Quantile Function Research Articles

Geographically weighted quantile regression for count Data

In recent years, a methodological framework known as geographically weighted quantile regression (GWQR) has emerged for spatial data analysis. This framework offers the abilities to simultaneously explore spatial heterogeneity or nonstationarity in regression relationships and to estimate various conditional quantile functions. However, the current configuration of GWQR is limited to the analysis of continuous dependent variables. Discrete count data are observed in many disciplines. Whenever modeling such outcomes is necessary, the conventional GWQR approach is inadequate and fails to provide comprehensive insights into count data. To address this issue, this study aims to extend the GWQR framework originally designed for continuous dependent variables to accommodate count outcomes. We introduce an approach called geographically weighted count quantile regression (GWCQR), wherein the model specification is based on the smoothing of count responses through a jittering procedure. A semiparametric counterpart that allows for the inclusion of both spatially varying and invariant coefficients is also discussed. Finally, the proposed techniques are applied to a dataset of dengue fever in Taiwan as an empirical illustration.

Symmetric and Asymmetric Expansion of the Weibull Distribution: Features and Applications to Complete, Upper Record, and Type-II Right-Censored Data

This paper introduces a new continuous lifetime model called the Odd Flexible Weibull-Weibull (OFW-W) distribution, which features three parameters. The new model is capable of modeling both symmetric and asymmetric datasets, regardless of whether they are positively or negatively skewed. Its hazard rate functions can exhibit various behaviors, including increasing, decreasing, unimodal, or bathtub-shaped. The key characteristics of the OFW-W model are discussed, including the quantile function, median, reliability and hazard rate functions, kurtosis and skewness, mean waiting (residual) lifetimes, moments, and entropies. The unknown parameters of the model are estimated using eight different techniques. A comprehensive simulation study evaluates the performance of these estimators based on bias, mean squared error (MSE), and mean relative error (MRE). The practical usefulness of the OFW-W distribution is demonstrated through four real datasets from the fields of engineering and medicine, including complete data, upper record data, and type-II right-censored data. Comparisons with five other lifetime distributions reveal that the OFW-W model exhibits superior flexibility and capability in fitting various data types, highlighting its advantages and improvements. In conclusion, we anticipate that the OFW-W model will prove valuable in various applications, including human health, environmental studies, reliability theory, actuarial science, and medical sciences, among others.

Representations of characteristic function via survival function and generalized inverse function

The moment generating function and moments of a real-valued random variable can be expressed in terms of the survival function or inverse survival function. In this article, we give the representation of characteristic function via the generalized inverse survival function or generalized inverse distribution function for arbitrary distributions. We extend existing studies, both univariate and multivariate, by giving formulae of the expectation of a more general function of a random variable. These results have potential applications in a variety of fields. To illustrate their applications, we calculate the characteristic functions of several distributions appear in distortion riskmetrics and weighted entropies.

Modeling of lifetime scenarios with non-monotonic failure rates.

The Weibull distribution is an important continuous distribution that is cardinal in reliability analysis and lifetime modeling. On the other hand, it has several limitations for practical applications, such as modeling lifetime scenarios with non-monotonic failure rates. However, accurate modeling of non-monotonic failure rates is essential for achieving more accurate predictions, better risk management, and informed decision-making in various domains where reliability and longevity are critical factors. For this reason, we introduce a new three parameter lifetime distribution-the Modified Kies Weibull distribution (MKWD)-that is able to model lifetime scenarios with non-monotonic failure rates. We analyze the statistical features of the MKWD, such as the quantile function, median, moments, mean, variance, skewness, kurtosis, coefficient of variation, index of dispersion, moment generating function, incomplete moments, conditional moments, Bonferroni, Lorenz, and Zenga curves, and order statistics. Various measures of uncertainty for the MKWD such as Rényi entropy, exponential entropy, Havrda and Charvat entropy, Arimoto entropy, Tsallis entropy, extropy, weighted extropy and residual extropy are computed. We discuss eight different parameter estimation methods and conduct a Monte Carlo simulation study to evaluate the performance of these different estimators. The simulation results show that the maximum likelihood method leads to the best results. The effectiveness of the newly suggested model is demonstrated through the examination of two different sets of real data. Regression analysis utilizing survival times data demonstrates that the MKWD model offers a superior match compared to other current distributions and regression models.

