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Response time models (RTMs) are of increasing interest in educational and psychological testing. This article focuses on the lognormal model for response times, which is one of the most popular RTMs. Several existing statistics for testing normality and the fit of factor analysis models are repurposed for testing the fit of the lognormal model. A simulation study and two real data examples demonstrate the usefulness of the statistics. The Shapiro–Wilk test of normality and a z-test for factor analysis models were the most powerful in assessing the misfit of the lognormal model.

Current insurance models, assuming that inter-arrival time of claims, are distributed randomly and thus well approximated by Poisson processes. Here we provide clear proof that the timing of inter-claims fits by non-Poisson patterns, marked by rapid events, separated by long periods of inactivity. The time of inter-arrival claims will be heavy tailed, most claims will be executed quickly, while a few will have very long waiting times. We will model and analysis of insurance based on claim inter-arrival time, the time interval between two successive claims and the ability to carry out such modeling was limited by a lack of ecologically relevant data collected on claims inter-arrival. We propose a structured process behavior model based on data from Egyptian fire insurance company. Our analysis shows that claim activities can be represented by non-Poisson processes and that the subsequent distribution of inter-arrival activity times follows the Pareto distribution. These results will help researchers understand daily behavioral trends and create more sophisticated predictive models of claims.

This paper analyzes credit rating default dependencies in a multisectoral framework. Using Mergent's FISD database, we study the default series in the U.S. over the last two decades, disaggregating defaults by industry-sector group. During this period, two main waves of default occurred: the implosion of the “dot-com” bubble and the global financial crisis. We estimate a Multivariate Autoregressive Conditional Poisson model according to the biweekly number of defaults that occurred in different sectors of the economy from 1996 to 2015. We discuss the contagion effect between sectors in two ways: the degree of transmission of the probability of default from one sector to another, i.e., the “infectivity” of the sector, and the degree of contagion of one sector from another, i.e., the “vulnerability” of the sector. Our results show differences between the sectors' relations during the first and second part of our sample. We add some exogenous variables to the analysis and evaluate their contribution to the goodness of fit.

In the quest to model various phenomena, the foundational importance of parameter identifiability to sound statistical modeling may be less well appreciated than goodness of fit. Identifiability concerns the quality of objective information in data to facilitate estimation of a parameter, while nonidentifiability means there are parameters in a model about which the data provide little or no information. In purely empirical models where parsimonious good fit is the chief concern, nonidentifiability (or parameter redundancy) implies overparameterization of the model. In contrast, nonidentifiability implies underinformativeness of available data in mechanistically derived models where parameters are interpreted as having strong practical meaning. This study explores illustrative examples of structural nonidentifiability and its implications using mechanistically derived models (for repeated presence/absence analyses and dose-response of Escherichia coli O157:H7 and norovirus) drawn from quantitative microbial risk assessment. Following algebraic proof of nonidentifiability in these examples, profile likelihood analysis and Bayesian Markov Chain Monte Carlo with uniform priors are illustrated as tools to help detect model parameters that are not strongly identifiable. It is shown that identifiability should be considered during experimental design and ethics approval to ensure generated data can yield strong objective information about all mechanistic parameters of interest. When Bayesian methods are applied to a nonidentifiable model, the subjective prior effectively fabricates information about any parameters about which the data carry no objective information. Finally, structural nonidentifiability can lead to spurious models that fit data well but can yield severely flawed inferences and predictions when they are interpreted or used inappropriately.

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokh...

Because model misspecification can lead to inconsistent and inefficient estimators and invalid tests of hypotheses, testing for misspecification is critically important. We focus here on several general purpose goodness-of-fit tests which can be applied to assess the adequacy of a wide variety of parametric models without specifying an alternative model. Parametric bootstrap is the method of choice for computing the p-values of these tests however the proof of its consistency has never been rigourously shown in this setting. Using properties of locally asymptotically normal parametric models, we prove that under quite general conditions, the parametric bootstrap provides a consistent estimate of the null distribution of the statistics under investigation.

This chapter concerns with goodness of fit tests that can be used to test whether a proposed model is consistent with data. In it the classical chi–square goodness of fit test and apply it to test for independence in contingency tables is presented. The chapter also introduces the Kolmogorov–Smirnov procedure for testing whether data comes from a specified continuous probability distribution. The classical approach to obtaining a goodness of fit test of a null hypothesis that a sample has a specified probability distribution is to partition the possible values of the random variables into a finite number of regions. The situation where each member of a population is classified according to two distinct characteristics is also considered and it is shown how to use analysis to test the hypothesis that the characteristics of a randomly chosen member of the population are independent. The chapter also provides a practice exercise and solved examples.

