Some general results on quantile functions for the generalized beta family

To generate random variates from an unknown continuous distribution via the inverse transform method, we present a flexible, computationally tractable procedure for estimating the associated inverse distribution function based on sample data. Previously proposed methods for estimating inverse distribution functions can fail in either the distribution-fitting or variate-generation stages of application. To avoid these difficulties, we have developed the procedure IDPF for estimating an Inverse Distribution with a Polynomial Filter. After a first-cut or reference distribution has been obtained by some standard technique, a front-end polynomial filter for the inverse of the reference distribution is estimated by constrained nonlinear regression so that the resulting inverse distribution has minimum distance from the empirical inverse distribution. The constraints on the regression ensure that the fitted inverse distribution function is nondefective and monotonically nondecreasing. A specific implementation of this procedure is based on well-known techniques for obtaining a reference fit from the Johnson translation system of distributions. We present the results of a Monte Carlo study to demonstrate the effectiveness of the method. Compared to the reference fit, procedure IDPF yields significantly better approximations not only to the empirical inverse distribution function but also to the underlying theoretical inverse distribution function.

FORTRAN 77 program for constructing and plotting confidence bands for the distribution and quantile functions for truncated data

This work is devoted to the study of rates of convergence of the empirical measures μ n = 1 n ∑ k = 1 n δ X k \mu _n = \frac {1}{n} \sum _{k=1}^n \delta _{X_k} , n ≥ 1 n \geq 1 , over a sample ( X k ) k ≥ 1 {(X_k)}_{k \geq 1} of independent identically distributed real-valued random variables towards the common distribution μ \mu in Kantorovich transport distances W p W_p . The focus is on finite range bounds on the expected Kantorovich distances E ( W p ( μ n , μ ) ) \mathbb {E}(W_p(\mu _n,\mu )) or [ E ( W p p ( μ n , μ ) ) ] 1 / p \big [ \mathbb {E}(W_p^p(\mu _n,\mu )) \big ]^{1/p} in terms of moments and analytic conditions on the measure μ \mu and its distribution function. The study describes a variety of rates, from the standard one 1 n \frac {1}{\sqrt n} to slower rates, and both lower and upper-bounds on E ( W p ( μ n , μ ) ) \mathbb {E}(W_p(\mu _n,\mu )) for fixed n n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

In this work, we provide an extended one parameter distribution into a new two parameter life model and called it the New Extended “A” (NEA) distribution. This new model provides the best description of the data sets in different sectors. Both of the probability distribution function (PDF) and the cumulative distribution function (CDF) of the NEA distribution are derived from one of the new families. The suggested distribution’s properties including moments, quantile function, survival function, hazard rate function, odds function and mode are deduced and acquired in explicit forms using Taylor expansion to simplify the integrals. The Rényi entropy of NEA distribution is derived and the maximum likelihood method for estimating its parameters is deduced. Numerical simulations are used to study the behavior of the parameters with different sizes of samples. According to the simulation results of this work, the maximum likelihood estimators are advised to be a suitable estimation method for estimating the parameters of the NEA distribution. We show the importance and flexibility of the NEA distribution by comparing it to some competing distributions using five real data sets from different sectors in behavioral sciences, ecophysiological, industry and health. Our new model is shown to be better than all competing distributions. Received: October 6, 2023Accepted: November 15, 2023

In this article, we introduce a new one-parameter model, which is named sine inverted exponential (SIE) distribution. The SIE distribution is a new extension of the inverse exponential (IE) distribution. The SIE distribution aims to provide the SIE model for data-fitting purposes. The SIE distribution is more flexible than the inverted exponential (IE) model, and it has many applications in physics, medicine, engineering, nanophysics, and nanoscience. The density function (PDFu) of SIE distribution can be unimodel shape and right skewed shape. The hazard rate function (HRFu) of SIE distribution can be J-shaped and increasing shaped. We investigated some fundamental statistical properties such as quantile function (QFu), moments (Mo), moment generating function (MGFu), incomplete moments (ICMo), conditional moments (CMo), and the SIE distribution parameters were estimated using the maximum likelihood (ML) method for estimation under censored samples (CS). Finally, the numerical results were investigated to evaluate the flexibility of the new model. The SIE distribution and the IE distribution are compared using two real datasets. The numerical results show the superiority of the SIE distribution.

In this article, a new generalization of the Topp-Leone distribution with a unit interval, namely Mixed Topp-Leone-Kumaraswamy distribution is defined and studied. The mathematical properties of this mixing distribution are described. Moments, quantile function, R?nyi entropy, incomplete moments and moments of residual are obtained for the new Mixed Topp-Leone - Kumaraswamy distribution. The maximum likelihood (MLE), Crans (CM) , Percentile (PM) and Particle Swarm Optimization(PSO) estimators of the parameters are derived. The percentile Method is more efficient method as compred to the others. Two real data sets are used to illustrate an application and superiority of the proposed distribution.

Let {Xn, n ≥ 1} be a sequence of stationary non-negative associated random variables with common marginal distribution function F(x) and quantile function Q(u), where Q(u) is defined as F(Q(u)) = u. Here we consider the smooth estimation of Q(u), adapted from generalized kernel smoothing (Cheng and Parzen J. Stat. Plann. Infer. 59, 291–307, 1997) of the empirical quantile function. Some asymptotic properties of the kernel quantile estimator, for associated sequences, are also established parallel to those in the i.i.d. case. Various estimators in this class of estimators are contrasted, through a simulation study, among themselves and with an indirect smooth quantile estimator obtained by inverting the Poisson weights based estimator of the distribution function studied in Chaubey et al. (Statist. Probab. Lett. 81, 267–276, 2011). The indirect smoothing estimator seems to be the best estimator on account of smaller MSE, however, a quantile estimator based on the Bernstein polynomials and that using the corrected Poisson weights turn out to be almost as good as the inverse distribution function estimator using Poisson weights.


