Pareto Distribution Research Articles

A New Statistical Approach Based on the Access of Electricity Application with Some Modified Control Charts

This article introduces a new probability model based on reflected parameter called the reflected Pareto (RP) distribution. The key properties of the RP model are investigated. A simulation study of the RP model is conducted to evaluate the performances of its estimators. A real-life application is considered to examine the performance of proposed model. The different criteria are discussed numerically as well as graphically to show the flexibility of the RP model. The exponential weighted moving average control charts based on the maximum likelihood and modified maximum likelihood estimators for the shape parameter of the RP distribution are obtained. Detailed simulation results of proposed charts are performed to examine and analyze the performance of these charts with three in-control average run length values and two sample sizes. Finally, the application of the proposed control charts is shown by considering a real-life data set.

On classes of consistent tests for the Type I Pareto distribution based on a characterization involving order statistics

We propose new classes of goodness-of-fit tests for the Pareto Type I distribution. These tests are based on a characterization of the Pareto distribution involving order statistics. We derive the limiting null distribution of the tests and also show that the tests are consistent against fixed alternatives. The finite-sample performance of the newly proposed tests are evaluated and compared to some of the existing tests, where it is found that the new tests are competitive in terms of powers. The paper concludes with an application to a real world data set, namely the earnings of the 22 highest paid participants in the inaugural season of LIV golf.

Impact of Drought in Northeastern Algeria: Comparative Study of the SPI and SPEI Índices

Purpose: The study aimed to evaluate drought trends in Annaba, Algeria, considering global warming and extreme weather, using quantitative indices for improved drought comprehension and prediction. Theoretical Framework:The study assesses changes in drought frequency and severity in the Annaba region of Algeria due to global warming and extreme weather conditions. It employs quantitative indices to enhance drought understanding and prediction. Methodology: The study analyzes temperature and precipitation data (1981-2021) from an Annaba weather station to assess drought characteristics.Standardized indices (SPI, SPEI) and extreme value analysis (Generalized Pareto Distribution) are used to quantify drought intensity, duration, and temperature impacts. Findings: Our analyses reveal a critical divergence in drought assessment tools. While stable monthly temperatures suggest a constant influence on evapotranspiration, the SPEI index (including temperature) indicates increasing drought compared to the precipitation-only SPI. This highlights the importance of considering temperature alongside precipitation for accurate drought assessment. Research Practical and Social Implication: This study emphasizes the significance of considering the interplay among temperature, precipitation, and evapotranspiration for better understanding and predicting drought changes in the Annaba region, Algeria. These insights are crucial for sustainable water resource management and climate change adaptation in the region. Originality/Value:This study distinguishes itself by its comprehensive analysis of long-term climate data to examine drought trends in a specific region (Annaba, Algeria) severely impacted by climate change. By comparing the effectiveness of two drought indices (SPEI and SPI), it offers valuable insights into how to improve drought assessments by considering the role of temperature in evapotranspiration. This research enriches the existing literature on drought management and climate change adaptation, providing a valuable case study for similar arid and semi-arid regions worldwide.

Mitigating against the between‐ensemble‐member precipitation bias in a lagged sub‐seasonal ensemble

AbstractThe Met Office GloSea5‐GC2 sub‐seasonal‐to‐seasonal 40‐member lagged ensemble consists of members who are up to 10 days different in age such that the between‐ensemble‐member bias is not internally consistent. Reforecasts tend to be used to convert these ensemble forecasts into anomalies from a normal state. These anomalies are however not that useful for applications where individual ensemble members are needed to drive downstream applications in the hazard and impact space. Here we explore whether there is a way of correcting for the within‐ensemble bias without using reforecasts. An investigation into the individual daily precipitation distributions from the JJAS 2019 Indian monsoon season, stratified by forecast horizon, highlights how the distribution changes, and shows that the model distribution is markedly different to the observed. Initial results suggest that it could be better to use recent model forecast distribution(s) as the reference for adjusting the model rainfall accumulations as a function of lead day horizon, that is, not attempting to correct the members to a vastly different (observed) distribution shape, but a more subtle shift towards the model's best guess of reality, rather than reality itself, to remove the between‐ensemble‐member bias. A combination of Exponential and Generalized Pareto distributions are used for parametric quantile mapping to remove this internal ensemble bias using computationally efficient pre‐computed lookup tables. Within‐ and out‐of‐sample results for the 2019 and 2020 monsoon seasons show that the method is effective in tightening precipitation gradients, with improvements in spread, accuracy and skill, especially for low accumulations.

