Alternative Distributions Research Articles

Reliability Analysis for Unknown Age Class of Lifetime Distribution with Real Applications in Medical Science

Analyzing the reliability of the aging class of life distribution provides important information about how long a product lasts and sustainability measures that are essential for determining the environmental impact and formulating resource-saving plans. The study emphasizes the goodness-of-fit technique of the nonparametric test for the NBRUmgf class because age data are crucial for applications. Evaluations were conducted using the test’s asymptotic properties and Pitman efficiency methodology for some selected asymmetric probability models, and the outcomes were compared with those of alternative methods. We assessed the test’s power against widely used reliability distributions for some well-known alternative asymmetric distributions, including the Weibull, Gamma, and linear failure rate (LFR) distributions, and provided percentiles for both censored and uncensored data. This study shows the efficacy of the test in various sectors using real-world datasets and comprehensive tables of test statistics.

A Study of the Unexplored Aspect of Energy Poverty: Overheating Risk Analysis

Energy poverty, defined as the inability of households to afford adequate heating or cooling, is worsened by anthropogenic climate change, leading to increased energy demand and indoor comfort challenges. This study addresses a gap in summer fuel poverty research by focusing on overheating concerns and energy subsidies. This study defines the indoor temperature and overheating risk of low-income households compared with general dwellings based on field measurements and walkthrough surveys. The difference in indoor temperature and overheating risk according to the provision of energy subsidy is analyzed. We evaluate the impact of energy subsidies on indoor temperature and the adequacy of current subsidy programs. Findings show that households without energy vouchers experience indoor temperatures around 29°C, while voucher recipients see a 2°C decrease. During Tropical Night periods, households with energy vouchers maintain similar indoor temperatures to the overall period, while those without vouchers tend to experience a 2°C increase. Although energy vouchers significantly mitigate overheating risks by approximately 95% during the summer, they are not as effective in fully eliminating summer overheating risks compared to previous winter results. This highlights the persisting issue of traditional policies providing integrated electricity subsidies. Hence, addressing indoor overheating during hot summers requires improvements in alternative subsidy distributions or cooling strategies warrants change to the energy voucher payment method or the cooling method.

Goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis

The aim of the paper is to examine goodness-of-fit testing for normality where alternative distributions have undefined or constant skewness and excess kurtosis. The first step involves collecting a set of normality-oriented goodness-of-fit tests (GoFTs) recommended for use in the source literature. The second step is to create a family of distributions with undefined or constant skewness and excess kurtosis. For greater clarity, the alternative distributions have been divided into three groups. Group I consisted of symmetric alternatives with undefined excess kurtosis, group II of symmetric alternatives with constant non-zero excess kurtosis, and group III comprised asymmetric alternatives with constants non-zero skewness and non-zero excess kurtosis. An appropriate similarity measure (SM) of a given alternative distribution to the normal distribution was applied. The third step involved a Monte Carlo simulation. The obtained results indicate that the GoFT power for the analysed alternatives and for a given sample size remain relatively unchanged despite the significant decrease in SM. Several GoFTs proved incapable of making a distinction between normal distribution and the alternatives. This situation occurs even when SM equals 0.15. If SM equals 1, then the probability density functions (PDFs) are identical. The best tests for group I are the Gel-Miao-Gastwirth (SJ) and Robust Jarque- Bera (RJB) tests (positive excess kurtosis) as well as the Coin and Chen-Shapiro (CS) tests (negative excess kurtosis), for group II the SJ, 1st Hosking and RJB tests, while the Zhang-Wu (ZA) and CS tests are best for group III.

Introduction of a general class of entropy-based control charts: The Φ-chart

We introduce a new class of Shewhart control charts, namely the f-chart. This new class is based on the cumulative paired f-divergence that generalizes both the cumulative (residual) entropy and the differential entropy. The f-chart contains several subclasses; of which one has a special case, the G-chart, which uses Gini’s mean difference as a measure of dispersion. We investigate the performance of three of the subclasses of f-charts in a showcase scenario, comparing its average run length under the Gaussian and several alternative distributions relevant to process control. We find especially the new Leik control chart to outperform classical Shewhart charts, which are based on ranks, standard deviation, or Gini’s mean difference. The results imply that monitoring a production process using f-charts results in faster detection of out-of-control processes, which can be crucial for a variety of application areas.