With tremendous associated economic and social costs of crashes, researchers have been trying not only to identify the factors affecting crashes, but also to estimate those coefficients in the most accurate ways. Estimating model coefficients without accounting for a correct distribution would result in biased and erroneous results. This risk especially holds true when modeling skewed equivalent property damage only (EPDO) crashes with a preponderance of zeroes. The distribution of EPDO is known for not being modeled with known distributions such as Poisson or negative binomial distributions. This issue is highlighted in particular for a mountainous state like Wyoming that has very low traffic levels and a severely high crash rate. In addition, we included barriers in the model that did not experience any crashes but did suffer from being under-designed by geometric architects, thereby adding to the number of zero count observations. Various models with different distributional characteristics were considered and compared in this study. Comparisons were not just made across models in terms of their goodness of fit, but the estimated coefficients were also compared to see the impact of considering the wrong distributional assumptions on model parameter estimates. As the objectives of this study are to implement the identified results for optimization purposes and locate hazardous locations that could host future crashes, the results highlight accurate model estimations and the consequences of a failure to account for the right distributions. After conducting different goodness-of-fit measures, a hurdle model was proposed in this study to accommodate observations with zero crashes, and to account for a sparse distribution of EPDO crashes in the state of Wyoming. For the hurdle model, binary logistic regression was used to account for observations with zero crashes, while the negative binomial method was considered for non-zero observations. The findings of this study have direct implications on the allocation of limited funds for policymakers in Wyoming, as optimization could be conducted on the geometric characteristics of traffic barriers in future studies.

Power law distributions have previously been observed in data like city-size distributions (Zipf’s Law), income distributions, and financial asset prices. In this paper, we explore the distribution of real estate prices in Charleston County, South Carolina. We fit power law, lognormal and exponential distributions to the data and compare the goodness of fit among the distributions. We find that the best fit distribution lies somewhere between the lognormal and power law distributions. We estimate how the power law exponent changes over time and find a potential relationship between the shape of the power law distribution and the bursting of the real estate bubble in 2007.

We propose a nonparametric estimator of the empirical distribution function (EDF) of the latent spot variance of the log-price of a financial asset. We show that over a fixed time span our realized EDF (or REDF) – inferred from noisy high-frequency data – is consistent as the mesh of the observation grid goes to zero. In a double-asymptotic framework, with time also increasing to infinity, the REDF converges to the cumulative distribution function of volatility, if it exists. We exploit these results to construct some new goodness-of-fit tests for stochastic volatility models. In a Monte Carlo study, the REDF is found to be accurate over the entire support of volatility. This leads to goodness-of-fit tests that are both correctly sized and relatively powerful against common alternatives. In an empirical application, we recover the REDF from stock market high-frequency data. We inspect the goodness-of-fit of several two-parameter marginal distributions that are inherent in standard stochastic volatility models. The inverse Gaussian offers the best overall description of random equity variation, but the fit is less than perfect. This suggests an extra parameter (as available in, e.g., the generalized inverse Gaussian) is required to model stochastic variance.

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov–Smirnov (KS) statistics is provided under Type-I censoring. The alt...

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump-diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump-diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models. Keywords: Option pricing model, Stochastic interest rate, Stochastic volatility, Double exponential jump, Markov Chain Monte Carlo with Latent Variable

Soybean yield in the south region of Brazil was analyzed by using Gaussian spatial linear models to account for the spatial dependence between the observations. Some special cases of Gaussian spatio-temporal linear models were also considered. The data set was collected over five years and we regarded four explanatory variables. Each year was considered to be a realization of the process. We checked the goodness of fit and since we noted the presence of an atypical year with lower productivity than in the other years, we conducted influence diagnostics analysis. Moreover, we considered two appropriate perturbation schemes in the response variable and case weight perturbation. The results showed that the methods were effective for detecting influential observations.

In recent years, with a growing interest in big or large datasets, there has been a rise in the application of large graphs and networks to financial big data. Much of this research has focused on the construction and analysis of the network structure of stock markets, based on the relationships between stock prices. Motivated by Boginski et al. (2005), who studied the characteristics of a network structure of the US stock market, we construct network graphs of the UK stock market using same method. We fit four distributions to the degree density of the vertices from these graphs, the Pareto I, Frechet, lognormal, and generalised Pareto distributions, and assess the goodness of fit. Our results show that the degree density of the complements of the market graphs, constructed using a negative threshold value close to zero, can be fitted well with the Frechet and lognormal distributions.