QVF: Incorporating quantile value function factorization into cooperative multi-agent reinforcement learning

QVF: Incorporating quantile value function factorization into cooperative multi-agent reinforcement learning

Modeling Extreme Healthcare Costs Using the Neutrosophic Cauchy Distribution

Real data modelling of extreme events, such as rainfall, temperature, financial costs is very important in neutrosophic statistical methods. The Cauchy distribution is one of statistical models used for modelling such extreme events in natural processes. In cases of imprecise data which most often involve vague, incomplete and ambiguous information, standard statistical methods cannot fully describe the spectrum of uncertainty. In this study, we have considered a new Cauchy distribution under neutrosophic context to deal with uncertain data. The proposed neutrosophic Cauchy distribution (NCD) may analysis extreme events data involving incomplete observations. We provide basic mathematical characteristics and important statistical functions of the Cauchy model under neutrosophic framework. A complete procedure of random numbers generation using neutrosophic quantile function is discussed. The unknown parameters of the proposed are estimated using the maximum likelihood approach. Numerical results show that the proposed model adequately fits the data involving extreme and imprecise values. The performance and flexibility of the model are also supported by an application to a real data set.

Optimize Decision-Making in the Industrial Sector under Uncertainty: A Neutrosophic Inverse Exponential Distribution Approach

The most widely used distribution for risk management data for modeling longevity is the one-parameter inverse exponential distribution. Among alternative models, we suggest the neutrosophic inverse exponential (NIE) model, which generalizes the extended inverse exponential distributions and the classical structure. For the suggested model, we derive explicit formulations for the quantile functions, median, mode, cumulative distribution function, and probability density function. Data generating process of the proposed model under neutrosophic environment is discussed. To estimate the model parameters, we use the maximum likelihood approach. Using the proposed model, we run the simulation setup for randomly generated data. A genuine data set is also used to support the proposed model applicability.

On the Equivalence of Likelihood‐Based Confidence Bands for Fatigue‐Life and Fatigue‐Strength Distributions

ABSTRACTFatigue data arise in many research and applied areas, and there have been statistical methods developed to model and analyze such data. The distributions of fatigue life and fatigue strength are often of interest to engineers designing products that might fail due to fatigue from cyclic‐stress loading. Based on a specified statistical model and the maximum likelihood method, the cumulative distribution function (cdf) and quantile function (qf) can be estimated for the fatigue‐life and fatigue‐strength distributions. Likelihood‐based confidence bands can then be obtained for the cdf and qf. This paper provides equivalence results for confidence bands for fatigue‐life and fatigue‐strength models. These results are useful for data analysis and computing implementation. We show (a) the equivalence of the confidence bands for the fatigue‐life cdf and the fatigue‐life qf, (b) the equivalence of confidence bands for the fatigue‐strength cdf and the fatigue‐strength qf, and (c) the equivalence of confidence bands for the fatigue‐life qf and the fatigue‐strength qf. Then we illustrate the usefulness of those equivalence results with two examples using experimental fatigue data.

A Three-Parameter Exponential Frechet Distribution: Theory and Application

A novel three-parameter distribution of the Exponential G-family distribution called the Exponential Frechet Distribution has been proposed. Some properties of the model like moment generating function, characteristic function, quantile function, survival function and hazard function were derived. Bayesian and MLE approach was used to estimate the parameter. The Exponential Frechet Distribution flexibility was demonstrated by both simulation and application to remission time of bladder cancer patients. The results revealed that the Exponential Frechet Distribution yielded a better fit in comparison with other extended distribution.

Gradient-induced variable selection in reproducing kernel Hilbert space for survival analysis

The analysis of survival data is often hampered by a potentially very large number of covariates compounded with incomplete data. This paper considers the problem of variable selection where the response is subject to random (right) censoring. We introduce a model-free variable selection procedure via learning the gradients of quantile regression functions with two popular censoring weighting schemes. The key advantage of the proposed approach is that it does not require explicit model assumptions and is computationally efficient. Besides, we also prove its asymptotic consistency and examine its finite sample performance via numerical studies. As an empirical example, the proposed procedure is applied for the analysis of a breast cancer data set.

Learning about tail risk: Machine learning and combination with regularization in market risk management

High-quality risk management is the key to ensuring the safe, efficient, and stable operation of the financial system. The current Basel Accord requires financial institutions to regularly calculate and disclose Value at Risk (VaR) and Expected Shortfall (ES) measures. However, the inaccuracy and instability of traditional risk models have reduced users' confidence. Therefore, we propose two new probabilistic deep learning frameworks for estimating VaR and ES. The trained first framework can output expectiles that are more sensitive to tail risks to map VaR and ES measures. In the second framework, we propose to approximate VaR and ES measures with spline quantile function and estimate the parameters by designing various deep learning architectures. To ensure the effectiveness of the proposed architectures, we derived the training loss and constraints for them. In addition, we solve the problem that existing machine learning risk models are difficult to estimate ES. In this way, combining various individual risk models has great potential for risk management. Therefore, we propose a regularization-based combination framework that adaptively selects and shrinks individual risk models. The developed individual methods and combinations outperform existing methods in backtesting, assisting financial institutions to allocate capital more effectively according to the Basel Capital Accord.

