Stability of characterizations of normal distributions based on the conditional expected values of the sample skewness and the sample kurtosis

Characterizations of normal distributions given by Nguyen and Dinh (1998) based on conditional expected values of the sample skewness and the sample kurtosis, given the sample mean and the sample variance, are shown to be stable. Received: September 1998     

On distances and goodness-of-fit tests for detecting multimodal distributions

Various distances between distributions and between densities are considered. The corresponding goodness-of-fit tests derived from them are examined for their abilities to detect multimodal alternatives. It is found that many well known techniques fail to detect such alternatives, while others do better in terms of their power results. These are mainly the tests derived from the variational metric which are based on spacings and gaps.     

Extended Neyman smooth goodness-of-fit tests,applied to competing heavy-tailed distributions

A simplified version of the Neyman (1937) “Smooth” goodness-of-fit test is extended to account for the presence of estimated model parameters, thereby removing overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyman greatly simplifies the calculations. Polynomials, splines, and the step function of Pearson’s test are compared as alternative perturbations to the theoretical uniform distribution. The extended tests have negligible size distortion and more power than standard tests. The tests are applied to competing symmetric leptokurtic distributions with US stock return data. These are generally rejected, primarily because of the presence of skewness.     

KMT-type inequalities and goodness-of-fit tests

The implications of the probability inequality of Komløs, Major and Tusnády (1975) for the theory of goodness-of-fit tests, especially tests based on stochastic integrals with respect to the basic martingale in the random censoring model, are discussed. Choices of the integrand of the stochastic integral which yield highly efficient generalized rank and supremum type tests are given for the simple as well as the composite null hypothesis.     

Asymptotic properties of MLE and UMVUE in one-parameter truncated distributions

Consider one-parameter families of continuous distributions whose range depend on an unknown parameter. In case a single sufficient and complete statistic exists, we obtain the limiting distributions of MLE and UMVUE. Both distributions are different transformations of a standard exponential variable.     

Characterizations of the Pareto and related distributions

In this note, a class of Pareto distributions is characterized based on the Shannon entropy of k-record statistics. As a consequence of that characterizations of the uniform and exponential distributions are given. Received: October 1999     

Exact confidence sets and goodness-of-fit methods for stable distributions

Usual inference methods for stable distributions are typically based on limit distributions. But asymptotic approximations can easily be unreliable in such cases, for standard regularity conditions may not apply or may hold only weakly. This paper proposes finite-sample tests and confidence sets for tail thickness and asymmetry parameters (α$\alpha$ and β$\beta$) of stable distributions. The confidence sets are built by inverting exact goodness-of-fit tests for hypotheses which assign specific values to these parameters. We propose extensions of the Kolmogorov–Smirnov, Shapiro–Wilk and Filliben criteria, as well as the quantile-based statistics proposed by McCulloch (1986) in order to better capture tail behavior. The suggested criteria compare empirical goodness-of-fit or quantile-based measures with their hypothesized values. Since the distributions involved are quite complex and non-standard, the relevant hypothetical measures are approximated by simulation, and p$p$-values are obtained using Monte Carlo (MC) test techniques. The properties of the proposed procedures are investigated by simulation. In contrast with conventional wisdom, we find reliable results with sample sizes as small as 25. The proposed methodology is applied to daily electricity price data in the US over the period 2001–2006. The results show clearly that heavy kurtosis and asymmetry are prevalent in these series.     

Characterizations of negative multinomial distributions based on conditional distributions

In this paper characterizations of negative multinomial distributions based on conditional distributions have been studied.     

Determining the optimal process means under mixture normal distributions

In this paper, we consider that the working environment has certain states, and in every state, the parameters of quality characteristics are different. Thus, if we set the characteristics parameters in a specified state, these parameters will change to another state. To describe this situation, we use a mixture of normal distributions, which comprise a flexible and powerful statistical-based modeling tool in practice. Under the step loss function and the piecewise linear loss function, we select the optimal means for the proposed manufacturing process.     

Characterizations of probability distributions via bivariate regression of record values

Bairamov et al. (Aust N Z J Stat 47:543–547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means.     

Distribution-free specification tests of conditional models

This article proposes a class of asymptotically distribution-free specification tests for parametric conditional distributions. These tests are based on a martingale transform of a proper sequential empirical process of conditionally transformed data. Standard continuous functionals of this martingale provide omnibus tests while linear combinations of the orthogonal components in its spectral representation form a basis for directional tests. Finally, Neyman-type smooth tests, a compromise between directional and omnibus tests, are discussed. As a special example we study in detail the construction of directional tests for the null hypothesis of conditional normality versus heteroskedastic contiguous alternatives. A small Monte Carlo study shows that our tests attain the nominal level already for small sample sizes.     

Parametric bootstrap tests for continuous and discrete distributions

Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

Robustness checks and robustness tests in applied economics

A common exercise in empirical studies is a “robustness check”, where the researcher examines how certain “core” regression coefficient estimates behave when the regression specification is modified by adding or removing regressors. If the coefficients are plausible and robust, this is commonly interpreted as evidence of structural validity. Here, we study when and how one can infer structural validity from coefficient robustness and plausibility. As we show, there are numerous pitfalls, as commonly implemented robustness checks give neither necessary nor sufficient evidence for structural validity. Indeed, if not conducted properly, robustness checks can be completely uninformative or entirely misleading. We discuss how critical and non-critical core variables can be properly specified and how non-core variables for the comparison regression can be chosen to ensure that robustness checks are indeed structurally informative. We provide a straightforward new Hausman (1978) type test of robustness for the critical core coefficients, additional diagnostics that can help explain why robustness test rejection occurs, and a new estimator, the Feasible Optimally combined GLS (FOGLeSs) estimator, that makes relatively efficient use of the robustness check regressions. A new procedure for Matlab, testrob, embodies these methods.     

A note on the distribution of the product of zero‐mean correlated normal random variables

The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.     

Predicting LGD distributions with mixed continuous and discrete ordinal outcomes

The loss given default (LGD) distribution is known to have a complex structure. Consequently, the parametric approach for its prediction by fitting a density function may suffer a loss of predictive power. To overcome this potential drawback, we use the cumulative probability model (CPM) to predict the LGD distribution. The CPM applies a transformed variable to model the LGD distribution. This transformed variable has a semiparametric structure. It models the predictor effects parametrically. The functional form of the transformation is unspecified. Thus, CPM provides more flexibility and simplicity in modeling the LGD distribution. To implement CPM, we collect a sample of defaulted debts from Moody’s Default and Recovery Database. Given this sample, we use an expanding rolling window approach to investigate the out-of-time performance of CPM and its alternatives. Our results confirm that CPM is better than its alternatives, in the sense of yielding more accurate LGD distribution predictions.     

Evaluating the accuracy of value-at-risk forecasts: New multilevel tests

We propose independence and conditional coverage tests which are aimed at evaluating the accuracy of Value-at-Risk (VaR) forecasts from the same model at different confidence levels. The proposed procedures are multilevel tests, i.e., joint tests of several quantiles corresponding to different confidence levels. In a comprehensive Monte Carlo exercise, we document the superiority of the proposed tests with respect to existing multilevel tests. In an empirical application, we illustrate the implementation of the tests using several VaR models and daily data for 15 MSCI world indices.