The paper is devoted to the supremum-type multivariate goodness-of-fit tests based on the empirical characteristic function. Particular attention is devoted to the composite hypothesis of normality and Gaussian distribution mixture model. An analytical way to approximate the null asymptotic distributions of the considered test statistics is discussed applying the theory of large excursions of differentiable Gaussian random fields. The produced comparative Monte Carlo power study shows that the considered tests are powerful competitors to the existing classical criteria, clearly dominating in verification of the goodness-of-fit hypotheses against the specific types of alternatives.

Threshold estimation in the Peaks Over Threshold (POT) method and the impact of the estimation method on the calculation of high return period quantiles and their uncertainty (or confidence intervals) are issues that are still unresolved. In the past, methods based on goodness of fit tests and EDF-statistics have yielded satisfactory results, but their use has not yet been systematized. This paper proposes a methodology for automatic threshold estimation, based on the Anderson-Darling EDF-statistic and goodness of fit test. When combined with bootstrapping techniques, this methodology can be used to quantify both the uncertainty of threshold estimation and its impact on the uncertainty of high return period quantiles. This methodology was applied to several simulated series and to four precipitation/river flow data series. The results obtained confirmed its robustness. For the measured series, the estimated thresholds corresponded to those obtained by nonautomatic methods. Moreover, even though the uncertainty of the threshold estimation was high, this did not have a significant effect on the width of the confidence intervals of high return period quantiles.

This paper fits the three-parameter log-logistic (3LL) distribution to sulphur dioxide (SO 2 ) monthly emissions in kilograms per gigawatt hour (kg/GWh) and in milligrams per cubic nano metre (mg/Nm 3 ), at 13 of Eskom’s coal fired power-generating stations in South Africa. The aim is to quantify and describe the emission of sulphur dioxide at these stations using a statistical distribution, and to also estimate the probabilities of extreme emissions and exceedances (emissions above a certain threshold). Using the 3LL distribution is proposed as such a distribution. The log-logistic distribution is a special form of a Burr-type distribution. Various goodness-of-fit measures, including the Kolmogorov Smirnov, the Anderson Darling and some graphical tests, are employed to test if the 3LL distribution is a good fit to the data. The maximum likelihood method is used to estimate the parameters. The distribution fit is important as it then becomes possible to quantify and manage the SO 2 emissions effectively. The 3LL distribution, which is compared with three other distributions, gave the best overall fit to most of the power stations. Keywords: emission, Eskom, log logistic distribution, goodness of fit, sulphur dioxide, Burr-type distribution Highlights Quantification of SO 2 emissions in terms of a statistical distribution Calculating the probability of SO 2 emissions exceeding certain specified limits Ranking power stations in terms of SO 2 emissions efficiency

Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.

Though rare, crashes at highway–railway crossings usually result in severe injuries and fatalities. Intersections along railway corridors share many conditions that support the use of spatial analyses, but so far the spatial effects are essentially unknown. Little research has been conducted about highway–railway crossings using spatial correlation. The study presented in this paper analyzed spatial correlation structures on crash frequency at railway crossings, particularly conditional autoregressive (CAR) and joint specification, with the use of Gaussian Kriging models. Full Bayesian Poisson–lognormal approaches were used to compare the effects of various models, including heterogeneity-only, spatial-only, and heterogeneity–spatial models. These methods were estimated with crash data from a low-speed passenger train service in Costa Rica. The deviance information criterion was used to compare the models. Heterogeneity–CAR models show a better goodness of fit than heterogeneity-only, joint Kriging, and CAR-only definitions. The second-order neighboring model (CAR-2) yielded the best fit and related to an average neighbor distance of about 700 m (0.44 mi). Robust semi-variograms based on joint Kriging models and CAR methods showed similar results. The proportion of variation in the data explained by spatial correlation methods was similar (29% of the total variation). These findings suggest that spatial correlation at highway–railway crossings should be considered in modeling of crash frequencies.

For comparing two cumulative hazard functions, we consider an extension of the Kullback-Leibler information to the cumulative hazard function, which is concerning the ratio of cumulative hazard functions. Then we consider its estimate as a goodness of fit test with the Type II censored data. For an exponential null distribution, the proposed test statistic is shown to outperform other test statistics based on the empirical distribution function in the heavy censoring case against the increasing hazard alternatives.

