Non-standard Distributions Research Articles

Joint Confidence Sets for Structural Impulse Responses

Many users of structural VAR models are primarily interested in learning about the shape of structural impulse response functions. This requires joint inference about sets of structural impulse responses, allowing for dependencies across time as well as across response functions. Such joint inference is complicated by the fact that the joint distribution of structural impulse response becomes degenerate when the number of structural impulse responses of interest exceeds the number of model parameters, as is often the case in applied work. This degeneracy may be overcome by transforming the estimator appropriately. We show that the joint Wald test is invariant to this transformation and converges to a nonstandard distribution, which can be approximated by the bootstrap, allowing the construction of asymptotically valid joint confidence sets for any subset of structural impulse responses, regardless of whether the joint distribution of the structural impulse responses is degenerate or not. We demonstrate by simulation the coverage accuracy of these sets in finite samples under realistic conditions. We make the case for representing these joint confidence sets in the form of shotgun plots rather than joint confidence bands for impulse response functions. Several empirical examples demonstrate that this approach not only conveys the same information as confidence bands about the statistical significance of response functions, but provides economically relevant additional information about the shape of response functions that is lost when reducing the joint confidence set to two-dimensional bands

Diagonalization matrix and its application in distribution theory

Some matrix representations of diverse diagonal arrays are studied in this work; the results allow new definitions of classes of elliptical distributions indexed by kernels mixing Hadamard and usual products. A number of applications are derived in the setting of prior densities from the Bayesian multivariate regression model and families of non-elliptical distributions, such as the matrix multivariate generalized Birnbaum–Saunders density. The philosophy of the research about matrix representations of quadratic and inverse quadratic forms can be extended as a methodology for exploring possible new applications in non-standard distributions, matrix transformations and inference.

Generalized cross-spectral test for nonlinear Granger causality with applications to money–output and price–volume relations

In this study, we propose a test statistic based on a generalized cross-spectral distribution function to test for linear and nonlinear Granger causality. The test statistic considers all time series lags and, at the same time, avoids the “curse of dimensionality” problem. Moreover, it avoids having to choose a kernel function and bandwidth parameter. Since the generalized cross-spectral distribution test statistic asymptotically converges to a nonstandard distribution, we propose a wild bootstrap approach to approximate its critical values. A Monte Carlo simulation shows that the generalized cross-spectral distribution test statistic has better finite sample performance than Hong's (2001) test. In the empirical analysis, we perform empirical tests for Granger causality between U.S. money and output and between the return and volume of the CSI 300 Index and show that the proposed test statistic succeeds in capturing nonlinear Granger causality.

Joint modeling of repeated multivariate cognitive measures and competing risks of dementia and death: a latent process and latent class approach.

Joint models initially dedicated to a single longitudinal marker and a single time-to-event need to be extended to account for the rich longitudinal data of cohort studies. Multiple causes of clinical progression are indeed usually observed, and multiple longitudinal markers are collected when the true latent trait of interest is hard to capture (e.g., quality of life, functional dependency, and cognitive level). These multivariate and longitudinal data also usually have nonstandard distributions (discrete, asymmetric, bounded, etc.). We propose a joint model based on a latent process and latent classes to analyze simultaneously such multiple longitudinal markers of different natures, and multiple causes of progression. A latent process model describes the latent trait of interest and links it to the observed longitudinal outcomes using flexible measurement models adapted to different types of data, and a latent class structure links the longitudinal and cause-specific survival models. The joint model is estimated in the maximum likelihood framework. A score test is developed to evaluate the assumption of conditional independence of the longitudinal markers and each cause of progression given the latent classes. In addition, individual dynamic cumulative incidences of each cause of progression based on the repeated marker data are derived. The methodology is validated in a simulation study and applied on real data about cognitive aging obtained from a large population-based study. The aim is to predict the risk of dementia by accounting for the competing death according to the profiles of semantic memory measured by two asymmetric psychometric tests.

