### Abstract

Minimum Kolmogorov distance estimates of arbitrary parameters are considered They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of order n^{-1/2}. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in the L_{1}-norm. The presented conditions for the existence, consistency, and consistency of order n^{-1/2} are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models.

Original language | English |
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Pages (from-to) | 237-255 |

Number of pages | 19 |

Journal | Metrika |

Volume | 43 |

Issue number | 3 |

Publication status | Published - 1996 |

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### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Metrika*,

*43*(3), 237-255.

**Minimum Kolmogorov distance estimates of parameters and parametrized distributions.** / Györfi, L.; Vajda, I.; Van Der Meulen, E.

Research output: Contribution to journal › Article

*Metrika*, vol. 43, no. 3, pp. 237-255.

}

TY - JOUR

T1 - Minimum Kolmogorov distance estimates of parameters and parametrized distributions

AU - Györfi, L.

AU - Vajda, I.

AU - Van Der Meulen, E.

PY - 1996

Y1 - 1996

N2 - Minimum Kolmogorov distance estimates of arbitrary parameters are considered They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of order n-1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in the L1-norm. The presented conditions for the existence, consistency, and consistency of order n-1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models.

AB - Minimum Kolmogorov distance estimates of arbitrary parameters are considered They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of order n-1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in the L1-norm. The presented conditions for the existence, consistency, and consistency of order n-1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models.

UR - http://www.scopus.com/inward/record.url?scp=0040655674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040655674&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040655674

VL - 43

SP - 237

EP - 255

JO - Metrika

JF - Metrika

SN - 0026-1335

IS - 3

ER -