Statistical independence test and validity of the CA (Cellular Automata) Markov process for projecting future land use and land cover (LULC) changes were carried out in this study. Predicting quantity and location changes have been analyzed, and statistically evaluated. Validity of the CA Markov process has been examined using various Kappa Index of Agreement (KIA or Kstandard) and related statistical variations on the KIA. Statistical test of independence ( K 2 ) was performed and markovian suitability has been checked using hypothesis of goodness of fit ( Xc 2 ). Hypothesis of statistical independence was rejected, which proved that land use land cover change trends are similar like previous development of land. With acceptance of the hypothesis of goodness of fit ( Xc 2 ) proved that actual transition probability of matrix is fitted with expected transition probability prepared using Markov chain method. Statistics indicates Kno, Klocation, Klocation Strata and Kstandard are 0.8347, 0.859, 0.8591 and 0.7928, respectively.

Substantial evidence shows that the frequency of hydrological extremes has been changing and is likely to continue to change in the near future. Non-stationary models for flood frequency analyses are one method of accounting for these changes in estimating design values. The objective of the present study is to compare four models in terms of goodness of fit, their uncertainties, the parameter estimation methods and the implications for estimating flood quantiles. Stationary and non-stationary models using the GEV distribution were considered, with parameters dependent on time and on annual precipitation. Furthermore, in order to study the influence of the parameter estimation approach on the results, the maximum likelihood (MLE) and Bayesian Monte Carlo Markov chain (MCMC) methods were compared. The methods were tested for two gauging stations in Slovenia that exhibit significantly increasing trends in annual maximum (AM) discharge series. The comparison of the models suggests that the stationary model tends to underestimate flood quantiles relative to the non-stationary models in recent years. The model with annual precipitation as a covariate exhibits the best goodness-of-fit performance. For a 10% increase in annual precipitation, the 10-year flood increases by 8%. Use of the model for design purposes requires scenarios of future annual precipitation. It is argued that these may be obtained more reliably than scenarios of extreme event precipitation which makes the proposed model more practically useful than alternative models.

Levy models are frequently used for asset log-returns. An important criterion is the distributional assumption on the increments. Candidates include, for example, the generalized hyperbolic, the normal inverse Gaussian, and the (skew) Student–Levy process. We perform a comparative study for multiple equity indices of different countries using different Levy models to determine the best fit using the Kolmogorov–Smirnov statistic, the Anderson–Darling statistic, and the Bayesian information criterion as goodness-of-fit measures. We fit Levy models both to daily and to hourly log-returns. To date, the literature has paid little attention to the question of whether these Levy models for daily returns also fit well at higher frequencies, that is, intraday returns, and vice versa. Eberlein and Ozkan (Quant Finance 3(1):40–50, 2003) called this “time consistency.” Our key finding is that there are time inconsistencies. This means that some models that fit well for daily returns, for example, the variance gamma model, fit poorly for hourly returns. We find that the Student–Levy process is a more appropriate alternative.

Parameter shrinkage applied optimally can always reduce error and projection variances from those of maximum likelihood estimation. Many variables that actuaries use are on numerical scales, like age or year, which require parameters at each point. Rather than shrinking these towards zero, nearby parameters are better shrunk towards each other. Semiparametric regression is a statistical discipline for building curves across parameter classes using shrinkage methodology. It is similar to but more parsimonious than cubic splines. We introduce it in the context of Bayesian shrinkage and apply it to joint mortality modeling for related populations. Bayesian shrinkage of slope changes of linear splines is an approach to semiparametric modeling that evolved in the actuarial literature. It has some theoretical and practical advantages, like closed-form curves, direct and transparent determination of degree of shrinkage and of placing knots for the splines, and quantifying goodness of fit. It is also relatively easy to apply to the many nonlinear models that arise in actuarial work. We find that it compares well to a more complex state-of-the-art statistical spline shrinkage approach on a popular example from that literature., Comment: 39 pages, 8 graphs

The application of the nonparametric Anderson–Darling, Cramer–Mises–Smirnov, Kuiper, Watson, Kolmogorov, and Zhang goodness-of-fit tests in verification of simple and composite hypotheses is considered. Based on an investigation of the power, it is shown for the first time that there exist pairs of competing hypotheses which these tests are not able to distinguish in the case of small sample sizes n and type 1 error probabilities. It is shown that the reason for this lies in the bias of the tests in corresponding situations.

