Deconvolution of ℙ(X<Y) with unknown error distributions

NONPARAMETRIC ESTIMATION OF P(X<Y) WITH LAPLACE ERROR DENSITIES

“…1 Let the observations be given by model (2). Let the quantity  be defined as in (1) and the estimator ˆ  be as in ( 4) with (0,1),…”

“…C only depends on . s Proposition 2.2 Let the observations be given by model (2). Let the quantity  be defined as in (1) and the estimator ˆ  be given by ( 4) with (0,1),   1. s  Suppose that , j g  and ,k g  are the Laplace density, 1, , ;…”

“…To prove Proposition 2.2, we need to prove Lemma 1 and Lemma 2. Lemma 1 Let the observations be given by model (2). Suppose that , j g  and ,k g  are the Laplace density, 1, , ,…”

“…Lemma 2 Let the observations be given by model (2). Suppose that , j g  and ,k g  are the Laplace density, 1, , ; 1, , .…”

“…This means that j  and k  have different distributions for 1 , j n 1 . k m Recently, Phuong-Thuy [2] concentrated on the case where the distribution of the random errors is unknown but symmetric around zero and can be estimated from some additional samples.…”