Showing papers by "Hammou El Barmi published in 2005"
TL;DR: In this article, the authors considered the k-sample case and derived the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction, with strict inequality in some cases.
Abstract: If X1 and X2 are random variables with distribution functions F1 and F2, then X1 is said to be stochastically larger than X2 if F1 ≤F2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F1, F2, … , Fk,k ≥ 2, and we assume that F1 ≤ F2 ≤ ˙˙˙ ≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If and are the empirical and our restricted estimators of Fi, then we show that, asymptotically, for all x and all u > 0, with strict inequality in some cases. This clearly show...
64 citations
TL;DR: In this paper, a nonparametric estimator of a continuous distribution function F under the restriction that it has Type I bias is proposed, and the weak convergence of the resulting process is used to test for symmetry against that type of bias.
Abstract: A random variable X has a symmetric distribution about a if and only if X − a and −X + a are identically distributed. By considering various types of partial orderings between the distributions of X − a and −X + a, one obtains various types of partial skewness or one-sided bias. For example, F has Type I bias about a if F¯(a + x) ≥ F((a − x)−) for all x > 0; here F¯ = 1 − F. In this article we assume that a = 0, and propose a nonparametric estimator of a continuous distribution function F under the restriction that it has Type I bias. We derive the weak convergence of the resulting process which is used to test for symmetry against that type of bias. The new estimator is then compared with the nonparametric likelihood estimator (NPMLE), [Fcirc] n , of F in terms of mean squared error. A simulation study seems to indicate that the new estimator outperforms the NPMLE uniformly at all the quantiles of the distributions that we have investigated. It turns out that the results developed here could be used to c...
3 citations