A test statistic based on ranked set sampling for two normal means

In this study, we considered a hypothesis test for the difference of two population means using ranked set sampling. We proposed a test statistic for this hypothesis test with more than one cycle under normality. We also investigate the performance of this test statistic, when the assumptions hold and are violated. For this reason, we investigate the type I error and power rates of tests under normality with equal and unequal variances, non-normality with equal and unequal variances. We also examine the performance of this test under imperfect ranking case. The simulation results show that derived test performs quite well.     

AN ANALYSIS OF CLASSICAL AND TIE-BREAKER TENNIS

Expressions for the probability that a player wins a game, set or match of classical or tie-breaker tennis are obtained. Also, expressions for the distribution, the mean and the variance of the number of points in a game, set or match of classical or tie-breaker tennis are obtained. These results explain the long matches often previously observed in classical tennis between two players each with very effective serves on fast grass-court surfaces. The methodology used to derive these expressions could be used to obtain corresponding expressions for other "nested-type" scoring systems. For example, tennis is 3-nested (game, set, match) whereas squash is only 2-nested (game, match).     

Soccer matches as experiments: how often does the ‘best’ team win?

Models in which the number of goals scored by a team in a soccer match follow a Poisson distribution, or a closely related one, have been widely discussed. We here consider a soccer match as an experiment to assess which of two teams is superior and examine the probability that the outcome of the experiment (match) truly represents the relative abilities of the two teams. Given a final score, it is possible by using a Bayesian approach to quantify the probability that it was or was not the case that ‘the best team won’. For typical scores, the probability of a misleading result is significant. Modifying the rules of the game to increase the typical number of goals scored would improve the situation, but a level of confidence that would normally be regarded as satisfactory could not be obtained unless the character of the game was radically changed.     

Non-parametric procedure for knockout tournaments

In a seeded knockout tournament, where teams have some preassigned strength, do we have any assurances that the best team in fact has won? Is there some insight to be gained by considering which teams beat which other teams solely examining the seeds? We pose an answer to these questions by using the difference in the seeds of the two players as the basis for a test statistic. We offer several models for the underlying probability structure to examine the null distribution and power functions and determine these for small tournaments (less than five teams). One structure each for 8 teams and 16 teams is examined, and we conjecture an asymptotic normal distribution for the test statistic.     

Performance of tests of association in misspecified generalized linear models

We examine the effects of modelling errors, such as underfitting and overfitting, on the asymptotic power of tests of association between an explanatory variable x and an outcome in the setting of generalized linear models. The regression function for x is approximated by a polynomial or another simple function, and a chi-square statistic is used to test whether the coefficients of the approximation are simultaneously equal to zero. Adding terms to the approximation increases asymptotic power if and only if the fit of the model increases by a certain quantifiable amount. Although a high degree of freedom approximation offers robustness to the shape of the unknown regression function, a low degree of freedom approximation can yield much higher asymptotic power even when the approximation is very poor. In practice, it is useful to compute the power of competing test statistics across the range of alternatives that are plausible a priori. This approach is illustrated through an application in epidemiology.     

A fiducial approach to multiple comparisons

Comparing treatment means from populations that follow independent normal distributions is a common statistical problem. Many frequentist solutions exist to test for significant differences amongst the treatment means. A different approach would be to determine how likely it is that particular means are grouped as equal. We developed a fiducial framework for this situation. Our method provides fiducial probabilities that any number of means are equal based on the data and the assumed normal distributions. This methodology was developed when there is constant and non-constant variance across populations. Simulations suggest that our method selects the correct grouping of means at a relatively high rate for small sample sizes and asymptotic calculations demonstrate good properties. Additionally, we have demonstrated the flexibility in the methods ability to calculate the fiducial probability for any number of equal means. This was done by analyzing a simulated data set and a data set measuring the nitrogen levels of red clover plants that were inoculated with different treatments.     

