Quantile Function Research Articles

A New Cubic Transmuted Inverse Weibull Distribution: Theory and Applications

This paper introduces a new cubic transmutation of the inverse Weibull distribution, known as a cubic transmuted inverse Weibull distribution. The model is thought to be useful for the analysis of complex life data, modeling failure times, accessing product reliability, and many other fields like economics, hydrology, biology, and engineering. Some statistical features of the proposed distribution are explored. These include moments, generating functions, quantile functions, reliability functions, and hazard rate functions. The distribution of order statistics for the proposed cubic transmuted inverse Weibull distribution is also studied. The maximum likelihood estimation approach is used to estimate the model parameters. The effectiveness of the estimation is investigated through extensive simulation study. The suitability of the proposed distribution has been studied by using five real-life datasets. It is found that the proposed distribution is the most suitable fit for the used data sets.

Investigating Growth-at-Risk Using a Multicountry Nonparametric Quantile Factor Model

We develop a nonparametric quantile panel regression model. Within each quantile, the quantile function is a combination of linear and nonlinear parts, which we approximate using Bayesian Additive Regression Trees (BART). Cross-sectional information is captured through a conditionally heteroscedastic latent factor. The nonparametric feature enhances flexibility, while the panel feature increases the number of observations in the tails. We develop Bayesian methods for inference and apply several versions of the model to study growth-at-risk dynamics in a panel of 11 advanced economies. Our framework usually improves upon single-country quantile models in recursive growth forecast comparisons.

Economic resilience:Measurement and assessment across time and space

This paper studies economic resilience as the ability of an economic system to respond to adverse shocks. We propose several measures of resilience based on a quantile function representing income dynamics. Applied to the evolution of per capita income, we evaluate the speed and nature of economic adjustments to adverse shocks across countries over the last two centuries. We find evidence of important income effects: low-income countries adjust to adverse shocks better in the short run; but high-income countries adjust better in the longer run. We show that the long run effects dominate: in terms of discounted present value, high-income countries have been able to reduce the effects of adverse shocks on expected future income better than low-income countries. Finally, we find that, over the last 50 years, most of the changes in resilience across countries can be attributed to income effects.

Smooth estimation of conditional quantile function with mixed covariates using Bernstein polynomials

ABSTRACT The conditional quantile function is a fundamental tool for understanding the relationship between covariates and the response variable in data analysis. In this study, we propose a novel estimator based on the Nadaraya–Watson discrete conditional distribution estimator, which uses Bernstein polynomials for smoothing. The consistency and asymptotic normality of the proposed estimator are established. Also, a simulation study to compare its performance with other existing estimators is carried out.

A new flexible distribution: Statistical inference with application

A flexible distribution for the analysis of bounded data is proposed. The new model is generalized through the transmuted approach, and the resultant model is named “transmuted modified Lehmann-type II distribution.” A comprehensive analysis of key characteristics is performed, including the shape of the model, survival, and hazard function, analytical expressions of mode, quantile function, ordinary moments, quantile function, and stress–strength reliability. Some famous entropy measures are also derived. The model parameters have been estimated using four distinct methods, including maximum likelihood estimation, Anderson Darling, Cramer–von Misses, and ordinary least squares. A detailed simulation study is used to evaluate the behavior of all derived estimators. Finally, a dataset was used to demonstrate the utility of the proposed distribution.

Marshall-Olkin Bilal distribution with associated minification process and acceptance sampling plans

In this paper, a new two parameters lifetime distribution, called Marshall-Olkin Bilal distribution is introduced and the structural properties are discussed. The proposed model results from the Marshall and Olkin class of distributions with the baseline model as Bilal distribution. We examined the statistical aspects like moments, quantile function, order statistics and entropy. The hazard function can model increasing and upside-down bathtub shaped data sets. The model parameter estimation is carried out by maximum likelihood estimation and a simulation study is performed. The flexibility of the proposed model is evaluated by two real data sets, compared with the competing models. Its application in time series is studied by the associated autoregressive minification process and the auto-correlation structure is derived. The acceptance sampling plans formulated for the proposed model and the characteristic results are illustrated.