Smoothed Weighted Quantile Regression for Censored Data in Survival Analysis

In this study, we propose a smoothed weighted quantile regression (SWQR), which combines convolution smoothing with a weighted framework to address the limitations. By smoothing the non-differentiable quantile regression loss function, SWQR can improve computational efficiency and allow for more stable model estimation in complex datasets. We construct an efficient optimization process based on gradient-based algorithms by introducing weight refinement and iterative parameter estimation methods to minimize the smoothed weighted quantile regression loss function. In the simulation studies, we compare the proposed method with two existing methods, including martingale-based quantile regression (MartingaleQR) and weighted quantile regression (WeightedQR). The results emphasize the superior computational efficiency of SWQR, outperforming other methods, particularly WeightedQR, by requiring significantly less runtime, especially in settings with large sample sizes. Additionally, SWQR maintains robust performance, achieving competitive accuracy and handling the challenges of right censoring effectively, particularly at higher quantiles. We further illustrate the proposed method using a real dataset on primary biliary cirrhosis, where it exhibits stable coefficient estimates and robust performance across quantile levels with different censoring rates. These findings highlight the potential of SWQR as a flexible and robust method for analyzing censored data in survival analysis, particularly in scenarios where computational efficiency is a key concern.

Introducing the unit Zeghdoudi distribution as a novel statistical model for analyzing proportional data

Unit distributions are essential in statistical modeling, providing a robust framework for understanding variables constrained within the unit interval [0,1]. This interval is crucial in public health, environmental studies, engineering, and finance, where measurements often represent proportions, probabilities, or rates. Despite numerous unit distributions derived by transforming existing distributions, there remains a pressing need for new distributions that can accommodate the unique characteristics of diverse datasets. In this study, we introduce the unit Zeghdoudi distribution (UZD), a novel transformation of the Zeghdoudi distribution (ZD). This new distribution retains the simplicity of the original ZD while offering enhanced flexibility and precision in modeling data confined to the unit interval. The UZD exhibits several noteworthy features, including both right and left-skewed probability density functions, a closed-form quantile function, easily defined moments, and manageable entropies. Using sixteen classical methods for parameter estimation, supported by a comprehensive simulation study, we demonstrate the efficiency and reliability of the UZD. Finally, the application of the UZD to three real-world proportional datasets related to (COVID-19, environmental, and radiation datasets) underscores its effectiveness and demonstrates its superiority over several established models.

Advancing Continuous Distribution Generation: An Exponentiated Odds Ratio Generator Approach.

This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the "Type 2 Gumbel Weibull-G family of distributions". We provide a comprehensive analysis of the mathematical properties of these distributions, including statistical properties such as density functions, moments, hazard rate and quantile functions, Rényi entropy, order statistics, and the concept of stochastic ordering. To test the robustness of our new model, we apply five distinct methods for parameter estimation. The practical applicability of the Type 2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These real-life applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications.

Estimation and Inference for Extreme Continuous Treatment Effects*

This paper studies estimation and inference for the treatment effect in deep tails of the potential outcome distributions corresponding to a continuously valued treatment, namely the extreme continuous treatment effect. We consider two measures for the tail characteristics: the quantile function and the tail mean function defined as the conditional mean beyond a quantile level. Then, for a quantile level close to 1, we define the extreme quantile treatment effect (EQTE) and extreme average treatment effect (EATE), which are, respectively, the ratios of the quantile and tail mean at different treatment statuses. We propose estimators for the EQTE and EATE based on tail approximations from the extreme value theory. Our limiting theory is for the EQTE and EATE processes indexed by a set of quantile levels and pairs of different treatment statuses. It facilitates uniform inference for the EQTE and EATE over multiple tail levels and multiple treatment effects. Simulations suggest that our method works well in finite samples, and an empirical study illustrates its practical merits.

