Probability Integral Transformation Research Articles

Hayate: photometric redshift estimation by hybridizing machine learning with template fitting

ABSTRACT Machine learning photo-z methods, trained directly on spectroscopic redshifts, provide a viable alternative to traditional template-fitting methods but may not generalize well on new data that deviates from that in the training set. In this work, we present a Hybrid Algorithm for WI(Y)de-range photo-z estimation with Artificial neural networks and TEmplate fitting (hayate), a novel photo-z method that combines template fitting and data-driven approaches and whose training loss is optimized in terms of both redshift point estimates and probability distributions. We produce artificial training data from low-redshift galaxy spectral energy distributions (SEDs) at z < 1.3, artificially redshifted up to z = 5. We test the model on data from the ZFOURGE surveys, demonstrating that hayate can function as a reliable emulator of eazy for the broad redshift range beyond the region of sufficient spectroscopic completeness. The network achieves precise photo-z estimations with smaller errors (σNMAD) than eazy in the initial low-z region (z < 1.3), while being comparable even in the high-z extrapolated regime (1.3 < z < 5). Meanwhile, it provides more robust photo-z estimations than eazy with the lower outlier rate ($\eta _{0.2}\lesssim 1~{{\ \rm per\ cent}}$) but runs ∼100 times faster than the original template-fitting method. We also demonstrate hayate offers more reliable redshift probability density functions, showing a flatter distribution of Probability Integral Transform scores than eazy. The performance is further improved using transfer learning with spec-z samples. We expect that future large surveys will benefit from our novel methodology applicable to observations over a wide redshift range.

Double Probability Integral Transform Residuals for Regression Models with Discrete Outcomes

The assessment of regression models with discrete outcomes is challenging and has many fundamental issues. With discrete outcomes, standard regression model assessment tools such as Pearson and deviance residuals do not follow the conventional reference distribution (normal) under the true model, calling into question the legitimacy of model assessment based on these tools. To fill this gap, we construct a new type of residuals for regression models with general discrete outcomes, including ordinal and count outcomes. The proposed residuals are based on two layers of probability integral transformation. When at least one continuous covariate is available, the proposed residuals closely follow a uniform distribution (or a normal distribution after transformation) under the correctly specified model. One can construct visualizations such as QQ plots to check the overall fit of a model straightforwardly, and the shape of QQ plots can further help identify possible causes of misspecification such as overdispersion. We provide theoretical justification for the proposed residuals by establishing their asymptotic properties. Moreover, in order to assess the mean structure and identify potential covariates, we develop an ordered curve as a supplementary tool, which is based on the comparison between the partial sum of outcomes and of fitted means. Through simulation, we demonstrate empirically that the proposed tools outperform commonly used residuals for various model assessment tasks. We also illustrate the workflow of model assessment using the proposed tools in data analysis. Supplementary materials for this article are available online.

Modelling spatiotemporal patterns of visceral leishmaniasis incidence in two endemic states in India using environment, bioclimatic and demographic data, 2013-2022.

