Simultaneous Joint Lower and Upper record values Probability Laws for Absolutely Continuous or Discrete Data

This paper investigates the probability density function (pdf) of the \((2n-1)\)-vector \((n\geq 1)\) of both lower and upper record values for a sequence of independent random variables with common \textit{pdf} \(f\) defined on the same probability space, provided that the lower and upper record times are finite up to \(n\). A lot is known about the lower or the upper record values when they are studied separately. When put together, the challenges are far complicated. The rare results in the literature still present some flaws. This paper begins a new and complete investigation with a few number of records: (n=2\) and \(n=3\). Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.

Bayesian Estimations under the Weighted LINEX Loss Function Based on Upper Record Values

The essential objective of this research is to develop a linear exponential (LINEX) loss function to estimate the parameters and reliability function of the Weibull distribution (WD) based on upper record values when both shape and scale parameters are unknown. We perform this by merging a weight into LINEX to produce a new loss function called the weighted linear exponential (WLINEX) loss function. Then, we utilized WLINEX to derive the parameters and reliability function of the WD. Next, we compared the performance of the proposed method (WLINEX) in this work with Bayesian estimation using the LINEX loss function, Bayesian estimation using the squared-error (SEL) loss function, and maximum likelihood estimation (MLE). The evaluation depended on the difference between the estimated parameters and the parameters of completed data. The results revealed that the proposed method is the best for estimating parameters and has good performance for estimating reliability.

Characterization of a New Family of Distribution Through Upper Record Values

In this paper, a new class of distribution has been characterized through the condi- tional expectations, conditioned on a non-adjacent upper record value. Also an equivalence between the unconditional and conditional expectation is used to characterize the new class of distribution.

Maximum Likelihood Estimations Based on Upper Record Values for Probability Density Function and Cumulative Distribution Function in Exponential Family and Investigating Some of Their Properties

A useful subfamily of the exponential family is considered. The ML estimation based on upper record values are calculated for the parameter, Cumulative Density Function, and Probability Density Function of the subfamily. The relationship between MLE based on record values and a random sample are discussed, along with some properties of these estimators, and its utility is shown for large samples.

Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.