Simple Hypothesis Testing Research Articles

On the favorable estimation for fitting heavy tailed data

Assessment of heavy tailed data and its compound sums has many applications in insurance, auditing and operational risk capital assessment among others. In this paper, we compare the classical estimators (maximum likelihood, QQ and moment estimators) with the recently introduced robust estimators of “generalized median”, “trimmed mean” and estimators based on t-score moments. We derive the exact distribution of the likelihood ratio tests of homogeneity and simple hypothesis on the tail index of a two-parameter Pareto model. Such exact tests support the assessment of the performance of estimators. In particular, we discuss some problems that one can encounter when misemploying the log-normal assumption based methods supported by the Basel II framework. Real data and simulated examples illustrate the methods.

HOW CAN AGRICULTURE RESEARCH MAKE A DIFFERENCE IN COUNTRIES EMERGING FROM CONFLICT?

SUMMARYConflict is the most common cause of food insecurity. Foreign aid to countries emerging from conflict often allows a funded but brief window for the confirmation-testing and diffusion of agricultural innovation in affected areas. This paper asks the question: what lessons has agricultural research learned through its involvement in this process in countries emerging from conflict? Drawing on experience from Afghanistan and other countries, this paper documents some cases in which it has been possible to inject an element of simple hypothesis testing, often in farmer-managed trials, into post-conflict plans leading to useful lessons. Agricultural researchers need to be cognizant of this approach so that the practice becomes more widely used and lessons recorded for future use.

Some restriction tests in a new class of regression models for proportions

The main purpose of this work is to study the behaviour of Skovgaard’s [Skovgaard, I.M., 2001. Likelihood asymptotics. Scandinavian Journal of Statistics 28, 3–32] adjusted likelihood ratio statistic in testing simple hypothesis in a new class of regression models proposed here. The proposed class of regression models considers Dirichlet distributed observations, and the parameters that index the Dirichlet distributions are related to covariates and unknown regression coefficients. This class is useful for modelling data consisting of multivariate positive observations summing to one and generalizes the beta regression model described in Vasconcellos and Cribari-Neto [Vasconcellos, K.L.P., Cribari-Neto, F., 2005. Improved maximum likelihood estimation in a new class of beta regression models. Brazilian Journal of Probability and Statistics 19, 13–31]. We show that, for our model, Skovgaard’s adjusted likelihood ratio statistics have a simple compact form that can be easily implemented in standard statistical software. The adjusted statistic is approximately chi-squared distributed with a high degree of accuracy. Some numerical simulations show that the modified test is more reliable in finite samples than the usual likelihood ratio procedure. An empirical application is also presented and discussed.

Goodness of fit and homogeneity tests on the basis of N-distances

In this paper we propose an application of N-distance theory [Klebanov, L.B., 2005. N-distances and their applications. Karolinum, Prague] for testing simple hypotheses of goodness of fit and homogeneity. The asymptotic null distribution of test statistics is established and coincides with the distribution of infinite quadratic form of independent standard normal random variables. A construction of multivariate free-of-distribution homogeneity test is considered. The power of proposed criteria is compared with classical tests using Monte-Carlo simulations.

Statistical eigen-inference from large Wishart matrices

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., "eigen-inference"). Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically, we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure. Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are "global" (in a sense that we describe) and exploit "global" information thereby overcoming the inherent biases that cripple classical inference procedures which are "local" and rely on "local" information.

Veto Players and Welfare State Change: What Delays Social Entitlement Bills?

AbstractIn contrast to the study of outcomes such as social spending, systematic comparative analysis of political processes underlying welfare state change is scarce. This study deals with the influence of government parties and second chambers as veto players in social entitlement legislation. It asks three questions regarding the duration and outcome of the legislative process at the parliamentary stage. Does the number of government parties or the ideological distance between them affect the passage of bills? Under which circumstances do second chambers have an influence? Does the ideological position of the leftmost governing party affect the speed of passage of bills in policy areas where there is pressure for retrenchment? The hypotheses are tested using an original dataset on social entitlement bills initiated in Belgium, Germany and the UK between 1987/88 and 2002/03. Event history analysis at the level of individual bills yields the following results: proposals initiated from among the government parties on the floor are delayed by a higher number of parties in government, by greater ideological distance between them, if the second chamber is controlled by the opposition and its approval is mandatory, if the left veto player is more rightwing and if the bills deal with expansionary or mixed policies. Cabinet bills, in contrast, are not affected by any of these factors. The results point to a number of further research questions and show that quantitative studies in comparative welfare state research can go beyond testing simple hypotheses with macro-level outcome data.

