Moment Generating Function of Current Records Based on Generalized Exponential Distribution - recurrence relations and moments

Sep 14, 2026

Uploaded by Franzy

A study develops formulas for the moment generating function of upper and lower current records drawn from generalized exponential distributed data, and derives the joint moment generating function between them. Using these results, the work obtains expressions for the moments of each record and the product moments of both upper and lower current records. The paper further establishes recurrence relations for many of the derived formulas and provides an integral form for moments of the record range, supported by a numerical simulation example and a concluding summary.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages15

Reading Time38 min

AccessFree to download

Introduction

Record definitions and recurrence setting

Preliminaries Results

Supporting results for the main derivations

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 97-111 ISSN 1307-5543 – [ejpam.com](ejpam.com) Published by New York Business Global Moment Generating Function of Current Records Based on Generalized Exponential Distribution with Some Recurrence Relations Ramy Abdelhamid Aldallal Department of Accounting, College of Business Administration in Hawtat bani Tamim, Prince Sattam bin Abdulaziz University, Saudi Arabia Abstract. This paper tries to find some formulas for calculating the moment generating function for upper and lower current records picked from generalized exponential distributed data and the joint moment generating function between them. After that, some formulas are derived from the previous ones to find the moments of each and the product moments of both upper and lower current records. Then, various recurrence relations are established for most of the mentioned formulas. After that, an integral form of the moments of record range is founded followed by a numerical example with simulated data to clarify the effectiveness of the formulas found in the study and how they can make the calculation process easier and faster. Finally, a conclusion part is added, to sum up what has been done and the results. 2020 Mathematics Subject Classifications: 62G30, 65Q30, 62E99, 60E10 Key Words and Phrases: Record range, lower and upper current records, moment generating function and moments, recurrence relations, generalized exponential distribution 1. Introduction Imagine that we have a sequence {Xj} that consists of identical and independent random variables such that each of which is distributed with a probability density function(pdf) f (x) and a cumulative distribution function(cdf) F (x) that is absolutely continuous. Xj as an observation is considered to be an upper record if Xj > Xi for every i < j. A similar definition has been established for the lower record values. In some situations, we tend to pick the smallest and largest X values detected as new lower or upper record values of either kind take place, and in such a situation, we name it current records. We will symbolize Ucn as the nth upper current records and Lcn as the nth lower current records of the sequence Xn , when any kind o the nth records appears. So, Ucn+1 = Ucn if Lcn+1 < Lcn and Lcn+1 = Lcn if Ucn+1 > Ucn , for all n = 1 , 2 , ... where by definition, Lc0 = Uc0 = X1 . We can define the record range as Rcn = Ucn − Lcn . And of DOI: [https://doi.org/10.29020/nybg.ejpam.v16i1.4641](https://doi.org/10.29020/nybg.ejpam.v16i1.4641) Email address: [dr.raldallal@gmail.com](dr.raldallal@gmail.com) (R.A.Aldallal) [https://www.ejpam.com](https://www.ejpam.com) 97 © 2023 EJPAM All rights reserved. R.A.Aldallal / Eur. J. Pure Appl. Math, 16 (1) (2023), 97-111 98 course, a new range is recorded at the moment a new upper or lower record has occurred. For example, let’s consider the following sequence of observations: 3 , 2 , 2.5 , 2.2 , 1 , 3.7 , 2.6 , 1.5 , 2.7 , 4 , 0.5 , 2.5 , 4.7.... Then we can determine the upper and lower current records along with the current record range as follows: Uc0 = 3, Uc1 = 3, Uc2 = 3, Uc3 = 3 .7, Uc4 = 4, Uc5 = 4, Uc6 = 4 .7 Lc0 = 3, Lc1 = 2, Lc2 = 1, Lc3 = 1, Lc4 = 1, Lc5 = 0 .5, Lc6 = 0 .5 Rc0 = 0, Rc1 = 1, Rc2 = 2, Rc3 = 2 .7, Rc4 = 3, Rc5 = 3 .5, Rc6 = 4 .2 There are cases in which it is interesting for the current records to be considered in real life. Such as whether data where lower and upper records are being taken together. Also, in the case of outlier detection and when choosing a fitted model in which the record range plays an important role in it (see, Basak [10]) . Again, it is very substantial to be used when we want to see if the production process is good enough to fall within the scope of the production’s specifications or not. Meaning, if the record range is greater than a specific value, then the products will not fit the specifications and vice versa.

FAQs

What are upper and lower current records in this paper?

Upper current records are new observations that exceed all previous ones, while lower current records are new observations smaller than all previous ones. Current records refer to the smallest and largest values selected as new lower or upper records during the sequence.

Which moment functions does the paper derive?

The paper derives moment generating function formulas for the nth upper and lower current records under the generalized exponential distribution, and also derives the joint moment generating function for the pair of records.

How are recurrence relations used in the study?

Recurrence relations are established to compute many of the mentioned moment generating function and moment expressions more efficiently. The paper also provides recurrence structure for integrated forms related to moments of the record range.

Sep 14, 2026

Uploaded by Franzy