As of 2021, the National Kala-azar Elimination Programme (NKAEP) in India has achieved visceral leishmaniasis (VL) elimination (<1 case / 10,000 population/year per block) in 625 of the 633 endemic blocks (subdistricts) in four states. The programme needs to sustain this achievement and target interventions in the remaining blocks to achieve the WHO 2030 target of VL elimination as a public health problem. An effective tool to analyse programme data and predict/ forecast the spatial and temporal trends of VL incidence, elimination threshold, and risk of resurgence will be of use to the programme management at this juncture. We employed spatiotemporal models incorporating environment, climatic and demographic factors as covariates to describe monthly VL cases for 8-years (2013-2020) in 491 and 27 endemic and non-endemic blocks of Bihar and Jharkhand states. We fitted 37 models of spatial, temporal, and spatiotemporal interaction random effects with covariates to monthly VL cases for 6-years (2013-2018, training data) using Bayesian inference via Integrated Nested Laplace Approximation (INLA) approach. The best-fitting model was selected based on deviance information criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC) and was validated with monthly cases for 2019-2020 (test data). The model could describe observed spatial and temporal patterns of VL incidence in the two states having widely differing incidence trajectories, with >93% and 99% coverage probability (proportion of observations falling inside 95% Bayesian credible interval for the predicted number of VL cases per month) during the training and testing periods. PIT (probability integral transform) histograms confirmed consistency between prediction and observation for the test period. Forecasting for 2021-2023 showed that the annual VL incidence is likely to exceed elimination threshold in 16-18 blocks in 4 districts of Jharkhand and 33-38 blocks in 10 districts of Bihar. The risk of VL in non-endemic neighbouring blocks of both Bihar and Jharkhand are less than 0.5 during the training and test periods, and for 2021-2023, the probability that the risk greater than 1 is negligible (P<0.1). Fitted model showed that VL occurrence was positively associated with mean temperature, minimum temperature, enhanced vegetation index, precipitation, and isothermality, and negatively with maximum temperature, land surface temperature, soil moisture and population density. The spatiotemporal model incorporating environmental, bioclimatic, and demographic factors demonstrated that the KAMIS database of the national programmme can be used for block level predictions of long-term spatial and temporal trends in VL incidence and risk of outbreak / resurgence in endemic and non-endemic settings. The database integrated with the modelling framework and a dashboard facility can facilitate such analysis and predictions. This could aid the programme to monitor progress of VL elimination at least one-year ahead, assess risk of resurgence or outbreak in post-elimination settings, and implement timely and targeted interventions or preventive measures so that the NKAEP meet the target of achieving elimination by 2030.

The weighted characteristic function of the multivariate PIT: Independence and goodness-of-fit tests

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector X∈Rd with cumulative distribution function FX is free of the marginal distributions. While most of these methods are based on the cdf of W=FX(X), this paper introduces the weighted characteristic function (WCf) of W. A sample version of the WCf of W based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset.

Validation of uncertainty predictions in digital soil mapping

It is quite common in digital soil mapping (DSM) to quantify the uncertainty of issued predictions, that is to make probabilistic predictions. Yet, little attention has been paid to its validation. Probabilistic predictions are only of value for end users if they are reliable and ideally also sharp. Reliability refers to the consistency between predicted conditional probabilities and observed frequencies of independent test data. Sharpness refers to the concentration of a conditional probability distribution function, i.e. its narrowness. The prediction interval coverage probability (PICP) is currently used in DSM to validate the reliability of prediction intervals but it is ignorant of a potential one-sided bias of its boundaries. Therefore, we propose to extend the current validation procedure with metrics used in the broader probabilistic literature. These metrics not only evaluate probabilistic predictions in prediction interval format but also quantiles or full conditional probability distributions. We suggest the quantile coverage probability (QCP) and probability integral transform (PIT) histogram as alternatives to PICP and proper scoring rules for relative comparisons of competing probabilistic models. As scoring rules, we present the interval score (IS) and the continuous ranked probability score (CRPS), which can be decomposed into a reliability part (RELI). We illustrated the use of these metrics in a case study using soil pH and soil organic carbon from the LUCAS-soil database. Thereby, probabilistic predictions of five different models were compared: a reference null model (NM), quantile regression forest (QRF), quantile regression post-processing of a random forest (QRPP RF), kriging with external drift (KED) and quantile regression neural network (QRNN). For KED and QRNN, one-sided bias was found. This was not apparent from PICP but was shown by use of the PIT histogram and QCP. RELI summarized the trends found in QCP, PICP and PIT histograms to one numerical value. CRPS and IS were especially harsh to outliers and low sharpness. According to CRPS and IS, the best probabilistic predictions were obtained by QRF and QRPP RF and the worst by NM.

