Simple Null Hypothesis Research Articles

Cladogenesis, coalescence and the evolution of the three domains of life

In this article, we explore the large-scale structure of the tree of life by using a simple model with a constant number of species and rates of speciation that equal the rates of extinction. In addition, we discuss the consequences of horizontal gene transfer for the concept of a most recent common ancestor of all living organisms (cenancestor). A simple null hypothesis based on coalescence theory explains some features of the observed topologies of the tree of life. Simulations of genes and organismal lineages suggest that there was no single common ancestor that contained all the genes ancestral to those shared among the three domains of life. Each contemporary molecule has its own history that traces back to an individual molecular cenancestor. However, these molecular ancestors were likely to be present in different organisms and at different times.

Local maximin properties of tests in Gaussian shift experiments

The local behavior of the power of weighted χ2-tests and Bayes tests is studied for simple null hypothesis in Gaussian shift experiments. A second order expansion of the power function is given. This expansion provides a shrinking family of ellipsoids (δE)0

Parametrization-invariant Wald tests

Parametrization-invariant versions of Wald statistics are introduced, based on yoke geometry. The construction is illustrated by several examples. In the case of simple null hypotheses, formulae are given for generalized Bartlett corrections of these geometric Wald statistics which bring their null distributions close to their large-sample asymptotic distributions.

Binning methodology for nonparametric goodness-of-fit test

In this paper, we develop a simple procedure for goodness-of-fit test. One feature of its construction is the use of "binning" transform of the sample observations. This enables us to define suitably a design variable and a response variable and they are shown to follow asymptotically a nonparametric regression model, under which the regression function corresponds to the population density function. As a result, the problem of testing density is formulated as assessing the parametric fits of a regression function against its nonparametric alternatives. Many nonparametric tests based on curve estimation technique could be incorporated, and we shall extend here the tests (Fan et al. , 2001; Zhang, 2001) based on local polynomial smoother combined with nonparametric likelihood-ratio. The resulting procedure is capable of testing not only the conventional simple null hypotheses, but also certain types of composite null hypotheses. Simulation studies illustrate the power of this procedure.

Chi-squared Goodness-of-fit Theory under Proportional Censorship

In this work we introduce chi-squared goodness-of-fit tests for censored data. In particular, we consider the Koziol–Green model of proportional censorship. The null hypothesis is that F equals a specified continuous distribution (simple null hypothesis) or that F belongs to a given parametric family of continuous distributions (composite null hypothesis). We use the proportional censorship relation for introducing primal and dual tests, the latter turning out to be quite simpler than the former. We analyze the performance of dual Pearson-type and Akritas-type tests in a small simulation study, which also considers the test by Akritas (1988) for randomly censored data. The benefits of including the proportional censorship relation in the construction of the test statistic are not always clearly seen.

Some approximations to power functions of ϕ-divergence tests in parametric models

In Morales et al (1997) phi-divergence statistics were proposed for testing hypotheses in general populations. In this paper we present some approximations to the power function of the new tests statistics, introduced in the cited paper, for the cases of simple null hypotheses and composite null hypotheses

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X2 statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h2 distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G2 (the power-divergence statistic of index u = 0).

Minimax Hypothesis Testing about the Density Support

The paper is concerned with testing nonparametric hypotheses about the underlying support G of independent and identically distributed observations. It is assumed that G belongs to a class .Cf' of compact sets with smooth upper surface called boundary fragments. It is required to distinguish the simple null hypothesis specified by a known set Go in ~ against nonparametric alternatives that G belongs to a class obtained by removing a certain neighbourhood of Go in ~. Using the asymptotic minimax approach, the problem is to determine the order of the smallest distance between the null hypothesis Ho and the alternatives for which one is able to test the null hypothesis against the alternatives with a given summarized error.

