Fisher, Neyman-Pearson Or NHST? a Tutorial for Teaching Data Testing
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Hypothesis Testing and Likelihood Ratio Tests
Hypottthesiiis tttestttiiing and llliiikellliiihood ratttiiio tttesttts Y We will adopt the following model for observed data. The distribution of Y = (Y1, ..., Yn) is parameter considered known except for some paramett er ç, which may be a vector ç = (ç1, ..., çk); ç“Ç, the paramettter space. The parameter space will usually be an open set. If Y is a continuous random variable, its probabiiillliiittty densiiittty functttiiion (pdf) will de denoted f(yy;ç) . If Y is y probability mass function y Y y discrete then f(yy;ç) represents the probabii ll ii tt y mass functt ii on (pmf); f(yy;ç) = Pç(YY=yy). A stttatttiiistttiiicalll hypottthesiiis is a statement about the value of ç. We are interested in testing the null hypothesis H0: ç“Ç0 versus the alternative hypothesis H1: ç“Ç1. Where Ç0 and Ç1 ¶ Ç. hypothesis test Naturally Ç0 § Ç1 = ∅, but we need not have Ç0 ∞ Ç1 = Ç. A hypott hesii s tt estt is a procedure critical region for deciding between H0 and H1 based on the sample data. It is equivalent to a crii tt ii call regii on: a critical region is a set C ¶ Rn y such that if y = (y1, ..., yn) “ C, H0 is rejected. Typically C is expressed in terms of the value of some tttesttt stttatttiiistttiiic, a function of the sample data. For µ example, we might have C = {(y , ..., y ): y – 0 ≥ 3.324}. The number 3.324 here is called a 1 n s/ n µ criiitttiiicalll valllue of the test statistic Y – 0 . S/ n If y“C but ç“Ç 0, we have committed a Type I error. -
Analysis of Imbalance Strategies Recommendation Using a Meta-Learning Approach
7th ICML Workshop on Automated Machine Learning (2020) Analysis of Imbalance Strategies Recommendation using a Meta-Learning Approach Afonso Jos´eCosta [email protected] CISUC, Department of Informatics Engineering, University of Coimbra, Portugal Miriam Seoane Santos [email protected] CISUC, Department of Informatics Engineering, University of Coimbra, Portugal Carlos Soares [email protected] Fraunhofer AICOS and LIACC, Faculty of Engineering, University of Porto, Portugal Pedro Henriques Abreu [email protected] CISUC, Department of Informatics Engineering, University of Coimbra, Portugal Abstract Class imbalance is known to degrade the performance of classifiers. To address this is- sue, imbalance strategies, such as oversampling or undersampling, are often employed to balance the class distribution of datasets. However, although several strategies have been proposed throughout the years, there is no one-fits-all solution for imbalanced datasets. In this context, meta-learning arises a way to recommend proper strategies to handle class imbalance, based on data characteristics. Nonetheless, in related work, recommendation- based systems do not focus on how the recommendation process is conducted, thus lacking interpretable knowledge. In this paper, several meta-characteristics are identified in order to provide interpretable knowledge to guide the selection of imbalance strategies, using Ex- ceptional Preferences Mining. The experimental setup considers 163 real-world imbalanced datasets, preprocessed with 9 well-known resampling algorithms. Our findings elaborate on the identification of meta-characteristics that demonstrate when using preprocessing algo- rithms is advantageous for the problem and when maintaining the original dataset benefits classification performance. 1. Introduction Class imbalance is known to severely affect the data quality. -
Particle Filtering-Based Maximum Likelihood Estimation for Financial Parameter Estimation
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Spiral - Imperial College Digital Repository Particle Filtering-based Maximum Likelihood Estimation for Financial Parameter Estimation Jinzhe Yang Binghuan Lin Wayne Luk Terence Nahar Imperial College & SWIP Techila Technologies Ltd Imperial College SWIP London & Edinburgh, UK Tampere University of Technology London, UK London, UK [email protected] Tampere, Finland [email protected] [email protected] [email protected] Abstract—This paper presents a novel method for estimating result, which cannot meet runtime requirement in practice. In parameters of financial models with jump diffusions. It is a addition, high performance computing (HPC) technologies are Particle Filter based Maximum Likelihood Estimation process, still new to the finance industry. We provide a PF based MLE which uses particle streams to enable efficient evaluation of con- straints and weights. We also provide a CPU-FPGA collaborative solution to estimate parameters of jump diffusion models, and design for parameter estimation of Stochastic Volatility with we utilise HPC technology to speedup the estimation scheme Correlated and Contemporaneous Jumps model as a case study. so that it would be acceptable for practical use. The result is evaluated by comparing with a CPU and a cloud This paper presents the algorithm development, as well computing platform. We show 14 times speed up for the FPGA as the pipeline design solution in FPGA technology, of PF design compared with the CPU, and similar speedup but better convergence compared with an alternative parallelisation scheme based MLE for parameter estimation of Stochastic Volatility using Techila Middleware on a multi-CPU environment. -