Estimation of Tensile Strength of Carbon Fibers Using Exponentiated Exponential Weibull-Dagum Distribution Model (EEWD) and Its Properties

The tensile strength of Carbon fibers was investigated. Tested EEWD distribution ability to fit with data and observed the skewness of data. The T-R{.} framework has been recently used to generalize various distributions, but the viability of using Dagum distribution has not been investigated. Three distributions are combined in the T-R{.}, through one serving as a baseline distribution. The combined potency of each distribution, which is a weighted hazard function of the baseline distribution, would have more parameters but would also be highly flexible in handling bimodality in datasets. Thus, this paper used the quantile function of the Weibull distribution to generalize the Dagum distribution. In this present research work a novel generalized 6P (six parameter) model called Exponentiated Exponential Weibull Dagum Distribution (EEWD) has been introduced. Appropriate distributions including PDF, CDF, moments, Moment Generating Function (MGF) of EEWD distribution, stochastic ordering, Cumulant Generating Function (CGF) of EEWD distribution, mean, mean deviations and their sub-models have been given. Further EEWD model has been applied in the real-time data admissible.

Exponentiated gamma Burr-type X distribution: model, theory, and applications

Several extended Burr-type X distributions have been formed in the past decade. These distributions are widely used in modeling lifetime data as their hazard functions can fit various shapes, such as bathtub, decreasing, and increasing. However, certain extended Burr-type X distributions may not adequately fit the unimodal hazard function. Thus, this paper proposes a new extended distribution with greater flexibility to solve this deficiency: exponentiated gamma Burr-type X distribution. We provide the expressions for the probability density and cumulative distribution functions of the proposed distribution, along with its statistical properties, such as limit behavior, quantile function, moment function, moment-generating function, Renyi entropy, and order statistics. To estimate the model parameters, we employ the maximum likelihood estimation method, and we assess its performance through a simulation study with different sample sizes and parameter values. Finally, to demonstrate the application of this new distribution, we apply it to a real dataset concerning the failure times of aircraft windshields. The results indicate that the new distribution provides a superior fit compared to its submodels and the extended Burr-type X distributions. Moreover, it proves to be highly competitive and can serve as an alternative to certain nonnested models. In summary, the new distribution is highly flexible, capable of modeling a variety of hazard-function shapes, including decreasing, increasing, bathtub, and unimodal patterns.

Gumbel–Logistic Unit Distribution with Application in Telecommunications Data Modeling

The manuscript deals with a new unit distribution that depends on two positive parameters. The distribution itself was obtained from the Gumbel distribution, i.e., by its transformation, using generalized logistic mapping, into a unit interval. In this way, the so-called Gumbel-logistic unit (abbr. GLU) distribution is obtained, and its key properties, such as cumulative distribution function, modality, hazard and quantile function, moment-based characteristics, Bayesian inferences and entropy, have been investigated in detail. Among others, it is shown that the GLU distribution, unlike the Gumbel one which is always positively asymmetric, can take both asymmetric forms. An estimation of the parameters of the GLU distribution, based on its quantiles, is also performed, together with asymptotic properties of the estimates thus obtained and their numerical simulation. Finally, the GLU distribution has been applied in modeling the empirical distributions of some real-world data related to telecommunications.

A new Marshall-Olkin generalized-k family of distributions with applications

Abstract In this article, we describe a new Marshall-Olkin family of distributions based on the Marshall-Olkin family of distributions by employing a different generator. We have derived several statistical properties of the new Marshall-Olkin family, including survival function, hazard rate function, quantile function, moments, and linear representation. We have taken the Weibull distribution as a baseline distribution and found that the new Marshall-Olkin Weibull distribution exhibits a variety of shapes for its hazard rates and densities. We have tested the performance of various estimation methods for the new Marshall-Olkin family of distributions, including eight different estimation methods. Additionally, we conducted extensive simulations to assess the robustness of the estimation methods across various scenarios and determine which methods are most effective for varying sample sizes. To further validate the new Marshall-Olkin Weibull distribution, we have used two different data sets to compare the fitted new Marshall-Olkin Weibull distribution with the well-known comparative models. By testing the performance of different estimation methods and using multiple data sets, we can demonstrate the versatility and usefulness of the new Marshall-Olkin family of distributions for modeling a wide range of phenomena.

Transmuted Unit Exponentiated Half-Logistic Distribution and its Applications

A novel distribution, termed the transmuted unit exponentiated half-logistic distribution, was proposed using the unit exponential half-logistic distribution, a member of the proportional hazard rate model family, as the base distribution. The statistical characteristics of the proposed distribution, including moments, moment-generating function, quantile function, and stress-strength reliability, were thoroughly examined in this study. The maximum likelihood estimation method is discussed for statistical inference of the distribution parameters. A simulation study based on the new distribution was conducted to investigate the behavior of maximum likelihood estimates under various conditions. In addition, a numerical example is presented to illustrate the performance of the distribution on a failure-time dataset.