A flexible method for estimating inverse distribution functions in simulation experiments

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.

Quantile function or inverse cumulative distribution function (CDF) is heavily utilized in modelling, simulation, reliability analysis and random number generation. The use is often limited if the inversion method fails to estimate it from the cumulative distribution function. As a result, approximation becomes the other feasible option. The failure of the inversion method is often due to the intractable nature of the CDF of the distribution. Approximation may come in the form of series expansions, closed form or functional approximation, numerical algorithm and the closed form expression drafted in terms of the quantile function of another distribution. This work used the cubic spline interpolation to obtain the closed form of the inverse cumulative distribution function of the Nakagami-m distribution. Consequently, the closed form of the quantile function obtained for the selected parameters of the distribution serves as an approximation which compares favourably with the R software values. The result obtained was a significant improvement over some results surveyed from literature for four reasons. Firstly, the approximates produced better results in simulation as evidenced by the some values of the root mean square error of this work when compared with others. Secondly, the result obtained at the extreme tail of the distribution is better than others selected from the literature. Thirdly, the closed form estimates are easy to compute and save computation time. The closed form of the quantile function obtained in this work can be used in simulating Nakagami random variables which are used in modelling attenuation and fading channels in wireless communications and ultrasonic tissue characterization.

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.

Various mathematical approaches can be used to analyze the performance of decision-makers. Our parametric model is based on comparing the distribution functions of the public green spaces areas (PGSA) of urban significance with a lognormal distribution. The data from the St. Petersburg Law No. 430-85 of October 08, 2007, were taken as the initial data. “On public green spaces.” The values of the logarithms of the urban PGSA were taken to construct the empirical and theoretical distribution functions. The maximum moduli of difference are calculated to compare empirical functions with each other, theoretical functions with each other, and theoretical and empirical functions. A comparison of areas using our model showed some correlations between trends in the formation of PGSA of urban significance and morphotypes of housing development.

The asymptotic properties of a family of minimum quantile function distance estimators are considered. These procedures take as the parameter estimates that vector which minimizes a weighted $L^2$ distance between the empirical quantile function and an assumed parametric family of quantile functions. Regularity conditions needed for these estimators to be consistent and asymptotically normal are presented. For single parameter families of distributions, the optimal form of the weight function is presented.

Let X 1,…, X n be i.i.d. random variables from an unknown cumulative distribution function F defined on the real line ℝ. No assumptions are made on the unknown F. The problem is to predict the empirical distribution function of a future sample Y = (Y 1,…,Y m ) from the distribution function F on the basis of the observations X 1,…, X n . To solve the problem, a decision theoretic approach is used. In certain situations, when for example a weighted quadratic loss function is used in the prediction problem, the problem of finding the minimax predictor of the empirical distribution function can be reduced to the problem of minimax estimation of the cumulative distribution function F under a composite weighted quadratic loss function. In such a case, a general method of finding minimax predictors of is presented. The method is also useful when no equalizer rule is available. The problem of prediction considered refers also to prediction of the empirical lifetime distribution function and may have some practical significance. For example, the results enable to predict, under the minimax criterion, the number of devices (from among all the m ones) which will fail up to any fixed time t 0.

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. 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. 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. 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.

Assessing goodness of fit to a given distribution plays an important role in computational statistics. The probability integral transformation (PIT) can be used to convert the question of whether a given sample originates from a reference distribution into a problem of testing for uniformity. We present new simulation- and optimization-based methods to obtain simultaneous confidence bands for the whole empirical cumulative distribution function (ECDF) of the PIT values under the assumption of uniformity. Simultaneous confidence bands correspond to such confidence intervals at each point that jointly satisfy a desired coverage. These methods can also be applied in cases where the reference distribution is represented only by a finite sample, which is useful, for example, for simulation-based calibration. The confidence bands provide an intuitive ECDF-based graphical test for uniformity, which also provides useful information on the quality of the discrepancy. We further extend the simulation and optimization methods to determine simultaneous confidence bands for testing whether multiple samples come from the same underlying distribution. This multiple sample comparison test is useful, for example, as a complementary diagnostic in multi-chain Markov chain Monte Carlo (MCMC) convergence diagnostics, where most currently used convergence diagnostics provide a single diagnostic value, but do not usually offer insight into the nature of the deviation. We provide numerical experiments to assess the properties of the tests using both simulated and real-world data and give recommendations on their practical application in computational statistics workflows.