In this study Variance-Gamma (VG) and Normal-Inverse Gaussian (NIG) distributions are compared with the benchmark of generalized hyperbolic distribution in terms of their fit to the empirical distribution of high-frequency stock market index returns in China. First, we estimate the considered models in a Markov regime switching framework for the identification of different volatility regimes. Second, the goodness-of-fit results are compared at different time scales of log-returns. Third, the goodness-of-fit results are validated through bootstrapping experiments. Our results show that as the time scale of log-returns decrease NIG model outperforms the VG model consistently and the difference between the goodness-of-fit statistics increase. For high-frequency Chinese index returns, NIG model is more robust and provides a better fit to the empirical distributions of returns at different time scales.

This study investigated the effects of including spatial heterogeneity in multivariate random parameters models and their influence on different collision severity levels. The models were developed for severe (injury and fatal) and no-injury collisions using three years of collision data from the city of Vancouver. Three different modeling formulations were applied to measure the effects of spatial heterogeneity in a multivariate random parameters model. The proposed models were estimated in a Full Bayesian (FB) context using Markov Chain Monte Carlo (MCMC) simulation. The Deviance Information Criteria (DIC) values indicated that all the models were comparable to one another. Therefore, no particular model can be distinctly preferred over others. According to parameter estimates, a variety of traffic and road geometric covariates were found to significantly influence collision severities. The variance for spatial heterogeneity was higher than the variance for heterogeneous effects. The correlation between severe and no-injury collisions for the total random effects (heterogeneous and spatial) was significant and quite high, indicating that higher no-injury collisions are associated with higher severe collisions. These results support the incorporation of spatial heterogeneity in multivariate random parameters models. Furthermore, the multivariate random parameters spatial models were compared with two independent univariate random parameters spatial models with respect to model inference and goodness of fit. The multivariate spatial models outperformed the two univariate spatial models with a very significant drop in the DIC value.

Power-law distributions can be generated by a variety of theoretical processes and have been found in many areas of economics and finance. This paper demonstrates that the distribution of cumulative oil and gas recovery in Texas is best described by a power-law distribution, with exponent approximately equal to 1.1 for oil production and 1.6 for gas production. Estimation is carried out using lease and well-level data from a cross section of over 600,000 observations gathered by DrillingInfo. The goodness of fit of the hypothesized power law is verified with regression-based and likelihood ratio tests, as well as nested and composite hypotheses. Results are significant because they show that production data are heavy tailed, that empirical variance estimates do not converge, and that 1 % of oil leases are responsible for 70 % of cumulative recovery. The distribution has consequences for efficient management and implications about peak oil because wells close to the mean account for so little of the cumulative distribution of recovery.

Background. Calibration (estimation of model parameters) compares model outcomes with observed outcomes and explores possible model parameter values to identify the set of values that provides the best fit to the data. The goodness-of-fit (GOF) criterion quantifies the difference between model and observed outcomes. There is no consensus on the most appropriate GOF criterion, because a direct performance comparison of GOF criteria in model calibration is lacking. Methods. We systematically compared the performance of commonly used GOF criteria (sum of squared errors [SSE], Pearson chi-square, and a likelihood-based approach [Poisson and/or binomial deviance functions]) in the calibration of selected parameters of the MISCAN-Colon microsimulation model for colorectal cancer. The performance of each GOF criterion was assessed by comparing the 1) root mean squared prediction error (RMSPE) of the selected parameters, 2) computation time of the calibration procedure of various calibration scenarios, and 3) impact on estimated cost-effectiveness ratios. Results. The likelihood-based deviance resulted in the lowest RMSPE in 4 of 6 calibration scenarios and was close to best in the other 2. The SSE had a 25 times higher RMSPE in a scenario with considerable differences in the values of observed outcomes, whereas the Pearson chi-square had a 60 times higher RMSPE in a scenario with multiple studies measuring the same outcome. In all scenarios, the SSE required the most computation time. The likelihood-based approach estimated the cost-effectiveness ratio most accurately (up to −0.15% relative difference versus 0.44% [SSE] and 13% [Pearson chi-square]). Conclusions. The likelihood-based deviance criteria lead to accurate estimation of parameters under various circumstances. These criteria are recommended for calibration in microsimulation disease models in contrast with other commonly used criteria.

