An Alternative Goodness-of-fit Test for Normality with Unknown Parameters

Goodness-of-fit tests have been studied by many researchers. Among them, an alternative statistical test for uniformity was proposed by Chen and Ye (2009). The test was used by Xiong (2010) to test normality for the case that both location parameter and scale parameter of the normal distribution are known. The purpose of the present thesis is to extend the result to the case that the parameters are unknown. A table for the critical values of the test statistic is obtained using Monte Carlo simulation. The performance of the proposed test is compared with the Shapiro-Wilk test and the Kolmogorov-Smirnov test. Monte-Carlo simulation results show that proposed test performs better than the Kolmogorov-Smirnov test in many cases. The Shapiro Wilk test is still the most powerful test although in some cases the test proposed in the present research performs better.

Parametric VaR with goodness-of-fit tests based on EDF statistics for extreme returns

Parametric VaR (Value-at-Risk) is widely used due to its simplicity and easy calculation. However, the normality assumption, often used in the estimation of the parametric VaR, does not provide satisfactory estimates for risk exposure. Therefore, this study suggests a method for computing the parametric VaR based on goodness-of-fit tests using the empirical distribution function (EDF) for extreme returns, and compares the feasibility of this method for the banking sector in an emerging market and in a developed one. The paper also discusses possible theoretical contributions in related fields like enterprise risk management (ERM). © 2013 Elsevier Ltd.

A Stochastic Approach for Finding of Semantically Related Words

2000 Mathematics Subject Classification: 62P99, 68T50

MGOF: Stata module to perform goodness-of-fit tests for multinomial data

mgof computes goodness-of-fit tests for the distribution of a discrete (categorical, multinomial) variable. The default is to perform classical large sample chi-squared approximation tests based on Pearson's X2 statistic and the log likelihood ratio (G2) statistic or a statistic from the Cressie-Read family. Alternatively, mgof computes exact tests using Monte Carlo methods or exhaustive enumeration. A Kolmogorov-Smirnov test for discrete data is also provided. The moremata package, also available from SSC, is required.

Multinomial goodness-of-fit: large sample tests with survey design correction and exact tests for small samples

A new Stata command called -mgof- is introduced. The command is used to compute distributional tests for discrete (categorical, multinomial) variables. Apart from classic large sample $\chi^2$-approximation tests based on Pearson's $X^2$, the likelihood ratio, or any other statistic from the power-divergence family (Cressie and Read 1984), large sample tests for complex survey designs and exact tests for small samples are supported. The complex survey correction is based on the approach by Rao and Scott (1981) and parallels the survey design correction used for independence tests in -svy:tabulate-. The exact tests are computed using Monte Carlo methods or exhaustive enumeration. An exact Kolmogorov-Smirnov test for discrete data is also provided.

Multi-dimensional Goodness-of-Fit Tests for Spectrum Sensing Based on Stochastic Distances

In this paper, we study two multi-dimensional Goodness-of-Fit tests for spectrum sensing in cognitive radios. The multi-dimensional scenario refers to multiple CR nodes, each with multiple antennas, that record multiple observations from multiple primary users for spectrum sensing. These tests, viz., the Interpoint Distance (ID) based test and the h, f distance based tests are constructed based on the properties of stochastic distances. The ID test is studied in detail for a single CR node case, and a possible extension to handle multiple nodes is discussed. On the other hand, the h, f test is applicable in a multi-node setup. A robustness feature of the KL distance based test is discussed, which has connections with Middleton's class A model. Through Monte-Carlo simulations, the proposed tests are shown to outperform the existing techniques such as the eigenvalue ratio based test, John's test, and the sphericity test, in several scenarios.

Exact Skewness-Kurtosis Tests for Multivariate Normality and Goodness-of-fit in Multivariate Regressions with Application to Asset Pricing Models

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross-equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared to simulation-based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal, Student t; normal mixtures and stable error models. In the Gaussian case, finite-sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi-stage Monte Carlo test methods. For non-Gaussian distribution families involving nuisance parameters, confidence sets are derived for the the nuisance parameters and the error distribution. The procedures considered are evaluated in a small simulation experi-ment. Finally, the tests are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995.

Multinomial goodness-of-fit: large sample tests with survey design correction and exact tests for small samples

I introduce the new mgof command to compute distributional tests for discrete (categorical, multinomial) variables. The command supports largesample tests for complex survey designs and exact tests for small samples as well as classic large-sample x2-approximation tests based on Pearson’s X2, the likelihood ratio, or any other statistic from the power-divergence family (Cressie and Read, 1984, Journal of the Royal Statistical Society, Series B (Methodological) 46: 440–464). The complex survey correction is based on the approach by Rao and Scott (1981, Journal of the American Statistical Association 76: 221–230) and parallels the survey design correction used for independence tests in svy: tabulate. mgof computes the exact tests by using Monte Carlo methods or exhaustive enumeration. mgof also provides an exact one-sample Kolmogorov–Smirnov test for discrete data.

Stochastic volatility and the goodness-of-fit of the Heston model

Recently, Drǎgulescu and Yakovenko proposed an analytical formula for computing the probability density function of stock log returns, based on the Heston model, which they tested empirically. Their research design inadvertently favourably biased the fit of the data to the Heston model, thus overstating their empirical results. Furthermore, Drǎgulescu and Yakovenko did not perform any goodness-of-fit statistical tests. This study employs a research design that facilitates statistical tests of the goodness-of-fit of the Heston model to empirical returns. Robustness checks are also performed. In brief, the Heston model outperformed the Gaussian model only at high frequencies and even so does not provide a statistically acceptable fit to the data. The Gaussian model performed (marginally) better at medium and low frequencies, at which points the extra parameters of the Heston model have adverse impacts on the test statistics. © 2005 Taylor & Francis Group Ltd.

The role of non-factorizability in determining "pseudo-classical" non-separability

This article introduces a “pseudo classical” notion of modelling non-separability. This form of non-separability can be viewed as lying between separability and quantum-like non-separability. Non-separability is formalized in terms of the non-factorizabilty of the underlying joint probability distribution. A decision criterium for determining the non-factorizability of the joint distribution is related to determining the rank of a matrix as well as another approach based on the chi-square-goodness-of-fit test. This pseudo-classical notion of non-separability is discussed in terms of quantum games and concept combinations in human cognition.

Quantum-like non-separability of concept combinations, emergent associates and abduction

Consider the concept combination ‘pet human’. In word association experiments, human subjects produce the associate ‘slave’ in relation to this combination. The striking aspect of this associate is that it is not produced as an associate of ‘pet’, or ‘human’ in isolation. In other words, the associate ‘slave’ seems to be emergent. Such emergent associations sometimes have a creative character and cognitive science is largely silent about how we produce them. Departing from a dimensional model of human conceptual space, this article will explore concept combinations, and will argue that emergent associations are a result of abductive reasoning within conceptual space, that is, below the symbolic level of cognition. A tensor-based approach is used to model concept combinations allowing such combinations to be formalized as interacting quantum systems. Free association norm data is used to motivate the underlying basis of the conceptual space. It is shown by analogy how some concept combinations may behave like quantum-entangled (non-separable) particles. Two methods of analysis were presented for empirically validating the presence of non-separable concept combinations in human cognition. One method is based on quantum theory and another based on comparing a joint (true theoretic) probability distribution with another distribution based on a separability assumption using a chi-square goodness-of-fit test. Although these methods were inconclusive in relation to an empirical study of bi-ambiguous concept combinations, avenues for further refinement of these methods are identified.