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Kolmogorov distance; $\phi $-divergence; minimum distance estimator; consistency rate; computer simulation
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, Lévy, Cramér, or discrepancy distance is used. Consequently, $n^{-1/2}$ consistency rate in any bounded $\phi$-divergence is established for Kolmogorov, Lévy, 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.
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