An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

Bayesian goodness-of-fit test for censored data

We propose a Bayesian computation and inference method for the Pearson-type chi-squared goodness-of-fit test with right-censored survival data. Our test statistic is derived from the classical Pearson chi-squared test using the differences between the observed and expected counts in the partitioned bins. In the Bayesian paradigm, we generate posterior samples of the model parameter using the Markov chain Monte Carlo procedure. By replacing the maximum likelihood estimator in the quadratic form with a random observation from the posterior distribution of the model parameter, we can easily construct a chi-squared test statistic. The degrees of freedom of the test equal the number of bins and thus is independent of the dimensionality of the underlying parameter vector. The test statistic recovers the conventional Pearson-type chi-squared structure. Moreover, the proposed algorithm circumvents the burden of evaluating the Fisher information matrix, its inverse and the rank of the variance–covariance matrix. We examine the proposed model diagnostic method using simulation studies and illustrate it with a real data set from a prostate cancer study.     

An optimal search procedure

Consider the following problem. There are exactly two defective (unknown) elements in the set X={x₁, x₂,…,x_n}, all possibilities occuring with equal probabilities. We want to identify the unknown (defective) elements by testing some subsets A of X, and for each such set A determining whether A contains any of them. The test on an individual subset A informs us that either all elements of the tested set A are good, or that at least one of them is defective (but we do not know which ones or how many). A set containing at least one defective element is said to be defective. Our aim is to minimize the maximal number of tests. For the optimal strategy, let the maximal test length be denoted by l₂(n). We obtain the value of this function for an infinite sequence of values of n.     

Analysis of the importance of on-field covariates in the German Bundesliga

In modern football, various variables as, for example, the distance a team runs or its percentage of ball possession, are collected throughout a match. However, there is a lack of methods to make use of these on-field variables simultaneously and to connect them with the final result of the match. This paper considers data from the German Bundesliga season 2015/2016. The objective is to identify the on-field variables that are connected to the sportive success or failure of the single teams. An extended Bradley–Terry model for football matches is proposed that is able to take into account on-field covariates. Penalty terms are used to reduce the complexity of the model and to find clusters of teams with equal covariate effects. The model identifies the running distance to be the on-field covariate that is most strongly connected to the match outcome.     

Testing for the maximum mean in a mixture of normals

We consider the test of the null hypothesis that the largest mean in a mixture of an unknown number of normal components is less than or equal to a given threshold. This test is motivated by the problem of assessing whether the Soviet Union has been operating in compliance with the Nuclear Test Ban Treaty. In our analysis, the number of normal components is determined using Akaike's Information Criterion while the hypothesis test itself is based on asymptotic results given by Behboodian for a mixture of two normal components. A bootstrap approach is also considered for estimating the standard error of the largest estimated mean. The performance of the testa are examined through the use of simulation.     

Comparing Two Survival Time Distributions: An Investigation of Several Weight Functions for the Weighted Logrank Statistic

In survival analysis, it is often of interest to test whether or not two survival time distributions are equal, specifically in the presence of censored data. One very popular test statistic utilized in this testing procedure is the weighted logrank statistic. Much attention has been focused on finding flexible weight functions to use within the weighted logrank statistic, and we propose yet another. We demonstrate our weight function to be more stable than one of the most popular, which is given by Fleming and Harrington, by means of asymptotic normal tests, bootstrap tests and permutation tests performed on two datasets with a variety of characteristics.     

An omnibus test of goodness-of-fit for conditional distributions with applications to regression models

We introduce an omnibus goodness-of-fit test for statistical models for the conditional distribution of a random variable. In particular, this test is useful for assessing whether a regression model fits a data set on all its assumptions. The test is based on a generalization of the Cramér–von Mises statistic and involves a local polynomial estimator of the conditional distribution function. First, the uniform almost sure consistency of this estimator is established. Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives. The extension to the case where unknown parameters appear in the model is developed. A simulation study shows that the test has good power against some common departures encountered in regression models. Moreover, its power is comparable to that of other nonparametric tests designed to examine only specific departures.     

Testing Target-Zone Models Using Efficient Method of Moments

The objectives of this article are threefold—(1) to test target-zone models using more efficient and direct econometric methodology than previous research, (2) to identify an implicit band, if it exists, from observed data and to test target-zone models based on the estimated implicit band rather than the stated official band, and (3) to examine whether the exchange rate can be modeled as a managed float system with a central parity that lacks a band. We find strong evidence that a model with intramarginal intervention and a narrower implicit (unofficial) band can describe the dynamics of the French franc/Deutsche mark exchange rate from January 1, 1987–July 30, 1993.     