In this paper we present a semiparametric test of goodness of fit which is based on the method of L-moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L-moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L-moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.

This study uses truncated Poisson regression to correct endogeneity of discrete choice experiments (DCEs) with data from on-site sampling, which has been commonly used to collect samples in the field of tourism research. We find that goodness of fit of model is significantly improved with on-site sampling correction. Estimated parameters with onsite sampling correction also match the predictions of economic theory better. Estimated willingness to pay for attributes are substantially underestimated due to sample selection bias. These results question the use of DCEs without on-site sampling correction for tourism policy making.

This paper discusses a time series model which has generalized long memory in the mean process with stochastic volatility errors and develops a new Bayesian posterior simulator that couples advanced posterior maximisation techniques, as well as traditional latent stochastic volatility estimation procedures. Details are provided on the estimation process, data simulation, and out of sample performance measures. We conduct several rigorous simulation studies and verify our results for in and out of sample behaviour. We further compare the goodness of fit of the generalized process to the standard long memory model by considering two empirical studies on the US Consumer Price Index (CPI) and the US equity risk premium (ERP).

This paper introduces mortality dependence in multi-country mortality modeling using a dynamic copula approach. Specifically, we use time-varying copula models to capture the mortality dependence structure across countries, examining both symmetric and asymmetric dependence structures. In addition, to capture the phenomenon of a heavy tail for the multi-country mortality index, we consider not only the setting of Gaussian innovations but also non-Gaussian innovations under the Lee–Carter framework model. As tests of the goodness of fit of different dynamic copula models, the pattern of mortality dependence, and the distribution of the innovations, we used empirical mortality data from Finland, France, the Netherlands, and Sweden. To understand the effect of mortality dependence on longevity derivatives, we also built a valuation framework for pricing a survivor index swap, then investigated the fair swap rates of a survivor swap numerically. We demonstrate that failing to consider the dynamic copula mortality model and non-Gaussian innovations would lead to serious underestimations of the swap rates and loss reserves.

We introduce a new goodness-of-fit test for regular vine (R-vine) copula models, a flexible class of multivariate copulas based on a pair-copula construction (PCC). The test arises from the information matrix ratio and assumes fixed margins. The corresponding test statistic is derived and its asymptotic normality is shown. The test's power is investigated and compared to 14 other goodness-of-fit tests, adapted from the bivariate copula case, in a high-dimensional setting. The extensive simulation study on the copula level shows the excellent performance with respect to size and power as well as the superiority of the information matrix ratio based test against most other goodness-of-fit tests. The best performing tests are applied to a portfolio of stock indices and their related volatility indices validating different R-vine specifications. We present a new goodness-of-fit test for regular vine copula models.The test arises from the information matrix ratio.The test's power is investigated and compared to 14 other goodness-of-fit tests.A simulation study shows the excellent performance with respect to size and power.An application to stock indices and their related volatility indices is performed.

The logistic distribution has been used to model growth curves in survival analysis and biological studies. In this article, we propose a goodness-of-fit test for the logistic distribution based on the empirical likelihood ratio. The test is constructed based on the methodology introduced by Vexler and Gurevich [17]. In order to compute the test statistic, parameters of the distribution are estimated by the method of maximum likelihood. Power comparisons of the proposed test with some known competing tests are carried out via simulations. Finally, an illustrative example is presented and analyzed.

Some sigmoidal growth models have been commonly used for forecasting energy consumption. In this article, a new sigmoidal growth model, which is an extended form of the Boltzmann 4-parameter function, has been developed for describing the evolution of biogas consumption in rural China. Additionally, the other three sigmoidal growth models (the general form of the logistic model, the Boltzmann 4-parameter function, and the Gompertz-Laird model) have also been applied to the same historical data on biogas consumption in rural China. These models have been compared using their goodness of fit to the historical data and the available national forecasts. The results revealed that the model newly-proposed in this paper is a very suitable candidate for forecasting biogas consumption in rural China.

Many analyses in the epidemiological and the prognostic studies and in the studies of event history data require methods that allow for unobserved covariates or “frailties”. We consider the shared frailty model in the framework of parametric proportional hazard model. There are certain assumptions about the distribution of frailty and baseline distribution. The exponential distribution is the commonly used distribution for analyzing lifetime data. In this paper, we consider shared gamma frailty model with bivariate exponential of Marshall and Olkin (1967) distribution as baseline hazard for bivariate survival times. We solve the inferential problem in a Bayesian framework with the help of a comprehensive simulation study and real data example. We fit the model to the real-life bivariate survival data set of diabetic retinopathy data. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model and then compare the true...