Parametric estimation and tests through divergences and the duality technique

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending the maximum likelihood technique. Another view of the maximum likelihood approach, for estimation and tests, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given under both the null and the alternative hypotheses, and approximations of the power functions are deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on χ 2 -divergence on signed finite measures and the duality technique.

A probabilistic construction of model validation

We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise.

Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding

In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding.

Asymptotically optimal multistage tests of simple hypotheses

A family of variable stage size multistage tests of simple hypotheses is described, based on efficient multistage sampling procedures. Using a loss function that is a linear combination of sampling costs and error probabilities, these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero. A numerical study shows significant improvement over group sequential tests in a binomial testing problem.

Spatial and temporal variability of net ecosystem production in a tropical forest: testing the hypothesis of a significant carbon sink

AbstractTropical forest ecosystems play an important role in the global carbon balance. Depending on age and land use, they can act as carbon sources, sinks, or be in approximate balance, but it is uncertain if global environmental changes are forcing these ecosystems outside their natural range of variation. We asked the question of whether or not the net carbon flux of a tropical primary forest, which should be in balance over the long term, is within the expected range of natural variation. A simple Bayesian hypothesis testing method was used to address this question for primary forests in the Porce region of Colombia. Net ecosystem production (NEP) was measured in this forest in a set of 33 permanent plots from 2000 to 2002 in 2, 1‐year intervals. Our estimate of NEP ranged between −4.03 and 2.22 Mg C ha−1 yr−1 for the two intervals. This range was compared with a priori defined range of natural variation estimated from the ecosystem model STANDCARB, which estimated spatial and temporal variation due to gap dynamics. The prior range of variation was estimated between −1.5 and 1.5 Mg C ha−1 yr−1. The observed data on NEP did not provide sufficient evidence to reject the null hypothesis that these forests are in C balance. We concluded that the ecosystem is likely behaving within its range of natural variation, but measurement uncertainties were a major limitation to finding evidence to reject the null hypothesis. A literature review of C flux studies in the tropics revealed that about half of the observations could be explained by gap dynamics alone, while significant C sinks have only been observed during La Niña years, with contrasting results in other tropical forests. In conclusion, observational data of carbon fluxes do not appear to provide direct evidence for a significant carbon sink in some sites in the tropics.

Sequential testing of two composite statistical hypotheses

Consideration was given to the sequential testing of two composite statistical hypotheses. Each hypothesis is described by the density depending on the parameter which can belong to any of the two disjoint sets. A sequential procedure minimizing the Bayes risk which is maximal in the family of the a priori parameter distributions was proposed. The family of a priori distributions consists of all probabilistic distributions over the parametric set for which the a priori probability of validity of one of the composite hypotheses is equal to the given value. It was established that this procedure minimizes the parameter-maximal mean observation time under an additional condition and with the assumption of validity of any of the hypotheses in the class of rules for which the parameter-maximal error probabilities do not exceed the given values. The results obtained change over to the Wald classical results for the case of testing simple hypotheses.

Significance of Gene Ranking for Classification of Microarray Samples

Many methods for classification and gene selection with microarray data have been developed. These methods usually give a ranking of genes. Evaluating the statistical significance of the gene ranking is important for understanding the results and for further biological investigations, but this question has not been well addressed for machine learning methods in existing works. Here, we address this problem by formulating it in the framework of hypothesis testing and propose a solution based on resampling. The proposed r-test methods convert gene ranking results into position p-values to evaluate the significance of genes. The methods are tested on three real microarray data sets and three simulation data sets with support vector machines as the method of classification and gene selection. The obtained position p-values help to determine the number of genes to be selected and enable scientists to analyze selection results by sophisticated multivariate methods under the same statistical inference paradigm as for simple hypothesis testing methods.

How Online Learning Approaches Ornstein Uhlenbeck Processes

We show that under reasonable conditions, online learning near a local minimum is similar to a multivariate Ornstein Uhlenbeck process. This implies that the parameter state oscillates randomly around the minimum point, with a Gaussian limiting distribution. We also develop a simple hypothesis test that detects Ornstein Uhlenbeck properties without storing the history of the learning process.

Improving Estimates of Genetic Maps: A Maximum Likelihood Approach

As a result of previous large, multipoint linkage studies there is a substantial amount of existing marker data. Due to the increased sample size, genetic maps estimated from these data could be more accurate than publicly available maps. However, current methods for map estimation are restricted to data sets containing pedigrees with a small number of individuals, or cannot make full use of marker data that are observed at several loci on members of large, extended pedigrees. In this article, a maximum likelihood (ML) method for map estimation that can make full use of the marker data in a large, multipoint linkage study is described. The method is applied to replicate sets of simulated marker data involving seven linked loci, and pedigree structures based on the real multipoint linkage study of Abkevich et al. (2003, American Journal of Human Genetics 73, 1271-1281). The variance of the ML estimate is accurately estimated, and tests of both simple and composite null hypotheses are performed. An efficient procedure for combining map estimates over data sets is also suggested.