A New Family of Continuous Distributions with Applications

This article introduces a novel set of optimizing probability distributions known as the Survival Power-G (SP-G) family, which employs a specific approach to introduce an additional parameter with the survival function of the original distributions. The utilization of this family enhances the modelling capabilities of diverse existing continuous distributions. By applying this approach to the single-parameter exponential distribution, a new two-parameter Survival Power-Exponential (SP-E) distribution is generated. The statistical characteristics of this fresh distribution and the maximum likelihood estimator are established, and Monte Carlo simulation is utilized to explore the efficiency of the maximum likelihood estimator of the two parameters under varying sample sizes. Subsequently, the new distribution is employed in the analysis of three distinct sets of real data. Through comparison with alternative distributions on these datasets, it is demonstrated that the new distribution outperforms the other distributions.

Ditching the norm: Using alternative distributions for biological data analysis.

Most classical statistical tests assume data are normally distributed. If this assumption is not met, researchers often turn to non-parametric methods. These methods have some drawbacks, and if no suitable non-parametric test exists, a normal distribution may be used inappropriately instead. A better option is to select a distribution appropriate for the data from dozens available in modern software packages. Selecting a distribution that represents the data generating process is a crucial but overlooked step in analysing data. This paper discusses several alternative distributions and the types of data that they are suitable for.

Achieving economy‐wide gains from residential energy efficiency improvements: The importance of timing and funding approach in driving the transition

AbstractResidential energy efficiency improvements are generally considered integral to achieving climate change targets. Alongside the primary benefits of reducing energy use and consumer bills, there is increasing policy interest in the potential for energy efficiency programmes to deliver economy‐wide gains, measured by gross domestic product, employment, household real incomes and spending power etc. Our previous research shows that such sustained gains are likely over time. Here, we consider how transitory outcomes are likely to be heavily influenced by the timing of actions and who pays, how and when. Insight in this regard is crucial for policy makers considering the mix and timing of measures to reach net zero outcomes that are economically as well as technically feasible. We consider alternative funding, distributions and timeframes for residential retrofitting costs and projects using a UK economy‐wide scenario simulation model. The key insight is that while government support for the provision of low‐cost finance options is strategically important in alleviating budget constraints and mitigating potential short‐term negative impacts on household spending, producer responses to the wind down of retrofitting spending can disrupt the adjustment of the economy. Here we identify pros and cons of different trajectories of action towards high‐level energy efficiency policy targets.

A Study on One-Parameter Entropy-Transformed Exponential Distribution and Its Application

This study presents the Entropy-Transformed Exponential Distribution (EnTrED), in an attempt to enhancing the flexibility and applicability of the traditional exponential distribution. The study explores the statistical properties of the EnTrED, including mode, quantile function, reliability, moments, and hazard function. The parameters of the distribution were estimated using maximum likelihood estimation, and the stability of these estimates was thoroughly evaluated through extensive Monte Carlo simulation. The simulation results demonstrated that the maximum likelihood estimates of the model parameters were well-behaved. Additionally, Empirical assessments against alternative distributions underscores the robustness of the EnTrED as a superior model for analyzing life data.

Tests for Large-Dimensional Shape Matrices via Tyler’s M Estimators

Tyler’s M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler’s M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension p grows proportionally to the sample size n for general populations. By using the eigenvalues of Tyler’s M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio p/n has a limit in (0, 1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler’s M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Goodness of fit test for a skewed generalized normal distribution

ABSTRACT The skewed generalized normal (SGN) distribution with four parameters is a versatile distribution that can effectively model data with skewness and heavy or light tails. In this paper, we conduct two classes of goodness of fit tests for the SGN distribution based on the empirical distribution function (edf) and the sample correlation coefficient. The first class involves transforming the sample into approximately mixed gamma observations, and then applying five classical parametric bootstrap edf-based goodness of fit tests. The second class is based on the inverse probability transformation and utilizes the sample correlation coefficient as the test statistic. We compare the finite sample performances of the proposed tests for different sample sizes and alternative distributions by extensive numerical studies. The simulation results demonstrate that the proposed tests provide a valid alternative to the standard tests using the original data, and the analysis of real data illustrates its application.