Tile low-rank approximations of non-Gaussian space and space-time Tukey g-and-h random field likelihoods and predictions on large-scale systems

Large-scale statistical modeling has become necessary with the vast flood of geospace data coming from various sources. In space statistics, the Maximum Likelihood Estimation (MLE) is widely considered for modeling geospace data by estimating a set of statistical parameters related to a predefined covariance function. This covariance function describes the correlation between a set of geospace locations where the main goal is to model given data samples and impute missing data. Climate/weather modeling is a prevalent application for the MLE operation where data interpolation and forecasting are highly required. In the literature, the Gaussian random field is often used to describe geospace data as one of the most popular models for MLE. However, real-life datasets are often skewed and/or have extreme values, and non-Gaussian random field models are more appropriate for capturing such features. In this work, we provide an exact and approximate parallel implementation of the well-known Tukey g-and-h (TGH) non-Gaussian random field in the context of climate/weather applications. The proposed implementation alleviates the computation complexity of the log-likelihood function, which requires O(n2) storage and O(n3) operations, where N is the number of geospace locations, M is the number of time slots, and n=N×M. Based on tile low-rank (TLR) approximations, our implementation of the TGH model can tackle large-scale problems. Furthermore, we rely on task-based programming models and dynamic runtime systems to provide fast execution for the MLE operation in space and space-time cases. We assess the performance and accuracy of the proposed implementations using synthetic space and space-time datasets up to 800K. We also consider a 12-month precipitation dataset in Germany to demonstrate the advantage of using non-Gaussian over Gaussian random field models. We evaluate the prediction accuracy of the TGH model on the precipitation dataset using the Probability Integral Transformation (PIT) tool showing that the TGH model outperforms the Gaussian modeling in the real dataset. Moreover, our performance assessment indicates that TLR computations allow solving larger matrix sizes while preserving the required accuracy for prediction. The TLR-based approximation shows a speedup up to 7.29X and 2.96X over the exact solution.

Robust optimization with order statistic uncertainty set

In this paper, we propose a new uncertainty set for robust models of linear optimization problems. We first study data-free and distribution-free statistical properties of continuous and independent random variables using the Probability Integral Transform. Based on these properties, we construct a new uncertainty set by placing constraints on the order statistics of random variables. We utilize the quantiles of random variables to depict the uncertainties and then adopt the formulation of the assignment problem to develop a tractable formulation for the order statistic uncertainty set. We show that the robust optimization models with the interval uncertainty set, the budget uncertainty set, and the demand uncertainty set can be obtained as special cases of the robust optimization model with the order statistic uncertainty set. Finally, using a robust portfolio construction problem as an example, we show via numerical experiments that the order statistic uncertainty set has better performance than other uncertainty sets when the sample size is small and the correlation between random variables is low.

Weighted Verification Tools to Evaluate Univariate and Multivariate Probabilistic Forecasts for High-Impact Weather Events

Abstract To mitigate the impacts associated with adverse weather conditions, meteorological services issue weather warnings to the general public. These warnings rely heavily on forecasts issued by underlying prediction systems. When deciding which prediction system(s) to utilize when constructing warnings, it is important to compare systems in their ability to forecast the occurrence and severity of high-impact weather events. However, evaluating forecasts for particular outcomes is known to be a challenging task. This is exacerbated further by the fact that high-impact weather often manifests as a result of several confounding features, a realization that has led to considerable research on so-called compound weather events. Both univariate and multivariate methods are therefore required to evaluate forecasts for high-impact weather. In this paper, we discuss weighted verification tools, which allow particular outcomes to be emphasized during forecast evaluation. We review and compare different approaches to construct weighted scoring rules, both in a univariate and multivariate setting, and we leverage existing results on weighted scores to introduce conditional probability integral transform (PIT) histograms, allowing forecast calibration to be assessed conditionally on particular outcomes having occurred. To illustrate the practical benefit afforded by these weighted verification tools, they are employed in a case study to evaluate probabilistic forecasts for extreme heat events issued by the Swiss Federal Office of Meteorology and Climatology (MeteoSwiss).