Distribution test, a method to evaluate the goodness-of-fit, has been applied to automatic modulation clas-sification (AMC) in recent years. It compares the empirical cumulative distribution function (ECDF) of the sample signal with the theoretical cumulative distribution function (TCDF) under each candidate modulation format. Most existing works in MIMO systems assume the additive noise is Gaussian. However, additive noise often exhibits non-Gaussian characteristics in practical systems. The TCDF of each modulation developed for Gaussian noise can not perform well in practical system. To solve this issue, this paper proposes a practical MIMO system model, which assumes that the noise is Cauchy-Gaussian bi-parameter mixture and analyzes the TCDF corresponding to different modulation modes under this noise model. The Kolmogorov-Smirnov (KS) distribution test is used to AMC in flat-fading channel for both Quadrature Amplitude Modulation and Phase Shift Keying. Extensive simulation results demonstrate that, compared with two-sample KS test based classifier, one-sample KS test based classifier can achieve good recognition results under the non-Gaussian noise model.

The aim of this study was to address the issue of significant performance degradation in existing defogging algorithms under extreme fog conditions. Traditional Taylor series-based deformable convolutions are limited by local approximation errors, while the heavy-tailed characteristics of the Cauchy distribution can more successfully model outliers in fog images. The following improvements are made: (1) A displacement generator based on the inverse cumulative distribution function (ICDF) of the Cauchy distribution is designed to transform uniform noise into sampling points with a long-tailed distribution. A novel double-peak Cauchy ICDF is proposed to dynamically balance the heavy-tailed characteristics of the Cauchy ICDF, enhancing the modeling capability for sudden changes in fog concentration. (2) An innovative Cauchy–Gaussian fusion module is proposed to dynamically learn and generate hybrid coefficients, combining the complementary advantages of the two distributions to dynamically balance the representation of smooth regions and edge details. (3) Tree-based multi-path and cross-resolution feature aggregation is introduced, achieving local–global feature adaptive fusion through adjustable window sizes (3/5/7/11) for parallel paths. Experiments on the RESIDE dataset demonstrate that the proposed method achieves a 2.26 dB improvement in the peak signal-to-noise ratio compared to that obtained with the TaylorV2 expansion attention mechanism, with an improvement of 0.88 dB in heavily hazy regions (fog concentration > 0.8). Ablation studies validate the effectiveness of Cauchy distribution convolution in handling dense fog and conventional lighting conditions. This study provides a new theoretical perspective for modeling in computer vision tasks, introducing a novel attention mechanism and multi-path encoding approach.

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary hyperrectangle. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the hyperrectangle but is otherwise independent of the product measure.

In this paper, dissolved gas analysis (DGA) and statistical distribution model (SDM) were used to predict the health index (HI) of dissolved gas in transformer oil. First, the individual DGA data are classified according to transformer ages ranging from 1 to 4 years. Then, representative fitting models were selected and extrapolated from 5 to 25 years. The inverse cumulative distribution function (ICDF) of the selected distribution model was used to calculate the single conditional parameter data from 5 to 25 years. Finally, the traditional scoring method is used to estimate the future HI value. The results show that DGA parameters can be expressed by exponential equation based on statistical model. The predicted values of DGA health index of transformer oil from 1 to 7 years were basically consistent with the calculated values, and the DGA score was 100 points. By the 20th year, the DGA score had dropped to 75, requiring timely monitoring. The research results can provide powerful data support and theoretical reference for transformer life prediction.

Quantile regression neural networks: Implementation in R and application to precipitation downscaling

This paper proposes a damage detection method based on the train-induced responses of high-speed railway box girders. Under the coupling effects of bending and torsion, the traditional damage detection method based on the Euler beam theory cannot be applied. In this research, the box girder section is divided into different components based on the plate element analysis method. The strain responses were preprocessed based on the principal component analysis (PCA) method to remove the influence of train operation variation. The residual error of the autoregressive (AR) model was used as a potential index of damage features. The optimal order of the model was determined based on the Bayesian information criterion (BIC) criterion. Finally, the confidence boundary (CB) of damage features (DF) constituting outliers can be estimated by the Gaussian inverse cumulative distribution function (ICDF). The numerical simulation results show that the proposed method in this paper can effectively identify, locate and quantify the damage, which verifies the accuracy of the proposed method. The proposed method effectively identifies the early damage of all components on the key section by using four strain sensors, and it is helpful for developing effective maintenance strategies for high-speed railway box girders.