in this paper, we prove the strong uniform consistency and asymptotic normality of the kernel density estimator proposed by jones [12] for length-biased data.the approach is based on the invariance principle for the empirical processes proved by horváth [10]. all simulations are drawn for different cases to demonstrate both, consistency and asymptotic normality and the method is illustrated by ...

length-biased data are widely seen in applications. they are mostly applicable in epidemiological studies or survival analysis in medical researches. here we aim to propose a berry-esseen type bound for the kernel density estimator of this kind of data.the rate of normal convergence in the proposed berry-esseen type theorem is shown to be o(n^(-1/6) ) modulo logarithmic term as n tends to infin...

In simple random sampling, there is an equal chance of selection for each unit in the population. However, there are some practical situations where this might not be ideal. We consider the case of length-biased sampling, whereby the chance of selection of a unit is proportional to the length of the unit, by which the length-biased density is defined. Specifically we consider the case of a cont...

In this paper, we develop the length-biased form of the weighted Weibull distribution (WWD) named as length-biased weighted Weibull distribution (LBWWD). Well known distributions are generated by expanding suitable function of the parameters. Shape of the distribution is studied in detail. Various properties of length-biased WWD are discussed. Newby’s method along with method of moment has been...

Given a group Γ and a biased graph (G,B), we define a what is meant by a Γ-realization of (G,B) and a notion of equivalence of Γ-realizations. We prove that for a finite group Γ and t ≥ 3, that there are numbers n(Γ) and n(Γ, t) such that the number of Γ-realizations of a vertically 3-connected biased graph is at most n(Γ) and that the number of Γ-realizations of a nonseparable biased graph wit...