A Bayesian Variable Selection Method for Spatial Autoregressive Quantile Models

In this paper, a Bayesian variable selection method for spatial autoregressive (SAR) quantile models is proposed on the basis of spike and slab prior for regression parameters. The SAR quantile models, which are more generalized than SAR models and quantile regression models, are specified by adopting the asymmetric Laplace distribution for the error term in the classical SAR models. The proposed approach could perform simultaneously robust parametric estimation and variable selection in the context of SAR quantile models. Bayesian statistical inferences are implemented by a detailed Markov chain Monte Carlo (MCMC) procedure that combines Gibbs samplers with a probability integral transformation (PIT) algorithm. In the end, empirical numerical examples including several simulation studies and a Boston housing price data analysis are employed to demonstrate the newly developed methodologies.

Mapping variations of redshift distributions with probability integral transforms

ABSTRACT We present a method for mapping variations between probability distribution functions and apply this method within the context of measuring galaxy redshift distributions from imaging survey data. This method, which we name PITPZ for the probability integral transformations it relies on, uses a difference in curves between distribution functions in an ensemble as a transformation to apply to another distribution function, thus transferring the variation in the ensemble to the latter distribution function. This procedure is broadly applicable to the problem of uncertainty propagation. In the context of redshift distributions, for example, the uncertainty contribution due to certain effects can be studied effectively only in simulations, thus necessitating a transfer of variation measured in simulations to the redshift distributions measured from data. We illustrate the use of PITPZ by using the method to propagate photometric calibration uncertainty to redshift distributions of the Dark Energy Survey Year 3 weak lensing source galaxies. For this test case, we find that PITPZ yields a lensing amplitude uncertainty estimate due to photometric calibration error within 1 per cent of the truth, compared to as much as a 30 per cent underestimate when using traditional methods.

A Bayesian Hierarchical Model Combination Framework for Real‐Time Daily Ensemble Streamflow Forecasting Across a Rainfed River Basin

AbstractThe frequent occurrence of floods during the rainy season is one of the threats in rainfed river basins, especially in river basins of India. This study implemented a Bayesian hierarchical model combination (BHMC) framework to generate skillful and reliable real‐time daily ensemble streamflow forecast and peak flow and demonstrates its utility in the Narmada River basin in Central India for the peak monsoon season (July–August). The framework incorporates information from multiple sources (e.g., deterministic hydrological forecast, meteorological forecast, and observed data) as predictors. The forecasts were validated with a leave‐1‐year‐out cross‐validation using accuracy metrics such as BIAS and Pearson correlation coefficient (R) and probabilistic metrics such as continuous ranked probability skill score, probability integral transform (PIT) plots, and the average width of the 95% confidence intervals (AWCI) plots. The results show that the BHMC framework can increase the forecast skill by 40% and reduce absolute bias by at least 28% compared to the raw deterministic forecast from a physical model, the Variable Infiltration Capacity model. In addition, PIT and AWCI show that the framework can provide sharp and reliable streamflow forecast ensembles for short lead times (1–3‐day lead time) and provide useful skills beyond up to 5‐day lead time. These will be of immense help in emergency and disaster preparedness.

Reference class selection in similarity‐based forecasting of corporate sales growth

AbstractThis paper proposes a general method to handle forecasts exposed to behavioral bias by finding appropriate outside views, in our case corporate sales forecasts of analysts. The idea is to find reference classes, that is, peer groups, for each analyzed company separately that share similarities to the firm of interest with respect to a specific predictor. The classes are regarded to be optimal if the forecasted sales distributions match the actual distributions as closely as possible. The forecast quality is measured by applying goodness‐of‐fit tests on the estimated probability integral transformations and by comparing the predicted quantiles. The method is out‐of‐sample backtested on a data set consisting of 21,808 US firms over the time period 1950–2019, which is also descriptively analyzed. It appears that, in particular, the past operating margins are good predictors for the distribution of future sales. A case study compares the outside view of our distributional forecasts with actual analysts' forecasts and emphasizes the relevance of our approach in practice.

Is Jump Robust Two Times Scaled Estimator Superior among Realized Volatility Competitors?