Resampling Hypothesis Tests for Autocorrelated Fields
JANUARY 1997 WILKS 65 Resampling Hypothesis Tests for Autocorrelated Fields D. S. WILKS Department of Soil, Crop and Atmospheric Sciences, Cornell University, Ithaca, New York (Manuscript received 5 March 1996, in ®nal form 10 May 1996) ABSTRACT Presently employed hypothesis tests for multivariate geophysical data (e.g., climatic ®elds) require the as- sumption that either the data are serially uncorrelated, or spatially uncorrelated, or both. Good methods have been developed to deal with temporal correlation, but generalization of these methods to multivariate problems involving spatial correlation has been problematic, particularly when (as is often the case) sample sizes are small relative to the dimension of the data vectors. Spatial correlation has been handled successfully by resampling methods when the temporal correlation can be neglected, at least according to the null hypothesis. This paper describes the construction of resampling tests for differences of means that account simultaneously for temporal and spatial correlation. First, univariate tests are derived that respect temporal correlation in the data, using the relatively new concept of ``moving blocks'' bootstrap resampling. These tests perform accurately for small samples and are nearly as powerful as existing alternatives. Simultaneous application of these univariate resam- pling tests to elements of data vectors (or ®elds) yields a powerful (i.e., sensitive) multivariate test in which the cross correlation between elements of the data vectors is successfully captured by the resampling, rather than through explicit modeling and estimation. 1. Introduction dle portion of Fig. 1. First, standard tests, including the t test, rest on the assumption that the underlying data It is often of interest to compare two datasets for are composed of independent samples from their parent differences with respect to particular attributes. -
Sampling and Resampling
Sampling and Resampling Thomas Lumley Biostatistics 2005-11-3 Basic Problem (Frequentist) statistics is based on the sampling distribution of statistics: Given a statistic Tn and a true data distribution, what is the distribution of Tn • The mean of samples of size 10 from an Exponential • The median of samples of size 20 from a Cauchy • The difference in Pearson’s coefficient of skewness ((X¯ − m)/s) between two samples of size 40 from a Weibull(3,7). This is easy: you have written programs to do it. In 512-3 you learn how to do it analytically in some cases. But... We really want to know: What is the distribution of Tn in sampling from the true data distribution But we don’t know the true data distribution. Empirical distribution On the other hand, we do have an estimate of the true data distribution. It should look like the sample data distribution. (we write Fn for the sample data distribution and F for the true data distribution). The Glivenko–Cantelli theorem says that the cumulative distribution function of the sample converges uniformly to the true cumulative distribution. We can work out the sampling distribution of Tn(Fn) by simulation, and hope that this is close to that of Tn(F ). Simulating from Fn just involves taking a sample, with replace- ment, from the observed data. This is called the bootstrap. We ∗ write Fn for the data distribution of a resample. Too good to be true? There are obviously some limits to this • It requires large enough samples for Fn to be close to F . -
This Is Dr. Chumney. the Focus of This Lecture Is Hypothesis Testing –Both What It Is, How Hypothesis Tests Are Used, and How to Conduct Hypothesis Tests
TRANSCRIPT: This is Dr. Chumney. The focus of this lecture is hypothesis testing –both what it is, how hypothesis tests are used, and how to conduct hypothesis tests. 1 TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing. 2 TRANSCRIPT: Let’s being the lecture with a summary of the logic process that underlies hypothesis testing. 3 TRANSCRIPT: It is often impossible or otherwise not feasible to collect data on every individual within a population. Therefore, researchers rely on samples to help answer questions about populations. Hypothesis testing is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest. Hypothesis testing is one of the most commonly used inferential procedures. Hypothesis testing will combine many of the concepts we have already covered, including z‐scores, probability, and the distribution of sample means. To conduct a hypothesis test, we first state a hypothesis about a population, predict the characteristics of a sample of that population (that is, we predict that a sample will be representative of the population), obtain a sample, then collect data from that sample and analyze the data to see if it is consistent with our hypotheses. 4 TRANSCRIPT: The process of hypothesis testing begins by stating a hypothesis about the unknown population. Actually we state two opposing hypotheses. The first hypothesis we state –the most important one –is the null hypothesis. The null hypothesis states that the treatment has no effect. In general the null hypothesis states that there is no change, no difference, no effect, and otherwise no relationship between the independent and dependent variables. -