X-exponential-G Family of Distributions With Applications

A new family of continuous distributions called the X-exponential-G (XE-G) family is proposed. Explicit expressions are derived for the ordinary and incomplete moments, generating functions, mean deviation about the mean and median, Shannon and R\'{e}nyi entropies, and order statistics of this new family. Estimation of the parameters of the new family is done using the method of maximum likelihood. Assessment of the performance of the maximum likelihood estimates is carried out through a simulation study using the quantile function of the XE-G distribution. The usefulness of this new family is illustrated by modeling two real datasets.

Dependent conditional tail expectation for extreme levels

We consider the estimation of the dependent conditional tail expectation, defined for a random vector (X,Y) with X≥0 as E(X|X>QX(1−p),Y>QY(1−p)), when E(X)<∞, and where QX and QY denote the quantile functions of X and Y, respectively. The distribution of X is assumed to be of Pareto-type while the distribution of Y is kept general. Using extreme-value arguments we introduce an estimator for this risk measure for the situation p≤1/n, where n is the number of available observations, i.e., focus is on estimation with extrapolation. The convergence in distribution of our estimator is established and its finite sample performance is illustrated on a simulation study. The method is then applied on wind gusts data set.

A hybrid deep learning model for lithium-ion batteries state of charge estimation based on quantile regression and attention

Accurate estimation of the state of charge (SOC) is essential to ensure the safe and efficient utilization of lithium-ion batteries. However, external factors can affect the SOC, making it challenging to achieve precise estimations. To address this issue, we propose a deep learning model that integrates the parallel computing capabilities of temporal convolutional networks with the robust learning abilities of gated recurrent units. This model incorporates an attention mechanism to dynamically assign weights and focus on salient features based on correlations in historical information. Furthermore, we employ the quantile regression loss function during network training, thereby equipping the neural network model with the capability to directly generate interval estimates. The results show that the MAE and RMSE of the lithium iron phosphate battery, lithium cobalt oxide battery and ternary lithium battery dataset are below 1.45 and 1.98, 1.12 and 1.25, 0.61 and 0.75 respectively. The comparison analysis with other SOC estimation methods demonstrates its exceptional accuracy in point estimation and high reliability in confidence interval estimation. Furthermore, it showcases remarkable generalization ability across diverse temperatures, operating conditions, battery lifetimes, and battery types.

Generation of normal distributions revisited

AbstractNormally distributed random numbers are commonly used in scientific computing in various fields. It is important to generate a set of random numbers as close to a normal distribution as possible for reducing initial fluctuations. Two types of samples from a uniform distribution are examined as source samples for inverse transform sampling methods. Three types of inverse transform sampling methods with new approximations of inverse cumulative distribution functions are also discussed for converting uniformly distributed source samples to normally distributed samples.

The Generalized Odd Maxwell-Kumaraswamy Distribution: Its Properties and Applications

This study introduces a novel family of continuous probability distributions using Alzaatreh‘s technique to explore the Generalized Odd Maxwell-Generated (GOM-G) distribution family. Within the Generalized Odd Maxwell family, the GOM-Kumaraswamy distribution is presented as extended form of the Kumaraswamy distribution. The investigation thoroughly examines the cumulative distribution function, probability density function, hazard, and survival functions, as well as mixture representations of the Generalized Odd Maxwell-Kumaraswamy distribution. The suggested distribution's structural properties, including moments, skewness, kurtosis, probability-weighted moments, entropies, stress-strength models, and order statistics, are derived. Model parameters are estimated through the maximum likelihood method, and simulation studies evaluate the performance of maximum likelihood estimations using the quantile function. Employing two real-life datasets, goodness-of-fit measures such as Akaike Information Criterion (AIC), Corrected Akaike Information Criterion, Bayesian Information Criterion (BIC), Hannan-Quinn Information Criterion (HQIC), and chi-square goodness-of-fit tests demonstrate the adaptability and flexibility of the GOM-Kumaraswamy distribution against competing distributions, including Kumaraswamy-Kumaraswamy and Kumaraswamy-Burr III. The results reveal that the GOM-Kumaraswamy distribution exhibits the lowest AIC, CAIC, BIC, and HQIC values and the highest goodness-of-fit values compared to other models, suggesting its superiority as the preferred fit for both dataset forms. This proposed distribution contributes to practical applications, unveiling its potential significance in modeling real-world phenomena within the domains of hydrology and engineering.

