On δ-record observations: asymptotic rates for the counting process and elements of maximum likelihood estimation

Gouet, Raúl; López, F. Javier; Sanz, Gerardo

doi:10.1007/s11749-011-0242-6

Cited by 15 publications

(28 citation statements)

References 17 publications

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“…On the other hand, to avoid missing any potential events of interest, one relaxes the record condition and includes observations in the record sequence that are smaller than the previous record by at most δ. Both situations have been invoked to motivate the study of δ-records defined by the condition [5,6] X n > max{X 1 , X 2 , ..., X n−1 } + δ.…”

Section: Record Processesmentioning

confidence: 99%

δ-exceedance records and random adaptive walks

Park¹,

Krug²

2016

J. Phys. A: Math. Theor.

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Abstract. We study a modified record process where the k'th record in a series of independent and identically distributed random variables is defined recursively through the condition Y k > Y k−1 − δ k−1 with a deterministic sequence δ k > 0 called the handicap. For constant δ k ≡ δ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of δ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park et al 2015 Phys. Rev. E 91 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences δ k , focusing in particular on the case when δ k matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when δ k is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where δ k corresponds to the change of the deterministic fitness component after k mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum.

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Section: Record Processesmentioning

confidence: 99%

δ-exceedance records and random adaptive walks

Park¹,

Krug²

2016

J. Phys. A: Math. Theor.

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“…The latter were independently defined in Reference [9], as natural and tractable generalizations of records, which are as easily collected as records. Their probabilistic properties have been studied in References [9][10][11][12][13][14].Concerning the statistical applications of δ-records, their likelihood function for a continuous distribution was first published in Reference [10], with results on maximum likelihood estimation (MLE) for the exponential and Weibull distributions. In Section 4.3 of the above cited paper, a variant of the sequential stress-testing scheme is proposed to collect δ-records, which we briefly describe here.…”

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confidence: 99%

“…We perform comparative analyses of estimators and predictors based on δ-records, in a variety of settings. In particular, we show that the performance of estimators and predictors is improved when using δ-record data with respect to only record data.Regarding real data, we analyze cumulative rainfall information recorded at the Castellote weather station in Spain; see Reference [10].The paper is organized as follows: Section 2 is devoted to preliminary definitions and notation. MLE and MLP of future records are developed in Section 3; we show existence of estimators and predictors and prove the strong consistency of the MLE of the scale parameter, if the shape parameter is known.…”

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confidence: 99%

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“…Regarding real data, we analyze cumulative rainfall information recorded at the Castellote weather station in Spain; see Reference [10].…”

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

et al. 2019

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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.

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