The aim of this paper is to study distribution of ratios of GENERALIZED ORDER statistics from pareto distribution. parameter estimation of Pareto distribution based on GENERALIZED ORDER statistics and ratios of them have been obtained. Inferences using method of moments and unbiased estimator have been obtained to develop point estimations. Consistency of unbiased estimator has been illustrated.To compare the performances of the employed methods, numerical results have been computed. Ilustrative example using real data is also given.

Let $G$ be a finite group and $k$ a fixed positive integer. We define the GENERALIZED ORDER divisor graph of $G$ to be a graph whose vertex set is the group $G$ and in which two vertices $a$ and $b$ are adjacent if and only if the ORDERs $o(a^k)$ and $o(b^k)$ are different and either $o(a^k)$ divides $o(b^k)$ or $o(b^k)$ divides $o(a^k)$. This generalizes the ORDER divisor graphs of finite groups. Some properties of our graph are introduced, and we investigate the structure of the GENERALIZED ORDER divisor graphs of finite cyclic groups.

The concept of GENERALIZED ORDER statistics (GOSs) was introduced as a unified approach to a variety of models of ORDERed random variables. The purpose of this paper is to investigate conditions on the underlying distribution functions and the parameters on which GOSs are based, to establish Shaked-Shanthikumar multivariate dispersive ORDERing of GOSs from one sample and Khaledi-Kochar multivariate dispersive ORDERing of GOSs from two samples. Some applications are also given.

The goal of this paper is to generalize the recently introduced summability method and introduce Il-double statistical convergence of ORDER a by using ideal. We also investigate certain properties of this convergence.

A universal formulation is obtained for the construction of a variational principle for a general third-ORDER differential equation, regardless of its self-adjoint condition. Three illustrative examples are provided to demonstrate the convenience and efficacy of the proposed formulation.

The purpose of this paper is to investigate conditions on the underlying distribution functions and the parameters, on which the GENERALIZED ORDER statistics are based on, to establish the usual stochastic and the likelihood ratio ORDERings of general p-spacings by conditioning on the right tail of another lower indexed GENERALIZED ORDER statistics. Some potential applications are also given.

The dual GENERALIZED ORDER statistics is a unified model which contains the well known decreasingly ORDERed random data such as (reversed ORDERed) ORDER statistics and lower record values. In the present paper, some characterization results on the power function distribution based on the properties of dual GENERALIZED ORDER statistics are provided. The results are proved without any restriction on the parameters of the model of dual GOS.

In this paper we specify the conditions on the parameters of pairs of gOS’s under which the corresponding GENERALIZED ORDER statistics are ORDERed according to usual stochastic ORDERing, hazard rate ORDERing, likelihood ratio ORDERing and dispersive ORDERing. We consider this problem in one-sample as well as two-sample problems. We show that some of the results obtained by Franco et al. [Probab.Engrg. Inform. Sci. 2002, 16, 471-484] and Belzunce et al. [Probab.Engrg. Inform. Sci. (2004), to appear] for stochastic ORDERings of gOS’s are contained in our new results.

In this paper, we discuss the new GENERALIZED fractional operator. This operator similarly conformable derivative satisfies properties such as the sum, product/quotient, and chain rule. Laplace transform is defined in this case, and some of its properties are stated. In the following, the Sturm-Liouville problems are investigated, and also eigenvalues and eigenfunctions are obtained.