On the Statistical Properties of the Remkan Distribution

The Remkan distribution is a two-parameter lifetime distribution that has been introduced into the literature to meet the ever-growing demand for the development of new lifetime distributions to meet the goodness of fit demand of complex datasets. The mathematical properties of the Remkan distribution have been derived in the literature. This study therefore aims to derive important statistical properties including the mode, quantile function, order statistics, entropy, stochastic ordering, average absolute deviation, and mid-point and Reliability indices such as the survivorship or existence measurement function, risk measurement function, and average residual measurement lifetime function.

New approach to measuring income inequality

We show that the conventional income inequality indexes assess income inequality incorrectly because of three problems. The unequally distributed (UD) income-based approach solves the problems, decomposes income inequality into two kinds of departure from equality, and provides two indexes. The comprehensive assessment of income inequality requires the integration of two kinds of departure. This paper proposes the relative UD (RUD) income-based approach. The RUD income-based approach combines the cumulative distribution function and quantile function of the RUD income and produces a new index integrating two kinds of departure. We investigate the properties of the new index and demonstrate its applicability through example income distributions.

An exponentiated XLindley distribution with properties, inference and applications

In this paper, we propose exponentiated XLindley (EXL) distribution. The novel model is adaptable due to the mixt shapes of its density and failure rate functions. The following key statistical properties of EXL distribution are derived: quantile function, moments, hazard function, mean residual life, and Rényi entropy. The parameters are estimated using the maximum likelihood, Anderson Darling, Cramer von Misses, maximum product spacing, ordinary and weighted least square estimation procedures. To examine the behavior of the estimate, Monte Carlo simulation is used. Further Bayesian technique is also utilized to estimate the EXL parameters. The traceplot and Geweke diagnostics are used to track the convergence of simulated processes. The applicability of the EXL distribution is demonstrated by three datasets from different domains such as mortality rate due to COVID-19, precipitation in inches, and failure time for repairable items. The proposed distribution provides efficient results as compared to renowned competitive distributions.

A scalar-on-quantile-function approach for estimating short-term health effects of environmental exposures.

Environmental epidemiologic studies routinely utilize aggregate health outcomes to estimate effects of short-term (eg, daily) exposures that are available at increasingly fine spatial resolutions. However, areal averages are typically used to derive population-level exposure, which cannot capture the spatial variation and individual heterogeneity in exposures that may occur within the spatial and temporal unit of interest (eg, within a day or ZIP code). We propose a general modeling approach to incorporate within-unit exposure heterogeneity in health analyses via exposure quantile functions. Furthermore, by viewing the exposure quantile function as a functional covariate, our approach provides additional flexibility in characterizing associations at different quantile levels. We apply the proposed approach to an analysis of air pollution and emergency department (ED) visits in Atlanta over 4 years. The analysis utilizes daily ZIP code-level distributions of personal exposures to 4 traffic-related ambient air pollutants simulated from the Stochastic Human Exposure and Dose Simulator. Our analyses find that effects of carbon monoxide on respiratory and cardiovascular disease ED visits are more pronounced with changes in lower quantiles of the population's exposure. Software for implement is provided in the R package nbRegQF.