Binary quantile regression with local polynomial smoothing

Manski (1975, 1985) proposed the maximum score estimator for the binary choice model under a weak conditional median restriction that converges at the rate of n−1/3 and the standardized version has a nonstandard distribution. By imposing additional smoothness conditions, Horowitz (1992) proposed a smoothed maximum score estimator that often has large finite sample biases and is quite sensitive to the choice of smoothing parameter. In this paper we propose a novel framework that leads to a local polynomial smoothing based estimator. Our estimator possesses finite sample and asymptotic properties typically associated with the local polynomial regression. In addition, our local polynomial regression-based estimator can be extended to the panel data setting. Simulation results suggest that our estimators may offer significant improvement over the smoothed maximum score estimators.

Comment on testing for spurious and cointegrated regressions: a wavelet approach

In a recent paper, Leong and Huang [6] proposed a wavelet-correlation-based approach to test for cointegration between two time series. However, correlation and cointegration are two different concepts even when wavelet analysis is used. It is known that statistics based on non-stationary integrated variables have non-standard asymptotic distributions. However, wavelet analysis offsets the integrating order of non-stationary series so that traditional asymptotics on stationary variables suffices to ascertain the statistical properties of wavelet-based statistics. Based on this, this note shows that wavelet correlations cannot be used as a test of cointegration.

SPATIAL VARIABILITY AND VITALITY OF EPIGEOUS TERMITE MOUNDS IN PASTURES OF MATO GROSSO DO SUL, BRAZIL

Epigeous termite mounds are frequently observed in pasture areas, but the processes regulating their population dynamics are poorly known. This study evaluated epigeous termite mounds in cultivated grasslands used as pastures, assessing their spatial distribution by means of geostatistics and evaluating their vitality. The study was conducted in the Cerrado biome in the municipality of Rio Brilhante, Mato Grosso do Sul, Brazil. In two pasture areas (Pasture 1 and Pasture 2), epigeous mounds (nests) were georeferenced and analyzed for height, circumference and vitality (inhabited or not). The area occupied by the mounds was calculated and termite specimens were collected for taxonomic identification. The spatial distribution pattern of the mounds was analyzed with geostatistical procedures. In both pasture areas, all epigeous mounds were built by the same species, Cornitermes cumulans. The mean number of mounds per hectare was 68 in Pasture 1 and 127 in Pasture 2, representing 0.4 and 1 % of the entire area, respectively. A large majority of the mounds were active (vitality), 91 % in Pasture 1 and 84 % in Pasture 2. A “pure nugget effect” was observed in the semivariograms of height and nest circumference in both pastures reflecting randomized spatial distribution and confirming that the distribution of termite mounds in pastures had a non-standard distribution.

Dynamics of non-Markovian exclusion processes

Driven diffusive systems are often used as simple discrete models of collective transport phenomena in physics, biology or social sciences. Restricting attention to one-dimensional geometries, the asymmetric simple exclusion process (ASEP) plays a paradigmatic role to describe noise-activated driven motion of entities subject to an excluded volume interaction and many variants have been studied in recent years. While in the standard ASEP the noise is Poissonian and the process is therefore Markovian, in many applications the statistics of the activating noise has a non-standard distribution with possible memory effects resulting from internal degrees of freedom or external sources. This leads to temporal correlations and can significantly affect the shape of the current-density relation as has been studied recently for a number of scenarios. In this paper we report a general framework to derive the fundamental diagram of ASEPs driven by non-Poissonian noise by using effectively only two simple quantities, viz., the mean residual lifetime of the jump distribution and a suitably defined temporal correlation length. We corroborate our results by detailed numerical studies for various noise statistics under periodic boundary conditions and discuss how our approach can be applied to more general driven diffusive systems.

