On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution

Ekawati, Dian; Warsono, Warsono; Kurniasari, Dian

doi:10.12962/j20882033.v25i3.574

Cited by 7 publications

(3 citation statements)

References 4 publications

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“…The "location" is a scalar parameter that determines the displacement of the probability distribution, meaning that with its increment, the probability density shifts to the right, maintaining its shape (Ekawati, Warsono, & Kurniasari, 2015). The scale parameter is directly proportional to the spread of the probability distribution (Bhattacharya & Bhattacharjee, 2010).…”

Section: Data Acquisition and Processingmentioning

confidence: 99%

Spatio-temporal assessment of extreme precipitation events in southeastern Brazil

Simões

Pedra

Alves

et al. 2023

Preprint

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Climate change is a phenomenon directly attributed to human activity due to greenhouse gas emissions and changes in land use and land cover. In this way, climate change has and will have direct and indirect impacts on the components of the climate systems, including the hydrological cycle, through an increase in the frequency and intensity of extreme events, such as heavy rains or droughts, which impact natural systems, groups, and human systems, as well as on economic activity, such as the energy, agricultural and water sectors that are directly dependent on the variables of the hydrological cycle. The southeast region of Brazil (São Paulo, Minas Gerais, Rio de Janeiro, and Espírito Santo) produces 60\% of the national wealth and much of the country's electricity, in addition to housing 80 million people. Extreme weather events have hit this region in recent decades, causing an increasing number of natural disasters. Good examples are the 2014/15 water crisis and the occurrence of heavy rains, particularly in the summers. Given the importance of this theme, the present study aims to analyze the spatial and temporal variability of precipitation over southeast Brazil through the Generalized Extreme Values (GEV) distribution. To this end, seasonal variability and changes in the patterns of precipitation extremes in the period 1981-2020 were analyzed, based on data obtained from the CPC (Climate Predict Center). This research revealed that the frequency and intensity of extreme rainfall events have been changing, notably decreasing during winter.

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Section: Data Acquisition and Processingmentioning

confidence: 99%

Spatio-temporal assessment of extreme precipitation events in southeastern Brazil

Simões

Pedra

Alves

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Tested Formula F-Measure (%) Dataset Frechet [46][47][48], Gumbel [49,50], Laplace [51,52], Logistic [53,54], Log-Logistic [55][56][57], Log-Normal [58][59][60], Normal [61,62], Pareto [63,64], and Weibull [65,66]. Obviously, there are a lot more distributions that we could have used, but we considered these to be representative enough for our purposes.…”

Section: General Formulamentioning

confidence: 99%

On Generalizing Sarle’s Bimodality Coefficient as a Path towards a Newly Composite Bimodality Coefficient

2022

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Determining whether a distribution is bimodal is of great interest for many applications. Several tests have been developed, but the only ones that can be run extremely fast, in constant time on any variable-size signal window, are based on Sarle’s bimodality coefficient. We propose in this paper a generalization of this coefficient, to prove its validity, and show how each coefficient can be computed in a fast manner, in constant time, for random regions pertaining to a large dataset. We present some of the caveats of these coefficients and potential ways to circumvent them. We also propose a composite bimodality coefficient obtained as a product of the weighted generalized coefficients. We determine the potential best set of weights to associate with our composite coefficient when using up to three generalized coefficients. Finally, we prove that the composite coefficient outperforms any individual generalized coefficient.

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“…M is linear. Note that if T is a random variable with a cumulative distribution function F , then the linear mapping M can be chosen as the moment generating function

M_{T} false(t false)

or the real moments

E false(T^{t} false) = E false(\exp false(t \log (T) false)

M (F_{W} (.; α_{1}, β_{1}) = E (T^{t}) = {false(α_{1} false)}^{t} normalΓ (() \frac{t}{β_{1}} + 1) t \in] - β_{1}, + \infty [)

M (F_{L L} (.; α_{2}, β_{2}) = E (T^{t}) = {false(α_{2} false)}^{t} boldB (() \frac{β_{2} + t}{β_{2}}, \frac{β_{2} - t}{β_{2}}) t \in] - β_{2}, β_{2} [),

where

B (., .)

denote the beta function, see Ekawati et al 22 □…”

Section: The Weibull‐log‐logistic Mixture Distributionsmentioning

confidence: 99%

The Weibull log‐logistic mixture distributions: Model, theory and application to lifetime data

Azzaz

Boudrissa

2020

Quality & Reliability Eng

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Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull‐log‐logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull‐log‐logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log‐logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.

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On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution

Cited by 7 publications

References 4 publications

Spatio-temporal assessment of extreme precipitation events in southeastern Brazil

Spatio-temporal assessment of extreme precipitation events in southeastern Brazil

On Generalizing Sarle’s Bimodality Coefficient as a Path towards a Newly Composite Bimodality Coefficient

The Weibull log‐logistic mixture distributions: Model, theory and application to lifetime data

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