Large sample distribution of the likelihood ratio test for normal mixtures

This article concerns with the problem of testing whether a mixture of two normal distributions with bounded means and specific variance is simply a pure normal. The large sample behavior of the likelihood ratio test for the problem is carefully investigated. In the case of one mean parameter, it is shown that the large sample null distribution of the likelihood ratio test statistic is the squared supremum of a Gaussian process with zero mean and explicitly given covariances. In the case of two mean parameters, both the simple and composite hypotheses of normality are considered. Under the simple null hypothesis, the large sample null distribution is found to be an independent sum of a chi-square variable and the squared supremum of another Gaussian process whose covariance structure is slightly different from the one mean parameter case, while under the composite null hypothesis, the chi-square term is absent.

Extreme-Value Moment Goodness-of-Fit Tests

A general goodness-of-fit test for scale-parameter families of distributions is introduced, which is based on quotients of expected sample minima. The test is independent of the mean of the distribution, and, in applications to testing for exponentiality of data, compares favorably to other goodness-of-fit tests for exponentiality based on the empirical distribution function, regression methods and correlation statistics. The new minimal-moment method uses ratios of easily-calculated, unbiased, strongly consistent U-statistics, and the general technique can be used to test many standard composite null hypotheses such as exponentiality, normality or uniformity (as well as simple null hypotheses).

Likelihood-Based Inference with Singular Information Matrix

We consider likelihood-based asymptotic inference for a p-dimensional parameter θ of an identifiable parametric model with singular information matrix of rank p-1 at θ=θ* and likelihood differentiable up to a specific order. We derive the asymptotic distribution of the likelihood ratio test statistics for the simple null hypothesis that θ=θ* and of the maximum likelihood estimator (MLE) of θ when θ=θ*. We show that there exists a reparametrization such that the MLE of the last p-1 components of θ converges at rate O p ( n - 1/2 ) . For the first component θ1 of θ the rate of convergence depends on the order s of the first non-zero partial derivative of the log-likelihood with respect to θ1 evaluated at θ*. When s is odd the rate of convergence of the MLE of θ1 is O p ( n - 1/2s ) . When s is even, the rate of convergence of the MLE of | θ 1 -θ 1 * | is O p ( n - 1/2s ) and, moreover, the asymptotic distribution of the sign of the MLE of θ1-θ1* is non-standard. When p=1 it is determined by the sign of the sum of the residuals from the population least-squares regression of the (s+1)th derivative of the individual contributions to the log-likelihood on their derivatives of order s. For p>1, it is determined by a linear combination of the sum of residuals of a multivariate population least-squares regression involving partial and mixed derivatives of the log-likelihood of a specific order. Thus although the MLE of |θ1-θ1*| has a uniform rate of convergence of O p ( n - 1/2s ) , the uniform convergence rate for the MLE of θ1 in suitable shrinking neighbourhoods of θ1* is only O p ( n - 1/( 2s+2 ) ) .

On Burbea–Rao Divergence Based Goodness-of-Fit Tests for Multinomial Models

This paper investigates a new family of statistics based on Burbea–Rao divergence for testing goodness-of-fit. Under the simple and composite null hypotheses the asymptotic distribution of these tests is shown to be chi-squared. For composite hypothesis, the unspecified parameters are estimated by maximum likelihood as well as minimum Burbea–Rao divergence

The Interpretation of Short Climate Records, with Comments on the North Atlantic and Southern Oscillations

Abstract This pedagogical note reminds the reader that the interpretation of climate records is dependent upon understanding the behavior of stochastic processes. In particular, before concluding that one is seeing evidence for trends, shifts in the mean, or changes in oscillation periods, one must rule out the purely random fluctuations expected from stationary time series. The example of the North Atlantic oscillation (NAO) is mainly used here: the spectral density is nearly white (frequency power law ≈ s-0.2) with slight broadband features near 8 and 2.5 yr. By generating synthetic but stationary time series, one can see exhibited many of the features sometimes exciting attention as being of causal climate significance. Such a display does not disprove the hypothesis of climate change, but it provides a simple null hypothesis for what is seen. In addition, it is shown that the linear predictive skill for the NAO index must be very slight (less than 3% of the variance). A brief comparison with the South...