ARCSINE RAYLIEGH PARETO DISTRIBUTION: PROPERTIES AND APPLICATION TO CARBON FIBERS DATA SETS

In this paper, we introduce a new modified distribution called arcsine Rayliegh Pareto (ASRP) Distribution. We derived its mathematical and statistical properties, including survival function, hazard function, entropy, moment, moment generating function, and order statistics. We also used maximum likelihood estimation for estimating the parameters of the distribution. The plots of the cdf, pdf, hazard rate function, and survival function were illustrated with right skewed probability density function, cumulative distribution function with monotone increasing function, which converge at one. The value of some goodness of fit measure (i.e. AIC, AICc, and BIC) were computed, as well as the KS, A, and W statistic. Finally, we suggested that the new modified model outperform better than the other standard distribution using Carbon fibers data sets.

Graphical Diagnostics for Threshold Selection in Fitting the Generalized Pareto Distribution

Graphical Diagnostics for Threshold Selection in Fitting the Generalized Pareto Distribution

Advancements in reliability estimation for the exponentiated Pareto distribution: a comparison of classical and Bayesian methods with lower record values

Advancements in reliability estimation for the exponentiated Pareto distribution: a comparison of classical and Bayesian methods with lower record values

Addressing the financial impact of natural disasters in the era of climate change

The objective of our study is to predict the financial losses that may result from natural disasters, along with their level of volatility, over a period of 1 to 15 years. Volatility can lead to significant fluctuations in Profit and Loss (P&L) for companies that are affected by unexpected events. To achieve this goal, we created a novel two-factor square-root model that allows us to establish a correlation between the frequency of occurrences and volatility, using correlated Brownian motions. Moreover, we utilized a Generalized Pareto Distribution (GPD) to estimate the maximum potential loss in terms of Value at Risk (VaR) for each specific type of natural disaster. To ensure the reliability of our predictions, we compared our results to those of four reference models and conducted a backtesting analysis. This approach is particularly suitable for insurance companies seeking to maintain stable reserves, but it can also be adapted for any other type of business that is vulnerable to extreme events and aims to safeguard a consistent cash flow for their stakeholders.

On α-Pseudo Spiralike Functions Associated with Exponential Pareto Distribution (EPD) and Libera Integral Operator

The present study aims at investigating some characterizations of a new subclass Gα(μ,τ) and obtaining the bounds on the first two Taylor–Maclaurin coefficients for functions belonging to the newly introduced subclass. In order to achieve this, a compound function Lx,nσ(z) is derived from the convolution of the analytic function f(z) and a modified exponential Pareto distribution G(x) in conjunction with the famous Libera integral operator L(ζ). With the aid of the derived function, the aforementioned subclass Gα(μ,τ) is introduced, while some properties of functions belonging to this subclass are considered in the open unit disk.

Deformation of power law in the double Pareto distribution using uniformly distributed observation time

The double Pareto distribution is a heavy-tailed distribution with a power-law tail, that is generated via geometric Brownian motion with an exponentially distributed observation time. In this study, we examine a modified model wherein the exponential distribution of the observation time is replaced with a continuous uniform distribution. The probability density, complementary cumulative distribution, and moments of this model are exactly calculated. Furthermore, the validity of the analytical calculations is discussed in comparison with numerical simulations of stochastic processes.

Transformer encoder based self-supervised learning for HVAC fault detection with unlabeled data

Data driven methods are the most studied fault detection and diagnostics (FDD) type in buildings HVAC systems. However, most studies rely on labeled data for specific faults which are hard to find and collect for real systems. While the fault-free data is easier to collect, it is still time consuming to label for large systems operation. Moreover, most of the studies rely on the usage of supervised learning algorithms which do not generalize well beyond the training data making unseen faults hard to detect. In this paper, we define a methodology to use a self-supervised learning method for HVAC systems' FDD using a Transformer encoder, moreover, we tested it on a real case study. By strategically masking portions of the multivariate time-series data using Markov chain approach with two states. The model is trained by predicting these concealed segments. This approach, independent of labeled data, offers a scalable solution for practical HVAC applications. Anomalies are labeled using the Peak Over Threshold (POT) method, which dynamically determines thresholds by fitting reconstruction errors to a generalized Pareto distribution. Subsequent fault diagnostics emphasize features with pronounced reconstruction errors, pinpointing potential HVAC malfunctions. This methodology reduces dependence on labeled datasets and augments the model's generalization, facilitating detection of unobserved faults. This approach was applied to data from a real building. As a results multiple faults were detected mainly due to the malfunctioning of the monitoring system. The model demonstrates the ability to detect both sequential and individual faults. The period from October 19th to December 23rd was detected as a fault period due to the change in the trend of the data because of the monitoring system.