This paper is to use the method of test of goodness of fit test of hypothesis. By computing the maximum likelihood estimates of income, the results, to test scores of students to obey normal distribution, and confirm by MATLAB.

This study investigated the inclusion of spatial correlation in multivariate count data models of collision severity. The models were developed for severe (injury and fatal) and no-injury collisions using three years of collision data from the city of Richmond and the city of Vancouver. The proposed models were estimated in a Full Bayesian (FB) context via Markov Chain Monte Carlo (MCMC) simulation. The multivariate model with both heterogeneous effects and spatial correlation provided the best fit according to the Deviance Information Criteria (DIC). The results showed significant positive correlation between various road attributes and collision severities. For the Richmond dataset, the proportion of variance for spatial correlation was smaller than the proportion of variance for heterogeneous effects. Conversely, the spatial variance was greater than the heterogeneous variance for the Vancouver dataset. The correlation between severe and no-injury collisions for the total random effects (heterogeneous and spatial) was significant and quite high (0.905 for Richmond and 0.945 for Vancouver), indicating that a higher number of no-injury collisions is associated with a higher number of severe collisions. Furthermore, the multivariate spatial models were compared with two independent univariate Poisson lognormal (PLN) spatial models, with respect to model inference and goodness-of-fit. Multivariate spatial models provide a superior fit over the two univariate PLN spatial models with a significant drop in the DIC value (35.3 for Richmond and 116 for Vancouver). These results advocate the use of multivariate models with both heterogeneous effects and spatial correlation over univariate PLN spatial models.

A chi-square type test statistic for testing the uniformity hypothesis, based on selected values of the sample quantiles of the empirical distribution function, is presented. Critical values of the test statistics are given for several particular choices of the sample quantiles. A good agreement with their asymptotic counterparts occurs for sample sizes of the magnitude 30 and sometimes even for small sample sizes.

In recent years, stationary time series models based on copula functions became increasingly popular in econometrics to model nonlinear temporal and cross-sectional dependencies. Within these models, we consider the problem of testing the goodness-of-fit of the parametric form of the underlying copula. Our approach is based on a dependent multiplier bootstrap and it can be applied to any stationary, strongly mixing time series. The method extends recent i.i.d. results by Kojadinovic et al. (2011) and shares the same computational benefits compared to methods based on a parametric bootstrap. The finite-sample performance of our approach is investigated by Monte Carlo experiments for the case of copula-based Markovian time series models.

We studied the problem of testing a hypothesized distribution in survival regression models when the data is right censored and survival times are influenced by covariates. A modified chi-squared type test, known as Nikulin-Rao-Robson statistic, is applied for the comparison of accelerated failure time models. This statistic is used to test the goodness-of-fit for hypertabastic survival model and four other unimodal hazard rate functions. The results of simulation study showed that the hypertabastic distribution can be used as an alternative to log-logistic and log-normal distribution. In statistical modeling, because of its flexible shape of hazard functions, this distribution can also be used as a competitor of Birnbaum-Saunders and inverse Gaussian distributions. The results for the real data application are shown. Copyright © 2017 John Wiley & Sons, Ltd.

In the last two years, the world crude oil prices have dropped dramatically, and consequently the oil market has become very volatile and risky. Since energy markets play very important roles in the international economy and have led several global economic crises, risk management of energy products prices becomes very important for both academicians and market participants. We apply Schwartz and Smith?s model (2000) to calculate risk measures of Brent oil futures contracts and light sweet crude oil (WTI) futures contracts. The model includes a long-term factor and a short-term factor. We show that the two factors explain the Samuelson effect well and the model present well goodness of fit. Our backtesting results demonstrate that the models provide satisfactory risk measures for listed crude oil futures contracts. A simple estimation method possessing quick convergence is developed.