A fast and accurate kernel-based independence test with applications to high-dimensional and functional data

Testing the dependency between two random variables is an important inference problem in statistics since many statistical procedures rely on the assumption that the two samples are independent. To test whether two samples are independent, a so-called HSIC (Hilbert–Schmidt Independence Criterion)-based test has been proposed. Its null distribution is approximated either by permutation or a Gamma approximation. In this paper, a new HSIC-based test is proposed. Its asymptotic null and alternative distributions are established. It is shown that the proposed test is root-n consistent. A three-cumulant matched chi-squared-approximation is adopted to approximate the null distribution of the test statistic. By choosing a proper reproducing kernel, the proposed test can be applied to many different types of data including multivariate, high-dimensional, and functional data. Three simulation studies and two real data applications show that in terms of level accuracy, power, and computational cost, the proposed test outperforms several existing tests for multivariate, high-dimensional, and functional data.

Learning epistatic polygenic phenotypes with Boolean interactions.

Detecting epistatic drivers of human phenotypes is a considerable challenge. Traditional approaches use regression to sequentially test multiplicative interaction terms involving pairs of genetic variants. For higher-order interactions and genome-wide large-scale data, this strategy is computationally intractable. Moreover, multiplicative terms used in regression modeling may not capture the form of biological interactions. Building on the Predictability, Computability, Stability (PCS) framework, we introduce the epiTree pipeline to extract higher-order interactions from genomic data using tree-based models. The epiTree pipeline first selects a set of variants derived from tissue-specific estimates of gene expression. Next, it uses iterative random forests (iRF) to search training data for candidate Boolean interactions (pairwise and higher-order). We derive significance tests for interactions, based on a stabilized likelihood ratio test, by simulating Boolean tree-structured null (no epistasis) and alternative (epistasis) distributions on hold-out test data. Finally, our pipeline computes PCS epistasis p-values that probabilisticly quantify improvement in prediction accuracy via bootstrap sampling on the test set. We validate the epiTree pipeline in two case studies using data from the UK Biobank: predicting red hair and multiple sclerosis (MS). In the case of predicting red hair, epiTree recovers known epistatic interactions surrounding MC1R and novel interactions, representing non-linearities not captured by logistic regression models. In the case of predicting MS, a more complex phenotype than red hair, epiTree rankings prioritize novel interactions surrounding HLA-DRB1, a variant previously associated with MS in several populations. Taken together, these results highlight the potential for epiTree rankings to help reduce the design space for follow up experiments.

Comparison of probability distributions used for harnessing the wind energy potential: a case study from India

Modeling wind speed data is the prime requirement for harnessing the wind energy potential at a given site. While the Weibull distribution is the most commonly employed distribution in the literature and in practice, numerous scientific articles have proposed various alternative continuous probability distributions to model the wind speed at their convenient sites. Fitting the best distribution model to the data enables the practitioners to estimate the wind power density more accurately, which is required for wind power generation. In this paper we comprehensively review fourteen continuous probability distributions, and investigate their fitting capacities at seventeen locations of India covering the east and west offshore corner as well as the mainland, which represents a large variety of climatological scenarios. A first main finding is that wind speed varies a lot inside India and that one should treat each site individually for optimizing wind power generation. A second finding is that the wide acceptance of the Weibull distribution should at least be questioned, as it struggles to represent wind regimes with heterogeneous data sets exhibiting multimodality, high levels of skewness and/or kurtosis. Our study reveals that mixture distributions are very good alternative candidates that can model difficult shapes and yet do not require too many parameters.

