Inference for the unit-Gompertz model based on record values and inter-record times with an application
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Inference for the unit-Gompertz model based on record values and inter-record times with an application Devendra Kumar1 · Sanku Dey2 · Ehsan Ormoz3 · S. M. T. K. MirMostafaee4 Received: 31 May 2019 / Accepted: 12 November 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Mazucheli et al. (Statistica 79:25–43, 2019) introduced a new transformed model called the unit-Gompertz (UG) distribution which exhibits right-skewed (uni-modal) and reversed-J shaped density and its hazard rate function can be increasing and increasing-decreasingincreasing. They worked on the estimation of the model parameters based on complete data sets. In this paper, by using lower record values and inter-record times, we develop inference procedures for the estimation of the parameters and prediction of future record values for the UG distribution. First, we derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariances between two lower record values. Next, we obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Further, we obtain the Bayes estimators under the assumption that the model parameters follow a joint bivariate density function. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential) loss functions with the help of the Tierney–Kadane’s method and Metropolis–Hastings algorithm. Finally, we compute Bayesian point predictors for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction. Keywords Unit-Gompertz model · Maximum likelihood technique · Bayesian viewpoint · Lower record values Mathematics Subject Classification 62F10 · 62F15
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S. M. T. K. MirMostafaee [email protected]
1
Department of Statistics, Central University of Haryana, Mahendergarh, India
2
Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India
3
Department of Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
4
Department of Statistics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
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D. Kumar et al.
1 Introduction Mazucheli et al. [21] proposed a new distribution with unit interval i.e., in (0, 1), which is referred to as the unit-Gompertz (UG) distribution. The UG distribution is specified by the following probability density function (pdf) f (x; β, α) = β α x −(β+1) exp −α (x −β − 1) , (1) for x ∈ (0, 1), α > 0 and β > 0. We write X ∼ U G(β, α) if the pdf of X corresponds to (1). The corresponding cumulative distribution function (cdf) and hazard function are given by F(x; β, α) = exp −α (x −β − 1) , x ∈ (0, 1), α, β > 0, (2) and
α β x −(β+1) exp −α (x −β − 1) . x ∈ (0, 1), α, β > 0, h(x; β, α) = 1 − exp −α (x −β − 1)
respectively. There are many situations where the observations
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