Statistics

In this article, the strong uniform consistency of two nonparametric estimators for the quantile density function is established under length-biased sampling. The rate of the strong approximation of the resulting processes of these estimators will be presented as well. A Monte Carlo simulation study is carried out to compare the proposed estimators with each other in terms of mean squared errors.

Two measures of uncertainty, the cumulative residual entropy (CRE) and doubly truncated
Shannon entropy have been introduced respectively by Rao et al. (2004) and Sunoj et al.
(2009). In this paper we proposed the doubly truncated (interval) cumulative residual
entropy (ICRE), which is an extension of CRE and the dynamic CRE defined by Asadi
and Zohrevand (2007). We study some properties and characterization of this measure,
including its connections with doubly truncated Shannon entropy and mean residual
life. Also, the dual measure, doubly truncated (interval) cumulative past entropy (ICPE)
is defined and some of its properties are studied. Furthermore, their monotonicity and
associated aging classes of distributions are discussed and the upper (lower) bound for
them are obtained. Finally for exponential distribution all entropies are presented.

One of the most popular methods and algorithms to partition data to k clusters is k-means clustering algorithm. Since this method relies on some basic conditions such as, the existence of mean and finite variance, it is unsuitable for data that their variances are infinite such as data with heavy tailed distribution. Pitman Measure of Closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. In this article using PMC, based on k-means clustering, a new distance and clustering algorithm is developed for heavy tailed data.

Pitman measure of closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. We develop a new similarity measure which uses PMC on the basis of the similarity measure for hierarchical clustering algorithm that has better performance for heavy-tailed data.