We address the issue of lack-of-fit testing for a parametric quantile regression. We propose a simple test that involves one-dimensional kernel smoothing, so that the rate at which it detects local alternatives is independent of the number of covariates. The test has asymptotically gaussian critical values, and wild bootstrap can be applied to obtain more accurate ones in small samples. Our procedure appears to be competitive with existing ones in simulations. We illustrate the usefulness of our test on birthweight data., Comment: 32 pages, 2 figures

Specification tests are developed for the conditional distribution of a dependent process {Xi} in a family of nonlinear time-series models. The family includes several Generalized AutoRegressive Conditional Heteroscedastic [GARCH] models that are widely used in finance and economics. Such tests are essential for ensuring the validity of density forecasts that are based on the assumed model. The tests are implemented using a bootstrap procedure, because the test statistics are not asymptotically pivotal. A novel method is developed for verifying the validity of the bootstrap tests for a given model; this involves simultaneously embedding the process under the null hypothesis and the bootstrap process in a system of stochastic recurrence equations. The method is illustrated for a well-known GARCH model. Within the specification testing literature, a feature that distinguishes this paper is that the conditional distribution of Xi under the null model, depends on the unobservable past values {. . . , X−2, X−1} extending back to the infinite past. The theory for testing the specification of the conditional distribution of Xi in such parametric models has not been developed in the general setting considered in this paper. The tests performed well in a simulation study, and a data example illustrates the tests.

We adopt Schwartz and Smith’s model (2000) to calculate risk measures of Brent oil futures contracts and light sweet crude oil (WTI) futures contracts and Mirantes, Poblacion and Serna’s model (2012) to calculate risk measures of natural gas futures contracts, gasoil futures contracts, heating oil futures contracts, RBOB gasoline futures contracts, PJM western hub peak and off-peak electricity futures contracts. We show that the models present well goodness of fit and explain two stylized facts of the data: the Samuelson effect and the seasonality effect. Our backtesting results demonstrate that the models provide satisfactory risk measures for listed energy commodity futures contracts. A simple estimation method possessing quick convergence is developed.

Goodness-of-fit tests for the innovation distribution in GARCH models based on measuring deviations between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis have been proposed in the literature. The asymptotic distributions of these test statistics depend on unknown quantities, so their null distributions are usually estimated through parametric bootstrap (PB). Although easy to implement, the PB can become very computationally expensive for large sample sizes, which is typically the case in applications of these models. This work proposes to approximate the null distribution through a weighted bootstrap. The procedure is studied both theoretically and numerically. Its asymptotic properties are similar to those of the PB, but, from a computational point of view, it is more efficient.

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ2 tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.

EQ-5D-3L scoring algorithms vary amongst countries, not only in the values of regression coefficients but also in the independent variables included in the regression model (hereafter referred to as model specification). It is unclear how much of this variation is due to differences in health state selection, the relative frequencies with which health states were valued, and model diagnostics, rather than to genuine differences in population preferences. Using aggregate data from a recent review, we noted all model specifications that were used. For each country the country’s own model was re-fitted, as were all other model specifications. This was done twice: once using all valued health states for each country, and again using a common set of 17 health states for all countries. Goodness of fit was assessed using the following model diagnostics: mean absolute error (MAE), mean squared error (MSE) and rho (the Pearson correlation coefficient between predicted and observed mean utilities), both with and without leave-one-out cross-validation. Thirteen countries contributed data. Even when using a common set of health states, the preferred model varied across countries. However, choice of health states did impact the preferred model specification: when using cross-validation, the preferred specification changed in five of ten countries when moving from 17 health states to all valued health states. The relative frequency with which health states were valued had little impact on the preferred model. Variation in choices of health states to value is responsible for some, but not all, of the observed heterogeneity in model specification. Relative frequency of health state valuation and choice of model diagnostic has a limited impact on model preference, however, use of cross-validation has a substantial impact. The use of cross-validation, implemented through omitting health states rather than respondents, is recommended as one approach to assessing model fit.

Gaussian random fields are frequently used to model spatial and spatial–temporal data, particularly in geostatistical settings. As much of the attention of the statistics community has been focused on defining and estimating the mean and covariance functions of these processes, little effort has been devoted to developing goodness-of-fit tests to allow users to assess the models' adequacy. We describe a general goodness-of-fit test and related graphical diagnostics for assessing the fit of Bayesian Gaussian process models using pivotal discrepancy measures. Our method is applicable for both regularly and irregularly spaced observation locations on planar and spherical domains. The essential idea behind our method is to evaluate pivotal quantities defined for a realization of a Gaussian random field at parameter values drawn from the posterior distribution. Because the nominal distribution of the resulting pivotal discrepancy measures is known, it is possible to quantitatively assess model fit directly from the output of Markov chain Monte Carlo algorithms used to sample from the posterior distribution on the parameter space. We illustrate our method in a simulation study and in two applications. Copyright © 2014 John Wiley & Sons, Ltd.