Nth Record Value Research Articles

On a muted family of bivariate distributions

• Introduced a new family of bivariate distributions named as Muted Bivariate Distributions (MBD). • Identified the forms of distributions of two more concomitant statistics, each of which further helps to generate MBD. • Developed an inference procedure for MBD based on concomitant record ranked set sample (crrss). • Modeled a bivariate photometric data of galaxy luminosity in globular clusters by MBD with logistic marginals. • Constructed MBD using a data collected by concomitant record ranked set sampling from acacia trees. This paper introduces ‘muted bivariate distribution (MBD)’ as a new family of bivariate distributions possessing the property (P) that the concomitant of n th record value on one variable arising from any distribution of this family is distributed identically as the largest order statistic of a sample of size n arising from the marginal distribution of the same variable. Also if the property P is observed on any distribution, then we illustrate a process of impounding the probability density function (pdf) of the n th record value on one variable resulting due to P with the known form of the marginal pdf on the other variable to build up an appropriate MBD of the parent population. Some properties of the newly defined MBD are further derived. We identify the forms of the pdf of the concomitant of generalized upper record value and the pdf of the concomitant of the smallest order statistic of a sample of size n in order that each of the above pdf’s again can be used to generate MBD of the parent population, alternatively. We further observe MBD as a suitable model for describing an astronomical bivariate dataset relating to the galaxy luminosity measurements made under two conditions on globular clusters. For illustrative purposes, a real-life dataset on acacia trees obtained via concomitant record ranked set sampling methodology is considered as an application of this new family of bivariate distributions.

Kerridge measure of inaccuracy for record statistics

In this article, we introduce a measure of inaccuracy between distributions of the nth record value and parent random variable and discuss some properties of it. It is also shown that the proposed measure characterizes the distribution of parent random variable uniquely. Measure of inaccuracy for some specific distributions has also been studied.

Measures of inaccuracy in record values

ABSTRACTIn this paper , we consider a measure of inaccuracy between distributions of the nth record value and parent random variable. We also propose the measure of residual inaccuracy of record values and study characterization results of dynamic cumulative residual inaccuracy measure. We discuss some properties of the proposed measures.

Characterizations in a Random Record Model with a Non-identically Distributed Initial Record

We consider a sequence of random length M of independent absolutely continuous observations Xi, 1 = 2, have cdf F. Let N be the number of upper records and Rn, n >= 1, be the nth record value. We show that N is free of F if and only if G(x) = G0(F (x)) for some cdf G0 and that if E(|X2|) is finite so is E(|Rn|) for n >= 2 whenever N >= n or N = n. We prove that the distribution of N along with appropriately chosen subsequences of E(Rn) characterize F and G, and along with subsequences of E(Rn - Rn - 1) characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

On Record Values and Reliability Properties of Marshall�Olkin Extended Exponential Distribution

The Marshall–Olkin Extended Exponential distribution is introduced and reliability properties are studied. The p.d.f.’s of nth record value, joint p.d.f.’s, of mth and nth record values are derived to obtain the expression for mean, variance and covariance of reord values. The entropy of jth record value is derived.The stress strength analysis for the new model is carried out. We develop autoregressive processes and sample path properties are explored. The results are verified using simulations as well as graphical studies.The model is extended to higher orders also.

Characterizations in a random record model with a nonidentically distributed initial record

We consider a sequence, of random length M, of independent, continuous observations Xi, 1 ≤ i ≤ M, where M is geometric, X1 has cumulative distribution function (CDF) G, and Xi, i ≥ 2, have CDF F. Let N be the number of upper records and let Rn, n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G0(F(x)) for some CDF G0, and that if E(|X2|) is finite then so is E(|Rn|), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(Rn), characterize F and G and, along with subsequences of E(Rn - Rn-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

Characterizations in a random record model with a nonidentically distributed initial record

We consider a sequence, of random length M, of independent, continuous observations X i , 1 ≤ i ≤ M, where M is geometric, X 1 has cumulative distribution function (CDF) G, and X i , i ≥ 2, have CDF F. Let N be the number of upper records and let R n , n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G 0(F(x)) for some CDF G 0, and that if E(|X 2|) is finite then so is E(|R n |), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(R n ), characterize F and G and, along with subsequences of E(R n - R n-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

On the number and sum of near-record observations

Let X 1,X 2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables X i as near-nth-record observations if X i ∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

On the number and sum of near-record observations

Let X1,X2,… be a sequence of independent and identically distributed random variables with some continuous distribution function F. Let L(n) and X(n) denote the nth record time and the nth record value, respectively. We refer to the variables Xi as near-nth-record observations if Xi∈(X(n)-a,X(n)], with a>0, and L(n)<i<L(n+1). In this work we study asymptotic properties of the number of near-record observations. We also discuss sums of near-record observations.

Prediction intervals for the future record values from exponential distribution: comparative study

!n this paper we consider the predicf an problem of the future nth record value based an the first m (m < n) observed record values from one-parameter exponential distribution. We introduce four procedures for obtaining prediction intervals for the nth record value. The performance of the so obtained intervals is assessed through numerical and simulation studies. In these studies, we provide the means and standard errors of lower limits. upper limits and lengths of prediction intervals. Further, we check the validation of these intervals based on some point predictors.

Some partial ordering results on record values

It has been shown that if the nth record value Rn of a sequence of i.i.d. r.vs. has an increasing failure rate (IFR) distribution then so does Rn+1 On the other hand Rn-1 has decreasing failure rate (DFR) distribution if Rn is DFR. Some other partial ordering results for the record values have also been obtained.