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This article proposes a procedure to estimate the number of common factors k in a static approximate factor model. The building block of the analysis is the fact that the first k eigenvalues of the covariance matrix of the data diverge, while the others stay bounded. On the grounds of this, we propose a test for the null that the ith eigenvalue diverges, using a randomized test statistic based directly on the estimated eigenvalue. The test only requires minimal assumptions on the data, and no assumptions are required on factors, loadings or idiosyncratic errors. The randomized tests are then employed in a sequential procedure to determine k. Supplementary materials for this article are available online.

In this article we introduce randomized repeated significance tests (RSTs), based on scan statistics, for detecting a local change in a parameter of a distribution for one and two dimensional discrete data. When the size of the window where a local change in the parameter has occurred is known, a randomized RST based on a fixed window scan statistic is proposed. When the size of the window where a local change in the parameter has occurred is unknown, a randomized RST based on the minimum P -value scan statistic is developed. Simulation based methods are used to implement these randomized RSTs. Numerical results for one and two dimensional data, generated from Bernoulli and Poisson distributions, for selected values of model parameters, demonstrate the effectiveness of the randomized RSTs in detecting a local change in the parameter of the respective model.

This paper is concerned with fuzzy hypothesis testing in the framework of the randomized and non-randomized hypergeometric test for a proportion. Moreover, we differentiate between a test of significance and an alternative test to control the type I error or both error types simultaneously. In contrast to classical (non-)randomized hypothesis testing, fuzzy hypothesis testing provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. In particular, various types of hypotheses with user-specified membership functions can be formulated. Additionally, the proposed test methods are compared via a comprehensive case study, which demonstrates the high flexibility of fuzzy hypothesis testing in practical applications.

In the analysis of next-generation sequencing technology, massive discrete data are generated from short read counts with varying biological coverage. Conducting conditional hypothesis testing such as Fisher's Exact Test at every genomic region of interest thus leads to a heterogeneous multiple discrete testing problem. However, most existing multiple testing procedures for controlling the false discovery rate (FDR) assume that test statistics are continuous and become conservative for discrete tests. To overcome the conservativeness, in this article, we propose a novel multiple testing procedure for better FDR control on heterogeneous discrete tests. Our procedure makes decisions based on the marginal critical function (MCF) of randomized tests, which enables achieving a powerful and non-randomized multiple testing procedure. We provide upper bounds of the positive FDR (pFDR) and the positive false non-discovery rate (pFNR) corresponding to our procedure. We also prove that the set of detections made by our method contains every detection made by a naive application of the widely-used q-value method. We further demonstrate the improvement of our method over other existing multiple testing procedures by simulations and a real example of differentially methylated region (DMR) detection using whole-genome bisulfite sequencing (WGBS) data.

Background Colorectal cancer (CRC) screening by fecal occult blood test (FOBT) significantly reduces CRC mortality, and compliance rates directly influence the efficacy of this screening method. The aim of this study is to investigate whether stool collection strategies affect compliance with the FOBT. Methods/Design In total, 3,596 study participants aged between 50 and 74 years will be recruited. The study will be conducted using a randomized controlled trial, with a 2 × 2 factorial design resulting in four groups. The first factor is the method of stool-collection device distribution (mailing vs. visiting the clinic) and the second is the type of stool-collection device (sampling kit vs. conventional container). Participants will be randomly assigned to one of four groups: (1) sampling kit received by mail; (2) conventional container received by mail; (3) sampling kit received at the clinic; (4) conventional container received at the clinic (control group). The primary outcome will be the FOBT compliance rate; satisfaction and intention to be rescreened in the next screening round will also be evaluated. The rates of positive FOBT results and detection of advanced adenomas or cancers through colonoscopies will also be compared between the two collection containers. Discussion Identifying a method of FOBT that yields high compliance rates will be a key determinant of the success of CRC screening. The findings of this study will provide reliable information for health policy makers to develop evidence-based strategies for a high compliance rate. Trial registration CRIS: KCT0000803 Date of registration in primary registry: 9 January, 2013.

The optimal hypothesis tests for the binomial distribution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two-tailed randomized tests. Conventional confidence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of UMP and UMPU tests. We also introduce a new P-value notion, called fuzzy P-values or abstract randomized P-values, that also inherits the same exactness and optimality.

We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289—302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitanić and Karatzas.

Fuzzy multiple comparisons procedures are introduced as a solution to the problem of multiple comparisons for discrete test statistics. The critical function of the randomized p-values is proposed as a measure of evidence against the null hypotheses. The classical concept of randomized tests is extended to multiple comparisons. This approach makes all theory of multiple comparisons developed for continuously distributed statistics automatically applicable to the discrete case. Examples of familywise error rate and false discovery rate procedures are discussed and an application to linkage disequilibrium testing is given. Software for implementing the procedures is available.

We consider the problem of testing a statistical hypothesis where the scientifically meaningful test statistic is a function of latent variables. In particular, we consider detection of genetic linkage, where the latent variables are patterns of inheritance at specific genome locations. Introduced by Geyer & Meeden (2005), fuzzy p-values are random variables, described by their probability distributions, that are interpreted as p-values. For latent variable problems, we introduce the notion of a fuzzy p-value as having the conditional distribution of the latent p-value given the observed data, where the latent p-value is the random variable that would be the p-value if the latent variables were observed. The fuzzy p-value provides an exact test using two sets of simulations of the latent variables under the null hypothesis, one unconditional and the other conditional on the observed data. It provides not only an expression of the strength of the evidence against the null hypothesis but also an expression of the uncertainty in that expression owing to lack of knowledge of the latent variables. We illustrate these features with an example of simulated data mimicking a real example of the detection of genetic linkage.

According to the Statement on Auditing Standards (SAS) No. 39 (AU 350.01), audit sampling is defined as \"the application of an audit procedure to less than 100 % of the items within an account balance or class of transactions for the purpose of evaluating some characteristic of the balance or class\". The audit system develops in different steps: some are not susceptible to sampling procedures, while others may be held using sampling techniques. The auditor may also be interested in two types of accounting error: the number of incorrect records in the sample that overcome a given threshold (natural error rate), which may be indicative of possible fraud, and the mean amount of monetary errors found in incorrect records. The aim of this study is to monitor jointly both types of errors through an appropriate system of hypotheses, with particular attention to the second type error that indicates the risk of non-reporting errors overcoming the upper precision limits.