 
 
 
 The Exponential distribution is memoryless and has a constant failure rate which makes it unsuitable for real life problems. This paper introduces a new distribution powered by an exponential random variable which gives a more flexible model for modelling real-life data. This new extension of the Exponential Distribution is called “Lomax-Exponential distribution (LED)”. The extension of the new distribution became possible with the help of a Lomax generator proposed by [1]. This paper derives and studies some expressions for various statistical properties of the new distribution including distribution function, moments, quantile function, survival function and hazard function known as reliability functions. The inference for the Lomax-Exponentially distributed random variable were investigated based on some plots of the distribution and others revealed its behaviour and usefulness in real life situations. The parameters of the distribution are estimated using the method of maximum likelihood estimation. The performance of the new Lomax-Exponential distribution has been tested and compared to the Weibull-Exponential, Transmuted Exponential and the conventional Exponential distribution using three real life data sets.
 
 
 
 

ABSTRACTRecently, in the literature, the use of quantile functions in the place of distribution functions has provided new models, alternative methodology and easier algebraic manipulations. In this paper, we introduce new orders among the random variables in terms of their quantile functions like the reversed hazard quantile function, the reversed mean residual quantile function and the reversed variance residual quantile function orders. The relationships among the proposed orders and some existing orders are also discussed.

Norm-based inequality measures are developed within the unequally distributed/relative unequally distributed (UD/RUD) framework by applying the L1 norm and the squared L2 norm to the cumulative distribution and quantile function (CDQF), the quantile function (QF), and their combined representation. All six indices maintain key properties such as scale, replication, and translation invariance, as well as anonymity, and their behavior under Pigou-Dalton transfers and subgroup decomposition is examined analytically. The indices based on the cumulative distribution and quantile function combine the vertical and horizontal gaps, giving them representation decomposability and making them the most informative overall measures of inequality. A Monte Carlo study on six contrasting income distributions corroborates these theoretical advantages, confirming finite, stable values even for mixed-sign and heavy-tailed supports.

Methods for estimating extreme loads are used in design as well as risk assessment. Regression using maximum likelihood or least squares estimation is widely used in a univariate analysis but these methods favour solutions that fit observations in an average sense. Here we describe a new technique for estimating extremes using a quantile function model. A quantile of a distribution is most commonly termed a ‘return level’ in flood risk analysis. The quantile function of a random variable is the inverse function of its distribution function. Quantile function models are different from the conventional regression models, because a quantile function model estimates the quantiles of a variable conditional on some other variables, while a regression model studies the conditional mean of a variable. So quantile function models allow us to study the whole conditional distribution of a variable via its quantile function, whereas conventional regression models represent the average behaviour of a variable.Little work can be found in the literature about prediction from a quantile function model. This paper proposes a prediction method for quantile function models. We also compare different types of statistical models using sea level observations from Venice. Our study shows that quantile function models can be used to estimate directly the relationships between sea condition variables, and also to predict critical quantiles of a sea condition variable conditional on others. Our results show that the proposed quantile function model and the developed prediction method have the potential to be very useful in practice.

We develop a new continuous distribution called the Gamma-Burr type X (GBX) distribution that extends the Burr type X distribution that has increasing, decreasing and bathtub shapes for the hazard function. Various structural properties of this new distribution are provide, that includes the limit behavior, Quantile function and sub-models. From the generalization of the probability density function and cumulative distribution function of this distribution, the expression for the rth moment, moment generating function, Renyi entropy, and the order statistics can be established. We considered the maximum likelihood estimation to estimate the parameters. A real data set is applied to illustrate the usefulness of the GBX distribution. This new distribution will serve as an alternative model to other models available in the literature for modeling positive real data in many areas.

：A new distribution called the exponentiated Burr XII Weibull(EBW) distributions is proposed and presented. This distribution contains several new and known distributions such as exponentiated log-logistic Weibull, exponentiated log-logistic Rayleigh, exponentiated log-logistic exponential, exponentiated Lomax Weibull, exponentiated Lomax Rayleigh, exponentiated Lomax Exponential, Lomax Weibull, Lomax Rayleigh Lomax exponential, Weibull, Rayleigh, exponential and log-logistic distributions as special cases. A comprehensive investigation of the properties of this generalized distribution including series expansion of probability density function and cumulative distribution function, hazard and reverse hazard functions, quantile function, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, R´enyi entropy and distribution of order statistics are presented. Parameters of the model are estimated using maximum likelihood estimation technique and real data sets are used to illustrate the usefulness and applicability of the new generalized distribution compared with other distributions.

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

In this article, we have introduced a new probability distribution having three parameters using half Cauchy family of distribution named half Cauchy extended exponential distribution. The statistical properties and characteristics of the proposed distribution like the hazard rate function (HRF), cumulative hazard function, and the probability density function (PDF), and the cumulative distribution function (CDF), quantile function and the skewness, kurtosis are provided. The parameters of the proposed distribution are estimated using the Cramer-Von-Mises (CVM) least-square estimation (LSE), and maximum likelihood estimators (MLE) methods. A real data set is analyzed to test the goodness-of-fit of the proposed distribution. It is found that the half Cauchy extended exponential distribution performed well as compared to some competing distributions.