Spin discrimination of new heavy resonances at the LHC

If a new resonance is discovered at the LHC, the problem of discriminating between two spin hypotheses is in the category of simple hypothesis testing if the angular distributions of the decay products are fully specified by the models. We discuss the interpretation of the likelihood ratio statistic for discriminating between a Z' boson (spin 1) and a graviton (spin 2), contrasting this interpretation with previously published work. Inclusion of the possibility of a spin-0 resonance makes the unambiguous identification of either spin-1 or spin-2 resonances more difficult.

Comparative phylogeographic summary statistics for testing simultaneous vicariance

Testing for simultaneous vicariance across comparative phylogeographic data sets is a notoriously difficult problem hindered by mutational variance, the coalescent variance, and variability across pairs of sister taxa in parameters that affect genetic divergence. We simulate vicariance to characterize the behaviour of several commonly used summary statistics across a range of divergence times, and to characterize this behaviour in comparative phylogeographic datasets having multiple taxon-pairs. We found Tajima's D to be relatively uncorrelated with other summary statistics across divergence times, and using simple hypothesis testing of simultaneous vicariance given variable population sizes, we counter-intuitively found that the variance across taxon pairs in Nei and Li's net nucleotide divergence (pi(net)), a common measure of population divergence, is often inferior to using the variance in Tajima's D across taxon pairs as a test statistic to distinguish ancient simultaneous vicariance from variable vicariance histories. The opposite and more intuitive pattern is found for testing more recent simultaneous vicariance, and overall we found that depending on the timing of vicariance, one of these two test statistics can achieve high statistical power for rejecting simultaneous vicariance, given a reasonable number of intron loci (> 5 loci, 400 bp) and a range of conditions. These results suggest that components of these two composite summary statistics should be used in future simulation-based methods which can simultaneously use a pool of summary statistics to test comparative the phylogeographic hypotheses we consider here.

Goodness‐of‐fit Tests Based on the Kernel Density Estimator

Abstract. Given an i.i.d. sample drawn from a density f on the real line, the problem of testing whether f is in a given class of densities is considered. Testing procedures constructed on the basis of minimizing the L1‐distance between a kernel density estimate and any density in the hypothesized class are investigated. General non‐asymptotic bounds are derived for the power of the test. It is shown that the concentration of the data‐dependent smoothing factor and the ‘size’ of the hypothesized class of densities play a key role in the performance of the test. Consistency and non‐asymptotic performance bounds are established in several special cases, including testing simple hypotheses, translation/scale classes and symmetry. Simulations are also carried out to compare the behaviour of the method with the Kolmogorov‐Smirnov test and an L2 density‐based approach due to Fan [Econ. Theory10 (1994) 316].

Record Selection for Nonlinear Seismic Analysis of Structures

This study addresses the question of selection and amplitude scaling of accelerograms for predicting the nonlinear seismic response of structures. Despite the current practices of record selection according to a specific magnitude-distance scenario and scaling to a common level, neither aspect of this process has received significant research attention to ascertain the benefits or effects of these practices on the conclusions. This paper hypothesizes that neither these usual principal seismological characteristics nor scaling of records matters to the nonlinear response of structures. It then investigates under what conditions this hypothesis may not be sustainable. Two classes of records sets are compared in several case studies: one class is carefully chosen to represent a specific magnitude and distance scenario, the other is chosen randomly from a large catalog. Results of time-history analyses are formally compared by a simple statistical hypothesis test to assess the difference, if any, between nonlinear demands of the two classes of records. The effect of the degree of scaling (by first-mode spectral acceleration level) is investigated in the same way. Results here show (1) little evidence to support the need for a careful site-specific process of record selection by magnitude and distance, and (2) that concern over scenario-to-scenario record scaling, at least within the limits tested, may not be justified.

Error Exponents for Hypothesis Testing of the General Source

In this correspondence, we consider the simple hypothesis testing problems for general sources in the sense of Han and Verdu. Recently Han established a compact formula for the supremum of achievable exponents for the second-kind of error probability under the asymptotic constraint of the form mu <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ~ e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-nr</sup> (n rarr infin) on the first-kind of error probability mu <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , where r is a given positive number. We investigate the same hypothesis testing problems studied by Han. The aim of the correspondence is to give a new expression for the supremum of achievable error exponents. Our formula is expressed in terms of the divergences and given in quite different forms from Han's expression