Oral health research: A source for innovative new statistical developments

This article reviews the challenges and opportunities that oral health research may offer to a statistician. To illustrate this, we focus on the Signal Tandmobiel® study, a longitudinal oral health survey that triggered many statistical explorations and developments over the last two decades. For example, non-standard distributions are more the rule than the exception in oral health research. In addition, often measurement error problems need to be addressed. The hierarchical structure of the oral health data also poses non-standard challenges. For instance, caries experience in the mouth is spatially correlated with, however, a specific metric defining the distance between two occurrences. In addition, since caries experience in the context of an epidemiological study is only measured at intervals, analysis of survival involves interval-censoring. Finally, when analyzing a realistic data set in oral health, all the above issues may have to be dealt with simultaneously.

Approximately Distribution-Free Diagnostic Tests for Regressions With Survival Data

We consider a linear regression model in which responses are in the form of log-duration while the corresponding error is modeled by means of a nonstandard distribution. This lifetime model is estimated by maximum likelihood and the estimates are used as basis for omnibus goodness-of-fit tests for the error distribution. In particular, we suggest a specific transformation such that when it is applied to the residuals, the corresponding test of fit becomes nonparametric, in the sense that the null distribution of the test statistic is the same regardless of the specific distribution being tested or of the values of the parameters involved in the regression model (such as regression parameters or other parameters figuring in the error distribution). The proposed procedure is applied to regressions involving the Birnbaum–Saunders, the extended Weibull, and the generalized Weibull distributions.

Simulation Program to Determine Sample Size and Power for a Multiple Logistic Regression Model with Unspecified Covariate Distributions

Binary logistic regression models are commonly used to assess the association between outcomes and covariates. Many covariates are inherently continuous, and have a variety of distributions, including those that are heavily skewed to the left or right. Existing theoretical formulas, criteria, and simulation programs cannot accurately estimate the sample size and power of non-standard distributions. Therefore, we have developed a simulation program that uses Monte Carlo methods to estimate the exact power of a binary logistic regression model. This power calculation can be used for distributions of any shape and covariates of any type (continuous, ordinal, and nominal), and can account for nonlinear relationships between covariates and outcomes. For illustrative purposes, this simulation program is applied to real data obtained from a study on the influence of smoking on 90-day outcomes after acute atherothrombotic stroke. Our program is applicable to all effect sizes and makes it possible to apply various statistical methods, logistic regression and related simulations such as Bayesian inference with some modifications.

Irregularities in X(Y) from Y(X) in linear calibration

Let X be an input measurement and Y the output reading of a calibrated instrument, with Y(X) as the calibration curve. Solving X(Y) projects an instrumental reading back onto the scale of measurements as an object of pivotal interest. The arrays of instrumental readings are projected in this manner in practice, yielding arrays of calibrated measurements, typically subject to errors of calibration. The effects of calibration errors on the properties of calibrated measurements are examined here under linear calibration. Irregularities arise as induced dependencies, inflated variances, non-standard distributions, inconsistent sample means, the underestimation of measurement variance, and other unintended consequences. On the other hand, conventional properties are seen to remain largely in place in the use of selected regression diagnostics and in one-way comparative experiments using calibrated data.

Numerical distribution functions for seasonal unit root tests

It is often necessary to test for the presence of seasonal unit roots when working with time series data observed at intervals of less than a year. One of the most widely used methods for doing this is based on regressing the seasonal difference of the series over the transformations of the series by applying specific filters for each seasonal frequency. This provides test statistics with non-standard distributions. A generalisation of this method for any periodicity is presented and a response surface regressions approach is used to calculate the P-values of the statistics whatever the periodicity and sample size of the data. The algorithms are prepared with the Gretl open source econometrics package and two empirical examples are presented.

A COMPARISON OF ALTERNATIVE APPROACHES TO SUPREMUM-NORM GOODNESS-OF-FIT TESTS WITH ESTIMATED PARAMETERS

Goodness-of-fit tests based on parametric empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several analytic solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985, Journal of Applied Probability 22(1), 99–122) can be applied to conduct inference for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact that has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981, Theory of Probability and Its Applications26(2), 240–257) through the score function of the parametric model. Simulation experiments suggest that in small samples, Durbin-style adjustments result in tests that have higher power than tests based on transformed processes, and in some cases they have higher power than parametric bootstrap procedures.