Moments of undersampled distributions: Application to the size of epidemics

The total number of fatalities of an epidemic outbreak is a dramatic but extremely informative quantity. Knowledge of the statistics of this quantity allows the calculation of the mean total number of fatalities conditioned to the fact that the outbreak has surpassed a given number of fatalities, which is very relevant for risk assessment. However, the fact that the total number of fatalities seems to be characterized by a power-law tailed distribution with exponent (for the complementary cumulative distribution function) smaller than one poses an important theoretical difficulty, due to the non-existence of a mean value for such distributions. Cirillo and Taleb (2020) propose a transformation from a so-called dual variable, which displays a power-law tail, to the total number of fatalities, which becomes bounded by the total world population. Here, we (i) show that such a transformation is ad hoc and unphysical; (ii) propose alternative transformations and distributions (also ad hoc); (iii) argue that the right framework for this problem is statistical physics, through finite-size scaling; and (iv) demonstrate that the real problem is not the non-existence of the mean value for power-law tailed distributions but the fact that the tail of the different theoretical distributions (which is what distinguishes one model from the other) is far from being well sampled with the available number of empirical data. Our results are also valid for many other hazards displaying (apparent) power-law tails in their size.

Advancements in Laplace transform techniques: Performing non-parametric hypothesis testing on real-world data through statistical analysis

This research delves into the exploration of a statistical testing approach grounded in Laplace transform techniques specifically tailored for the New Better than Used Laplace transform order (NBUL) class of life distributions. The developed test exhibits versatility, accommodating both complete and censored data, and critical values are systematically calculated for its application. Beyond the methodological presentation, our study investigates the test’s statistical power and explores Pitman’s asymptotic efficiency concerning various alternative distributions. Comparative analyses with other tests within the same class contribute to a comprehensive understanding of the proposed approach. To demonstrate practical applicability, we apply this novel testing technique to authentic engineering and medical datasets. The outcomes of these applications serve as illustrative examples, showcasing the effectiveness and relevance of the proposed methodology in real-world scenarios.

Trend-Chasing Versus Minimalism: Selling Fewer, Better Products to Fashion-Sensitive Customers

Fashion sellers are sometimes critiqued for selling low durability products, resulting in waste. Blame is also directed at consumers, who purchase new fashions despite having accumulated a closet full of prior fashions. The “slow fashion” movement encourages sellers to produce higher durable products, thus supporting less frequent purchases by consumers. Using an infinite-time model and considering strategic consumer behavior, including their ability to accumulate a “closet” of varieties over time, we analyze the seller’s profit-maximizing price and product-durability decisions. We initially assume a static price but later analyze the potential profit gains from dynamic pricing. When analyzing a heterogeneous consumer market, we initially allow customers to vary (distributed uniformly) in their sensitivity to fashion. Subsequently, we explore alternative distributions for consumers’ fashion sensitivity and the correlation between their fashion sensitivities and product valuations. Using this framework, we show how the seller’s optimal price and durability decisions yield distinct shopping segments, which we refer to as the minimalist versus trend-chasers. We find that if the degree of fashion uncertainty is moderate, the seller’s optimal choice of product durability will support the emergence of both behaviors. As the variety uncertainty expands, if the seller’s costs are sufficiently low, it will support a throwaway culture via disposable products. Otherwise, given high costs, the seller optimally targets a slow fashion-type outcome, with consumers targeting reuse (with durability) rather than variety. Our findings shed light on consumers’ optimal purchasing behaviors in relation to both market parameters and the firm’s pricing and durability decisions.

A Cocktail of Bidirectional Tests for Power Symmetry and Repairable System Reliability

To a practitioner who chooses to test the robustness while keeping track of all the perspectives, angles, and elements of a study's conclusions in the data analysis, we offer an option between a cocktail of tests (a quintet) and a solo one-size-fits-all test as a needed antidote to power asymmetries evidenced by real data and simulations characterized by a variety of alternative distributions and two sampling schemes. The strategy is to be considered in equipment performance assessments of repairable systems and, by implication, other applications. Two bidirectional tests are first established to empower a basic pair of tests, documented in the literature as tests with asymmetrical performances. Unsurprisingly, we have seen either bidirectional device championing in one alternative setting but also being turned against the very setting altered with just one of the composed elements. Progressing by filtering out the bad and enhancing the good, we assemble a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> -value-induced dual bidirectional hybrid from the empowered pair to streamline the statistical data robustness analysis and restore power symmetries. Data elaborated by the maximum power deficits, made for facilitating the power comparisons and acknowledging the asymmetries, certify that the hybrid is robust with essential quality assurance in addition to a user-friendly version for the practitioners conducting the work.