This paper compares the empirical performance of the realized volatility estimators on an extensive high-frequency dataset of stock indices from four developed European markets with thick trading and intensive intraday activity. Even though the proposed estimators have distinctive properties, it is not clear which one has a better performance in terms of unbiasedness and consistency. Some of them are robust to microstructure noise only, and others are robust solely to price jumps, whereas a few of them are robust to both. Therefore, the main purpose is finding a benchmark estimator among alternative competitors, as the best proxy of integrated variance, and empirical demonstration of its superiority. The vast majority of the existing studies largely rely on developed US data or simulation data, but inferences obtained on such data might deviate from European developed markets. This study aims to fill in that niche. In particular, the optimal sampling frequency of proposed benchmark estimator is determined with respect to the trade-off between its bias and the variance of each stock index individually. Afterwards, probability integral transformation, Mincer–Zarnowitz regression and upper tail correlation from appropriate copula function are considered as an adequate pairwise comparison methods. Notable contributions of this paper include unambiguously proven superiority of robust two times scaled estimator for selected European developed markets within the range of optimal slow time frequency from 10 to 30 s. Finally, recommendations for research and practitioners regarding the usage of jump robust two times scaled estimator are given. In fact, asset managers, institutional investors as well as market regulators could benefit from proposed realized volatility benchmark in making long-term investment decisions, leading to sustainable finance.

Graphical test for discrete uniformity and its applications in goodness-of-fit evaluation and multiple sample comparison

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.

Fuzzy improved distribution function and order statistics

A new concept of fuzzy improved distribution function based on fuzzy ordering defined on the support set of a given random variable is introduced. A fuzzy improved random variable based on this distribution function is defined. The existence of such a fuzzy improved random variable follows from probability integral transformation. The order statistics based on the fuzzy ordering and fuzzy improved distribution function are considered and their distributions are studied. The need for fuzzy improved order statistics arises in reliability analysis of sensitive systems where the lifetimes of the components and the system are considered by taking into account some degree of smallness defined by membership function of a fuzzy set. Examples and graphical representations of fuzzy improved distribution functions are provided.

Financial Asset Risk Measurement Based on Smart Sensor Big Data Security Analysis and Bayesian Posterior Probability Model

Following the speeding up of a process of financial globalization, the risks faced by financial markets have become more complex and diversified. Correlated patterns among financial assets exhibit characteristics of nonlinearity, asymmetry, and tail correlation. The original linear correlation analysis method is no longer suitable, but relevant information describing financial risks. In order to confirm whether an asset is safe, the key is to study and master its volatility, and this is based on our mastery of volatility measurement skills. This article is based on smart sensor big data security analysis and Bayesian analysis. The risk measurement of financial assets based on the empirical probability model is studied. The GARCH-t(1,1) model is selected according to the Akaike information criterion (AIC) after the generalized autoregressive conditional heteroskedasticity (GARCH) model is established by the EViews software. According to the results of probability integral transformation, a series of correlation coefficients and degrees of freedom oft-copula are obtained by the maximum likelihood estimation method. This paper uses the risk-adjusted return on capital (RAROC) method to evaluate the risk performance of financial assets. Financial institutions can only retain and absorb the financial market risks that cannot be avoided and transferred. The edge user node sends the service request to the edge server node. The edge server uses the model proposed in this paper to evaluate the user’s trust and selects the corresponding service level according to the trust level corresponding to the calculated credibility results. The data show that the edge calculation takes 0.2581 seconds, while the linear search takes about 64 seconds. The results show that intelligent edge computing improves the accuracy and efficiency of financial asset risk measurement.

Copula Modelling to Analyse Financial Data

Copula modelling is a popular tool in analysing the dependencies between variables. Copula modelling allows the investigation of tail dependencies, which is of particular interest in risk and survival applications. Copula modelling is also of specific interest to economic and financial modelling as it can help in the prediction of financial contagion and periods of “boom” or “bust”. Bivariate copula modelling has a rich variety of copulas that may be chosen to represent the modelled dataset dependencies and possible extreme events that may lie within the dataset tails. Financial copula modelling tends to diverge as this richness of copula types within the literature may not be well realised with the two different types of modelling, one being non-time-series and the other being time-series, being undertaken differently. This paper investigates standard copula modelling and financial copula modelling and shows why the modelling strategies in using time-series and non-time-series copula modelling is undertaken using different methods. This difference, apart from the issues surrounding the time-series component, is mostly due to standard copula modelling having the ability to use empirical CDFs for the probability integral transformation. Financial time-series copula modelling uses pseudo-CDFs due to the standardized time-series residuals being centred around zero. The standardized residuals inhibit the estimation of the possible distributions required for constructing the copula model in the usual manner.