Redundancy in electronic health record corpora: analysis, impact on text mining performance and mitigation strategies

BackgroundThe increasing availability of Electronic Health Record (EHR) data and specifically free-text patient notes presents opportunities for phenotype extraction. Text-mining methods in particular can help disease modeling by mapping named-entities mentions to terminologies and clustering semantically related terms. EHR corpora, however, exhibit specific statistical and linguistic characteristics when compared with corpora in the biomedical literature domain. We focus on copy-and-paste redundancy: clinicians typically copy and paste information from previous notes when documenting a current patient encounter. Thus, within a longitudinal patient record, one expects to observe heavy redundancy. In this paper, we ask three research questions: (i) How can redundancy be quantified in large-scale text corpora? (ii) Conventional wisdom is that larger corpora yield better results in text mining. But how does the observed EHR redundancy affect text mining? Does such redundancy introduce a bias that distorts learned models? Or does the redundancy introduce benefits by highlighting stable and important subsets of the corpus? (iii) How can one mitigate the impact of redundancy on text mining?ResultsWe analyze a large-scale EHR corpus and quantify redundancy both in terms of word and semantic concept repetition. We observe redundancy levels of about 30% and non-standard distribution of both words and concepts. We measure the impact of redundancy on two standard text-mining applications: collocation identification and topic modeling. We compare the results of these methods on synthetic data with controlled levels of redundancy and observe significant performance variation. Finally, we compare two mitigation strategies to avoid redundancy-induced bias: (i) a baseline strategy, keeping only the last note for each patient in the corpus; (ii) removing redundant notes with an efficient fingerprinting-based algorithm. aFor text mining, preprocessing the EHR corpus with fingerprinting yields significantly better results.ConclusionsBefore applying text-mining techniques, one must pay careful attention to the structure of the analyzed corpora. While the importance of data cleaning has been known for low-level text characteristics (e.g., encoding and spelling), high-level and difficult-to-quantify corpus characteristics, such as naturally occurring redundancy, can also hurt text mining. Fingerprinting enables text-mining techniques to leverage available data in the EHR corpus, while avoiding the bias introduced by redundancy.

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty-fifty mixture of ... and ..., where H is the number of moment conditions, as opposed to a .... With more than two assets, this distribution lies between the ... and ... (p denotes the number of parameters). These results show that ignoring the lack of first-order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings. (ProQuest: ... denotes formulae/symbols omitted.)

Testing for Common GARCH Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that the Hansen’s (1982) J-test statistic is asymptotically distributed as the minimum of the limit of a certain empirical process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a half-half mixture of x_(H-1)^2and x_H^2, where H is the number of moment conditions, as opposed to a x_(H-1)^2. With more than two assets, this distribution lies between the x_(H-p)^2 and x_H^2 (p, the number of parameters). These results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with possibly increasing over-rejection rate with the number of assets. A Monte Carlo study illustrates these findings.

Some Studies on Preparation of Large Spray Deposited Aluminium Alloy Tube Preform

In this paper, preparation of large spray deposited aluminum alloy tube was investigated. The shape and particle size distribution of over-sprayed powders together with structure of deposits were analyzed. The results show that the over-sprayed powders resulted from particles missing, spurting and rebounding. The particle size distribution was non-standard normal distribution, the particle size of the great majority was within 150μm and the average size was 50μm. The residual pores in deposits came into being when inter-particle space could not be filled with enough liquid during spray deposition. The as-deposited preform with rough surface was a coalesced bulk of deposited particles. The rotate speed of substrate and scan speed of atomizer together with spray stripe width should satisfy any certain relationship to obtain a qualified tube preform. Substrate preheating and surface treatment and residual stress control were very important for preparing a large preform.

A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

Goodness of fit tests based on empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Simulation experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.

Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.