Statistical Inference for the Weibull Distribution Based on δ-Record Data

Abstract

We consider the maximum likelihood and Bayesian estimation of parameters and prediction of future records of the Weibull distribution from δ -record data, which consists of records and near-records. We discuss existence, consistency and numerical computation of estimators and predictors. The performance of the proposed methodology is assessed by Montecarlo simulations and the analysis of monthly rainfall series. Our conclusion is that inferences for the Weibull model, based on δ -record data, clearly improve inferences based solely on records. This methodology can be recommended, more so as near-records can be collected along with records, keeping essentially the same experimental design.

Exact and asymptotic properties of δ-records in the linear drift model

“…Our results open the door to the use of δ-records for statistical applications in the LDM. It has been shown that δ-records perform better than records in statistical inference, using trend-free data [44][45][46], so we expect that their use in the LDM is also advantageous.…”

“…setting, the distribution [41,42], process structure [43] and asymptotic properties [44] of δ-records have been studied. In the case δ < 0, where δ-records are more numerous than records, their use in statistical inference has been recently proposed and positively assessed; see [44][45][46]. In these articles, it is shown how the information of δ-records can be incorporated successfully into the likelihood of the sample, which is used for computing maximum likelihood and Bayes estimators and predictions of future records.…”

“…In the iid setting, the distribution [38,39], process structure [40] and asymptotic properties [41] of δ-records have been studied. In the case δ < 0, where δ-records are more numerous than records, their use in statistical inference has been recently proposed and positively assessed; see [41][42][43].…”

Near-Record Values in Discrete Random Sequences

“…Near-records were introduced in [5] for applications in finance, and their properties were analysed in [6,7]. In addition, the related concept of the δ-record was introduced in [8] and later studied in [9][10][11][12]. These objects have practical applications in the case of negative δ, since δ-records are more numerous than records.…”

“…So, by considering samples of δ-records for statistical inference, we address the problem of the scarcity of records while keeping the extremal nature of the data. Indeed, it has been shown in [9] and in references therein that inferences based on δ-records outperform those based on records only.…”