Photometric redshifts for the S-PLUS Survey: Is machine learning up to the task?

The Southern Photometric Local Universe Survey (S-PLUS) is a novel project that aims to map the Southern Hemisphere using a twelve filter system, comprising five broad-band SDSS-like filters and seven narrow-band filters optimized for important stellar features in the local universe. In this paper we use the photometry and morphological information from the first S-PLUS data release (S-PLUS DR1) cross-matched to unWISE data and spectroscopic redshifts from Sloan Digital Sky Survey DR15. We explore three different machine learning methods (Gaussian Processes with GPz and two Deep Learning models made with TensorFlow) and compare them with the currently used template-fitting method in the S-PLUS DR1 to address whether machine learning methods can take advantage of the twelve filter system for photometric redshift prediction. Using tests for accuracy for both single-point estimates such as the calculation of the scatter, bias, and outlier fraction, and probability distribution functions (PDFs) such as the Probability Integral Transform (PIT), the Continuous Ranked Probability Score (CRPS) and the Odds distribution, we conclude that a deep-learning method using a combination of a Bayesian Neural Network and a Mixture Density Network offers the most accurate photometric redshifts for the current test sample. It achieves single-point photometric redshifts with scatter ($\sigma_\text{NMAD}$) of 0.023, normalized bias of -0.001, and outlier fraction of 0.64% for galaxies with r-auto magnitudes between 16 and 21.

Assessing Goodness of Fit for Verifying Probabilistic Forecasts

The verification of probabilistic forecasts in hydro-climatology is integral to their development, use, and adoption. We propose here a means of utilizing goodness of fit measures for verifying the reliability of probabilistic forecasts. The difficulty in measuring the goodness of fit for a probabilistic prediction or forecast is that predicted probability distributions for a target variable are not stationary in time, meaning one observation alone exists to quantify goodness of fit for each prediction issued. Therefore, we suggest an additional dissociation that can dissociate target information from the other time variant part—the target to be verified in this study is the alignment of observations to the predicted probability distribution. For this dissociation, the probability integral transformation is used. To measure the goodness of fit for the predicted probability distributions, this study uses the root mean squared deviation metric. If the observations after the dissociation can be assumed to be independent, the mean square deviation metric becomes a chi-square test statistic, which enables statistically testing the hypothesis regarding whether the observations are from the same population as the predicted probability distributions. An illustration of our proposed rationale is provided using the multi-model ensemble prediction for El Niño–Southern Oscillation.

Adapting The Extended Neyman’s Smooth Test to Be Used in Accelerated Failure Time Models

Accelerated life testing (ALT) has gained greater importance because of dealing with high reliability units. As a result, there is a big need to use a goodness of fit (GOF) technique for testing the underlying lifetime distribution. But there is a difficulty due to the existence of several stress levels with different samples of units at each level. Then, the choice of a certain GOF technique is based on its capability to combine the failure times from all stress levels to reach a conclusion about the adequacy of a certain lifetime distribution at each stress level. In this paper, the extended Neyman’s smooth test (ENST) is chosen. It is then modified in order to be used in validating the distributional assumption of accelerated failure time (AFT) model. This modified method is called; the adapted extended Neyman’s smooth test (AENST). It is applied to test for both Weibull and exponential distributions in case of constant stress under complete sampling. To check the performance of the AENST, a comparison is made with the conditional probability integral transformation test (CPITT) via a simulation study. Moreover, a real data set is provided to illustrate the application of the introduced AENST. The results revealed that the AENST is a powerful test comparing with the CPITT. Thus, the AENST is recommended for testing the AFT models.