A weighted cramer-von mises statistic derived from a decomposition of the anderson-darling statistic

Two goodness of fit statistics with asymmetric weight function are derived from a decomposition of the Anderson-Darling statistic, For each one, the asymptotic null distribution is found for a simple null hypothesis and some upper percentties are calculated. The asymptotic power of the tests are obtained for some contiguous alternatives around a normal null hypothesis. The tests allow the user to choose to which tail to give more weight and it is intended to be used for that purpose. Therefore it should be not considered as a competitor of the classical goodness of fit tests.

A Critique of One-Tailed Hypothesis Test Procedures in Business and Economics Statistics Textbooks

The ability to conduct and correctly interpret the results of hypothesis tests is one of the most important skills that students can acquire in the introductory statistics course. Unfortunately, it is also one of the most difficult skills for them to learn.' Although this problem is widely recognized, textbook authors disagree about the best approach to use for conducting one-tailed hypothesis tests. In a recent survey of introductory business and economics statistics textbooks, we found that about half use the simple null hypothesis approach and about half use the composite null hypothesis approach for one-tailed hypothesis tests.2 Whereas both approaches are valid, the composite null hypothesis approach requires a more detailed statistical explanation than does the simple null hypothesis approach. Unfortunately, most textbooks that use the composite null hypothesis approach eschew the relevant explanation and use, instead, an explanation suitable only for the simple null hypothesis approach.

Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives

We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a spherical cone and a piecewise smooth cone.

Data-Driven Smooth Tests When the Hypothesis is Composite

In recent years several authors have recommended smooth tests for testing goodness of fit. However, the number of components in the smooth test statistic should be chosen well; otherwise, considerable loss of power may occur. Schwarz's selection rule provides one such good choice. Earlier results on simple null hypotheses are extended here to composite hypotheses, which tend to be of more practical interest. For general composite hypotheses, consistency of the data-driven smooth tests holds at essentially any alternative. Monte Carlo experiments on testing exponentiality and normality show that the data-driven version of Neyman's test compares well to other, even specialized, tests over a wide range of alternatives.

Data-Driven Smooth Tests When the Hypothesis Is Composite

In recent years several authors have recommended smooth tests for testing goodness of fit. However, the number of components in the smooth test statistic should be chosen well; otherwise, considerable loss of power may occur. Schwarz's selection rule provides one such good choice. Earlier results on simple null hypotheses are extended here to composite hypotheses, which tend to be of more practical interest. For general composite hypotheses, consistency of the data-driven smooth tests holds at essentially any alternative. Monte Carlo experiments on testing exponentiality and normality show that the data-driven version of Neyman's test compares well to other, even specialized, tests over a wide range of alternatives.

Imprecise Tests and Imprecise Hypotheses

This paper argues that theoretical predictions can only ever be expected to hold inexactly empirically, and constructs a method of inexact tests of exact hypotheses. Using a two‐parameter cost structure of ‘acceptance’ and ‘rejection’ of a simple null hypothesis, recognising a distinction between ‘statistical’ and ‘material’ significance, hypotheses are ‘accepted’ if they are only ‘slightly’ false. This may be contrasted with the exact testing of inexact hypotheses and the circumstances in which the two methods are equivalent are identified. The proposal facilitates informed decisions regarding the size and power of tests of simple hypotheses, and provides a transparent method of testing.

Bahadur efficiency and robustness of studentized score tests

We derive the exact Bahadur slopes of studentized score tests for a simple null hypothesis in a one-parameter family of distributions. The Student's t-test is included as a special case for which a recent result of Rukhin (1993, Sankhyā Ser. A, 55, 159–163) was improved upon. It is shown that locally optimal Bahadur efficiency for one-sample location models with a known or estimated scale parameter is attained within the class of studentized score tests. The studentized test has an asymptotic null distribution free of the scale parameter, and the optimality of likelihood scores does not depend on the existence of a moment generating function. We also consider the influence function and breakdown point of such tests as part of our robustness investigation. The influence of any studentized score test is bounded from above, indicating certain degree of robustness of validity, but a bounded score function is needed to cap the influence from below and to ensure a high power breakdown point. We find that the standard Huber-type score tests are not only locally minimax in Bahadur efficiency, but also very competitive in global efficiency at a variety of location models.