Estimating the hazard functions of two alternately occurring recurrent events

Often two recurrent events of equal importance can occur alternately. The life-time patterns of the two events can then be of considerable interest. In this paper, we consider two such events, the inclusion and exclusion of players in a team sport, and study whether there is any inherent pattern in the time-lengths between these events. The life-time distributions are modelled and methods of estimating the model parameters suggested taking into account any relationship in the pattern of recurrence. The results are then applied to the inclusion and exclusion of players in the Indian national cricket team. As further illustration, a simulation study is made. Broad application areas are identified both in the introduction and conclusion.     

A Class of Non-parametric Tests in the Competing Risks Model for Comparing Two Samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D₁, obtained when the process is under control. Our main goal is to test if the failure rates in D₁ are equal to or less than the failure rates in another data set D₂, against "undesirable" neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests     

An Improved Estimation in Regression Parameter Matrix in Multivariate Regression Model

We consider shrinkage and preliminary test estimation strategies for the matrix of regression parameters in multivariate multiple regression model in the presence of a natural linear constraint. We suggest a shrinkage and preliminary test estimation strategies for the parameter matrix. The goal of this article is to critically examine the relative performances of these estimators in the direction of the subspace and candidate subspace restricted type estimators. Our analytical and numerical results show that the proposed shrinkage and preliminary test estimators perform better than the benchmark estimator under candidate subspace and beyond. The methods are also applied on a real data set for illustrative purposes.     

Investigation of disease-free survival as a surrogate endpoint for survival in cancer clinical trials

A stochastic model wiuh exponential components is used to describe our data collected from a phase III cancer clinical trial. Criteria which guarantee that disease-free survival (DFS) can be used as a surrogate for overall survival are explored under this model. We examine several colorectal adjuvant clinical trials and find that these conditions are not satisfied. The relationship between the hazard ratio of DFS for an active treatment versus a control treatment and the cumulative hazard ratio of survival for the same two treatments is then explored. An almost linear relationship is found such that a hazard ratio for DFS of less than a threshold R corresponds to a non-null treatment effect on survival The threshold value R is determined for our colorectal adjuvant trial data. Based on this relationship, a one-sided test of equal hazard rate of survival is equivalent to a test of hazard ratio of DFS small than R This approach assumes that recurrence information is unbiasedly and accurately assessed; an assumpion which is sometimes difficult to ensure for multicenter clinical trials, particularly for interim analyses.     

An empirical goodness-of-fit test for multivariate distributions

An empirical test is presented as a tool for assessing whether a specified multivariate probability model is suitable to describe the underlying distribution of a set of observations. This test is based on the premise that, given any probability distribution, the Mahalanobis distances corresponding to data generated from that distribution will likewise follow a distinct distribution that can be estimated well by means of a large sample. We demonstrate the effectiveness of the test for detecting departures from several multivariate distributions. We then apply the test to a real multivariate data set to confirm that it is consistent with a multivariate beta model.     

Testing for lack of fit in inverse regression—with applications to biophotonic imaging

Summary.  We propose two test statistics for use in inverse regression problems Y = K θ + ɛ , where K is a given linear operator which cannot be continuously inverted. Thus, only noisy, indirect observations Y for the function θ are available. Both test statistics have a counterpart in classical hypothesis testing, where they are called the order selection test and the data-driven Neyman smooth test. We also introduce two model selection criteria which extend the classical Akaike information criterion and Bayes information criterion to inverse regression problems. In a simulation study we show that the inverse order selection and Neyman smooth tests outperform their direct counterparts in many cases. The theory is motivated by data arising in confocal fluorescence microscopy. Here, images are observed with blurring, modelled as convolution, and stochastic error at subsequent times. The aim is then to reduce the signal-to-noise ratio by averaging over the distinct images. In this context it is relevant to decide whether the images are still equal, or have changed by outside influences such as moving of the object table.