preserve consistency and simultaneously to have better robust properties than the Kolmogorov estimate.We perform an extensive simulation study to compare robustness of newly introduced estimates with the original Kolmogorov and Cramér-von Mises estimates.