A study on wave climate variability along the nearshore regions of Bohai Sea based on long term observation data

The Bohai Sea off the coast of northern China plays important economic and ecological roles. No studies have been conducted on wave characteristics in the Bohai Sea nearshore area using ocean station data in the past. Observational data (1998–2022) from eight ocean-stations in the Bohai Sea were used in the present study. Overall, wave height (Hs) was small; winter seasons had higher Hs than summer seasons, and the maximum Hs occurred in the Bohai Strait. Joint distribution analysis indicated that waves with Hs ranging from 0.2 to 0.4 m, and mean wave periods (Tm) ranging from 3 to 3.5 s, were most common. Hs in the Bohai Sea nearshore area showed a weak response to climate variability and was moderately correlated with the Ocean Niño Index (ONI) and the North Pacific Index (NPI) in winter, and with the Arctic Oscillation (AO) in summer. Under extreme conditions, Hs was best described by a Generalized Pareto distribution, Tm was best described by a GEV distribution, and wave directions mainly followed a t location-scale distribution. The maximum wave height (Hmax)/Hs was 1.51, and the ratio varied with location. Machine learning methods were used to calculate Hmax more precisely, and they yielded better results than statistical methods.

Connecting flow duration curve and precipitation duration curve based on the relationship deduced from machine learning in the watersheds of northern China

Obtaining reliable hydrological data for ungauged watersheds has always been a significant challenge in the water resources management. The flow duration curve (FDC) and precipitation duration curve (PDC) are classical methods to address this challenge. However, traditional FDC and PDC methods mostly rely on pre-assumed probability distribution function form, which may result in sub-optimal model parameters and consequently sub-optimal simulation results of hydrological data, and the common function forms are not applicable to deal with the data including zero-values, which often occurs in the discharge data of arid region and daily rainfall data. To address the issues in FDC and PDC , this study firstly determines the most suitable function form to describe FDC and PDC from various distribution functions, i.e., Log Normal, Generalized Pareto Distribution and H2018 function, from 1960 to 2010 in 7 watersheds in the north of China. The multiple methods including Random Forest, polynomial, Multilayer Perceptron and K-Nearest Neighbor are employed to simulate the parameters of the functions, aiming to deduce the optimal parameters of FDC functions. The results show that, H2018 function form has the best performance in simulating the PDC and FDC and in general, parameter a and b of H2018 function indicate an opposite trend, and parameter k declines in mostly condition at the annual scale from 1960 to 2010. The results of different methods indicate that the Random Forest method is the most efficient and accurate approach to deduce FDC parameters from measured precipitation and climate factors, compared with polynomial, Multilayer Perceptron and K-Nearest Neighbor. Based on the relationship between PDC parameters and FDC parameters, the FDC of ungauged basins can be deduced from the observed precipitation data. The research findings provide a new method to estimate FDC for the water resources management in the ungauged basins.

Bayesian Non-Parametric Inference for Multivariate Peaks-over-Threshold Models.

We consider a constructive definition of the multivariate Pareto that factorizes the random vector into a radial component and an independent angular component. The former follows a univariate Pareto distribution, and the latter is defined on the surface of the positive orthant of the infinity norm unit hypercube. We propose a method for inferring the distribution of the angular component by identifying its support as the limit of the positive orthant of the unit p-norm spheres and introduce a projected gamma family of distributions defined through the normalization of a vector of independent random gammas to the space. This serves to construct a flexible family of distributions obtained as a Dirichlet process mixture of projected gammas. For model assessment, we discuss scoring methods appropriate to distributions on the unit hypercube. In particular, working with the energy score criterion, we develop a kernel metric that produces a proper scoring rule and presents a simulation study to compare different modeling choices using the proposed metric. Using our approach, we describe the dependence structure of extreme values in the integrated vapor transport (IVT), data describing the flow of atmospheric moisture along the coast of California. We find clear but heterogeneous geographical dependence.