New Weighted Burr XII Distribution: Statistical Properties, Applications, and Regression

In this study, a three-parameter modification of the Burr XII distribution has been developed through the integration of the weighted version of the alpha power transformation family of distributions. This newly introduced model, termed the modified alpha power-transformed Burr XII distribution, exhibits the unique ability to effectively model decreasing, right-skewed, or unimodal densities. The paper systematically elucidates various statistical properties of the proposed distribution. The estimation of parameters was obtained using maximum likelihood estimation. The estimator has been evaluated for consistency through simulation studies. To gauge the practical applicability of the proposed distribution, two distinct datasets have been employed. Comparative analyses involving six alternative distributions unequivocally demonstrate that the modified alpha power-transformed Burr XII distribution provides a better fit. Additionally, a noteworthy extension is introduced in the form of a location-scale regression model known as the log-modified alpha power-transformed Burr XII model. This model is subsequently applied to a dataset related to stock market liquidity. The findings underscore the enhanced fitting capabilities of the proposed model in comparison to existing distributions, providing valuable insights for applications in financial modelling and analysis.

Poverty, redistribution, and the middle class: redistribution via probability distributions vs. redistribution via the linear income tax system.

It has been known for a long time that (1) when graphs of income amount on income relative rank for two income distributions intersect twice, three "transfer groups" are generated, with the poorest and richest both gaining under the same alternative income distribution and the middle group losing; and (2) the linear income tax system satisfies three fundamental principles of tax justice, namely, that as pretax income increases, three quantities should also increase-posttax income, tax amount, and tax rate. This paper links those two ideas, suggesting that the linear income tax system may be the natural and most effective way to guard against poverty reduction policies which, while helping the poorest, as urged by Rawls, may harm the middle, contributing to the weakening of the middle class, thought at least since Aristotle to be the backbone of society. This paper illustrates the two approaches with one initial distribution and three alternative final distributions, contrasting their minimum, median, proportion below the mean, and inequality. It also shows how to guard the linear income tax system against violating the tax amount principle of tax fairness when there is an injection of resources (e.g., from deficit spending or oil revenues) and how to empirically estimate the parameters (e.g., the marginal tax rate) of the linear income system that the population will regard as fair.

Wealth Redistribution to Extend Longevity in the US

The US is unique among wealthy countries in its degree of wealth inequality and its poor health outcomes. Wealth is known to be positively associated with longevity, but little is known about whether wealth redistribution might extend longevity. To examine the association between wealth and longevity and estimate the changes in longevity that could occur with simulated wealth distributions that were perfectly equal, similar to that observed in Japan (among the most equitable of Organisation for Economic Co-operation and Development [OECD] countries), generated by minimum inheritance proposals, and produced by baby bonds proposals. This longitudinal cohort study analyzed the association between wealth and survival among participants in the Health and Retirement Study (1992-2018), a nationally representative panel study of middle-aged and older (≥50 years) community-dwelling, noninstitutionalized US adults. The data analysis was performed between November 15, 2022, and September 24, 2023. Household wealth on study entry, calculated as the sum of all assets minus the value of debts and classified into deciles. Weibull survival models were used to estimate the association between per-person wealth decile and survival, adjusting for age, sex, marital status, household size, and race and ethnicity. Changes in longevity that might occur under alternative wealth distributions were then estimated. The sample included 35 164 participants (mean [SE] age at study entry, 59.1 [0.1] years; 50.1% female and 49.9% male [weighted]). The hazard of death generally decreased with increasing wealth, wherein participants in the highest wealth decile had a hazard ratio of 0.59 for death (95% CI, 0.53-0.66) compared with those in the lowest decile, corresponding to a 13.5-year difference in survival. A simulated wealth distribution of perfect equality would increase populationwide median longevity by 2.2 years (95% CI, 2.2-2.3 years), fully closing the mortality gap between the US and the OECD average. A simulated minimum inheritance proposal would increase populationwide median longevity by 1.7 years; a simulated wealth distribution similar to Japan's would increase populationwide median longevity by 1.2 years; and a simulated baby bonds proposal would increase populationwide median longevity by 1.0 year. These findings suggest that wealth inequality in the US is associated with significant inequities in survival. Wealth redistribution policies may substantially reduce those inequities and increase population longevity.