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XII: 23, 332-341, LNM 649 (1978)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Lois empiriques et distance de Prokhorov (Mathematical statistics)

Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones

Keywords: Empirical distribution function, Prohorov distance

Nature: Original

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XII: 24, 342-363, LNM 649 (1978)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Estimation des densités~: risque minimax (Mathematical statistics)

A sequel to the preceding paper 1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed

Comment: For a correction see 1360. An improved version appeared in (*Zeit. für W-theorie,* **47**, 1979)

Keywords: Empirical distribution function

Nature: Original

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XIII: 60, 647-647, LNM 721 (1979)

**BRETAGNOLLE, Jean**; **HUBER, Catherine**

Corrections à un exposé antérieur (Mathematical statistics)

Two misprints and a more substantial error (in the proof of proposition 1) of 1224 are corrected

Comment: A revised version appeared in (*Zeit. für W-Theorie,* **47**, 1979)

Keywords: Empirical distribution function, Prohorov distance

Nature: Correction

Retrieve article from Numdam

Lois empiriques et distance de Prokhorov (Mathematical statistics)

Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones

Keywords: Empirical distribution function, Prohorov distance

Nature: Original

Retrieve article from Numdam

XII: 24, 342-363, LNM 649 (1978)

Estimation des densités~: risque minimax (Mathematical statistics)

A sequel to the preceding paper 1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed

Comment: For a correction see 1360. An improved version appeared in (

Keywords: Empirical distribution function

Nature: Original

Retrieve article from Numdam

XIII: 60, 647-647, LNM 721 (1979)

Corrections à un exposé antérieur (Mathematical statistics)

Two misprints and a more substantial error (in the proof of proposition 1) of 1224 are corrected

Comment: A revised version appeared in (

Keywords: Empirical distribution function, Prohorov distance

Nature: Correction

Retrieve article from Numdam