Pareto and probability distributions

Although familiar with the developments in probability theory of his time, Vilfredo Pareto made little use of this tool in his writings, preferring theoretical constructions based on experimentation and observation. This article attempts to reconstruct Pareto’s overall approach to probability by examining his references to the distribution of income, an economic fact that lends itself to probabilistic investigation. The result of this research shows how Pareto alludes to the application of probability to income and social groups, but leaves the task to his followers.JEL classification B31, B4, C1.

An efficient MEWMA chart for Gumbel's bivariate Pareto distribution

Control charts play a vital role in process monitoring to ensure the product's desired standards. Due to rapid improvements in data collection methods, multivariate charts are preferred over univariate charts. This paper proposes a bivariate exponentially weighted moving average chart for the simultaneous monitoring of the mean vector of Gumbel's bivariate Pareto type II (also known as the Lomax distribution) time-between-events data. The performance of the proposed chart is assessed through average run length, median run length, and the standard deviation of the run length criteria. To show the implementation of the chart in the real world, illustrative examples are also presented.

Title: Modeling Normal and Imbalanced Neural Avalanches: A Computational Approach to Understanding Criticality in the Brain and Its Potential Role in Schizophrenia

Abstract: Neural avalanches, characterized by bursts of activity followed by periods of quiescence, have been observed in the brain and are thought to reflect the critical dynamics necessary for optimal information processing. Deviations from normal avalanche behavior have been hypothesized to underlie various neurological disorders, particularly schizophrenia. Schizophrenia is a complex psychiatric disorder associated with altered perception, cognition, and behavior, and recent theories suggest that disruptions in the brain's critical dynamics may contribute to its pathophysiology. In this study, we present a computational model to investigate the properties of normal and imbalanced neural avalanches, with a focus on understanding the potential role of criticality in schizophrenia. We generate avalanche sizes using the Pareto distribution with a power law exponent of -3/2, which is consistent with experimental observations. The model incorporates increasing avalanche sizes over time to simulate the growth of neural activity. We introduce imbalance by adding lateness or earliness to the avalanche sizes, mimicking the potential disruptions in critical dynamics that may occur in schizophrenia. The mean and standard deviation of avalanche sizes are calculated to characterize the normal and imbalanced behavior. The results are visualized using line plots, with shaded areas representing the standard deviation range. Our model provides a framework for understanding the differences between normal and imbalanced neural avalanches, offering insights into the potential mechanisms underlying the altered critical dynamics in schizophrenia. By exploring the relationship between neural avalanches and schizophrenia, this study contributes to the ongoing efforts to elucidate the neurobiological basis of this disorder and may inform future research on potential diagnostic markers and therapeutic interventions targeting the brain's critical dynamics.

Evaluation of survival weighted Pareto distribution: Analytical properties and applications to industrial and aeronautics data

In statistical modeling, lifetime datasets play an important and significant role. Weighted distributions can be useful to attain the essential purposes relating to the developed distributions’ flexibility and to improve further compliances in data forming; they provide the extension of distributions. The current study suggests a new best-fit probability distribution model based on Pareto distribution. The new model is called Survival Weighted Pareto (SWP) distribution model. Furthermore, the statistical expressions for different properties, which include moments, inverse moments, quantile function, and order statistics, are reported. Moreover, to check the performance and the behavior of different estimators using different sample sizes, a simulation study, which is based on SWP distribution, is performed. According to the simulation results, it is established that the proposed model is a most useful fitted probability model. In addition, an application of the SWP model for three datasets is provided to check the goodness of fit measures of the model.

Return Levels on Stationary Extreme Rainfall Series: A Comparative Study of Generalized Extreme Value and Generalized Pareto Distributions

Return Levels on Stationary Extreme Rainfall Series: A Comparative Study of Generalized Extreme Value and Generalized Pareto Distributions

Measuring and Analyzing Scholarly Literature Published on Diabetes Mellitus Type 1 with Special Reference to Bradford Law of Scattering and Leimkuhler Model: A Scientometric Study

The present paper deals with the applicability of Bradford’s law of Scattering to the literature on Diabetes Mellitus type 1 disease. The PubMed database has been selected to collect data for the present study. A total of 489 journals published 1542 articles during the year 2012 to 2021 on Diabetes Mellitus type 1 disease. It was found that Current Diabetes Reports was the top core journal that published the highest number of articles (100) along with citation 4550. Bradford's law was tested for applicability using a ranked list of journals and publishing output. The Diabetes Mellitus type 1 literature's journal distribution pattern corresponded to Bradford's distribution pattern. The applicability of Leimkuhler’s model was also tested and it is also fitted to the selected disease dataset of Diabetes Mellitus Type-1