A Novel Load Extrapolation Method for Multiple Non-Stationary Loads on the Drill Pipe of a Rotary Rig

The drill pipe of a rotary rig is subject to the dynamic influence of non-stationary loads, including rotation torque and applied force. In order to address the challenge of simultaneously extrapolating multiple non-stationary loads, a novel extrapolation framework is proposed. This framework utilizes rainflow counting to obtain mean and amplitude sequences of the loads. The extreme values of the amplitude sequence are fitted using the Generalized Pareto Distribution (GPD), while the median values are fitted using the Double Kernel Density Estimation (DKDE). By extrapolating the Inverse Cumulative Distribution Function (ICDF) based on the fitted distribution, a new amplitude sequence can be derived. The combination of this extrapolated amplitude sequence with the original mean sequence forms a new load spectrum. The results of applying the proposed extrapolation method to the drill pipe of a rotary rig demonstrate the ability of the method to yield conservative extrapolation results and accurately capture the variations in damage under the original working conditions.

Secant Kumaraswamy Family of Distributions: Properties, Regression Model, and Applications

In this study, Secant Kumaraswamy family of distributions is proposed and studied. This is motivated by the fact that no one distribution can model all types of data from different fields. Therefore, there is the need to develop distributions with desirable properties and flexible enough for modelling data exhibiting different characteristics. Some properties of the new family of distributions, including the quantile function, moments, moment generating function, and mean residual life function, are derived. Five special cases of the family of distributions are presented, and their flexibility is shown by the varying degrees of skewness and kurtosis and nonmonotonic hazard rates. The maximum likelihood estimation method is used to obtain estimators of the family of distributions. Two location-scale regression models are developed for the Secant Kumaraswamy Weibull distribution, which is a special case of the family of distributions. Six different real datasets are used to demonstrate the usefulness of the family of distributions and the regression models. The results show that the family of distributions can be used to model real datasets.

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A new reduced quantile function for generating families of distributions

In this paper, a variant of the T-X(Y) generator was developed by suppressing the scale parameter of the classical Lomax distribution in the quantile function. Uniquely, the reduction of the number of parameters essentially accounts for the parsimony of the attendant model. The study considered the Exponential distribution as the transformer and consequently obtained the New Reduced Quantile Exponential-G (NRQE-G) family. The Type-II Gumbel distribution was deployed as the baseline to obtain a special sub-model known as the New Reduced Quantile Exponential Type-II Gumbel (NRQE-T2G) model. Some functional properties of the distribution namely, moment and its related measures such as the mean, variance, second, third, and fourth moments were obtained. The Mode, skewness, Kurtosis, index of dispersion, coefficient of variation, order statistics, survival, hazard, and quantile function were also derived. The maximum likelihood estimation method was used to estimate its parameters. The model's credibility, applicability, and flexibility were proven using two real-life datasets.

Quantile Regression in Survival Analysis: Comparing Check-Based Modeling and the Minimum Distance Approach

Introduction: Quantile regression is a valuable alternative for survival data analysis, enabling flexible evaluations of covariate effects on survival outcomes with intuitive interpretations. It offers practical computation and reliability. However, challenges arise when applying quantile regression to censored data, particularly for upper quantiles. The minimum distance approach, utilizing dual-kernel estimation and the inverse cumulative distribution function, shows promise in addressing these challenges, especially with higher-dimensional covariates.&#x0D; Methods: This study contrasts two methods within the realm of quantile linear regression for survival analysis: check-based modeling and the minimum distance approach. Effectiveness is assessed across various scenarios through comprehensive simulation.&#x0D; Results: The simulation results showed that using the quantile regression model with the minimum distance approach reduces the percentage of root mean square error in parameter estimation compared to the quantile regression models based on the check loss function. Additionally, a larger sample size and reduced censoring percentage led to decreased root mean square error in parameter estimation.&#x0D; Conclusion: The research highlights the benefits of using the minimum distance approach for quantile regression. It reduces errors, improves model predictions, captures patterns, and optimizes parameters even with complete data. However, this approach has limitations. The accuracy of estimated quantiles can be influenced by the choice of distance metric and weighting scheme. The assumption of independence between censoring mechanism and survival time may not hold in real-world scenarios. Additionally, dealing with large datasets can be computationally complex.&#x0D; &#x0D;