The Scientific Method: Hypothesis Testing and Experimental Design
Appendix I The Scientific Method The study of science is different from other disciplines in many ways. Perhaps the most important aspect of “hard” science is its adherence to the principle of the scientific method: the posing of questions and the use of rigorous methods to answer those questions. I. Our Friend, the Null Hypothesis As a science major, you are probably no stranger to curiosity. It is the beginning of all scientific discovery. As you walk through the campus arboretum, you might wonder, “Why are trees green?” As you observe your peers in social groups at the cafeteria, you might ask yourself, “What subtle kinds of body language are those people using to communicate?” As you read an article about a new drug which promises to be an effective treatment for male pattern baldness, you think, “But how do they know it will work?” Asking such questions is the first step towards hypothesis formation. A scientific investigator does not begin the study of a biological phenomenon in a vacuum. If an investigator observes something interesting, s/he first asks a question about it, and then uses inductive reasoning (from the specific to the general) to generate an hypothesis based upon a logical set of expectations. To test the hypothesis, the investigator systematically collects data, either with field observations or a series of carefully designed experiments. By analyzing the data, the investigator uses deductive reasoning (from the general to the specific) to state a second hypothesis (it may be the same as or different from the original) about the observations. -
Efficient Computation of Adjusted P-Values for Resampling-Based Stepdown Multiple Testing
University of Zurich Department of Economics Working Paper Series ISSN 1664-7041 (print) ISSN 1664-705X (online) Working Paper No. 219 Efficient Computation of Adjusted p-Values for Resampling-Based Stepdown Multiple Testing Joseph P. Romano and Michael Wolf February 2016 Efficient Computation of Adjusted p-Values for Resampling-Based Stepdown Multiple Testing∗ Joseph P. Romano† Michael Wolf Departments of Statistics and Economics Department of Economics Stanford University University of Zurich Stanford, CA 94305, USA 8032 Zurich, Switzerland [email protected] [email protected] February 2016 Abstract There has been a recent interest in reporting p-values adjusted for resampling-based stepdown multiple testing procedures proposed in Romano and Wolf (2005a,b). The original papers only describe how to carry out multiple testing at a fixed significance level. Computing adjusted p-values instead in an efficient manner is not entirely trivial. Therefore, this paper fills an apparent gap by detailing such an algorithm. KEY WORDS: Adjusted p-values; Multiple testing; Resampling; Stepdown procedure. JEL classification code: C12. ∗We thank Henning M¨uller for helpful comments. †Research supported by NSF Grant DMS-1307973. 1 1 Introduction Romano and Wolf (2005a,b) propose resampling-based stepdown multiple testing procedures to control the familywise error rate (FWE); also see Romano et al. (2008, Section 3). The procedures as described are designed to be carried out at a fixed significance level α. Therefore, the result of applying such a procedure to a set of data will be a ‘list’ of binary decisions concerning the individual null hypotheses under study: reject or do not reject a given null hypothesis at the chosen significance level α. -
Hypothesis Testing – Examples and Case Studies
Hypothesis Testing – Chapter 23 Examples and Case Studies Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a test statistic. 3. Use the test statistic to determine the p-value. 4. The result is statistically significant if the p-value is less than or equal to the level of significance. Often media only presents results of step 4. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 2 23.2 Testing Hypotheses About Proportions and Means If the null and alternative hypotheses are expressed in terms of a population proportion, mean, or difference between two means and if the sample sizes are large … … the test statistic is simply the corresponding standardized score computed assuming the null hypothesis is true; and the p-value is found from a table of percentiles for standardized scores. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 3 Example 2: Weight Loss for Diet vs Exercise Did dieters lose more fat than the exercisers? Diet Only: sample mean = 5.9 kg sample standard deviation = 4.1 kg sample size = n = 42 standard error = SEM1 = 4.1/ √42 = 0.633 Exercise Only: sample mean = 4.1 kg sample standard deviation = 3.7 kg sample size = n = 47 standard error = SEM2 = 3.7/ √47 = 0.540 measure of variability = [(0.633)2 + (0.540)2] = 0.83 Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 4 Example 2: Weight Loss for Diet vs Exercise Step 1. -
Resampling Methods: Concepts, Applications, and Justification
Practical Assessment, Research, and Evaluation Volume 8 Volume 8, 2002-2003 Article 19 2002 Resampling methods: Concepts, Applications, and Justification Chong Ho Yu Follow this and additional works at: https://scholarworks.umass.edu/pare Recommended Citation Yu, Chong Ho (2002) "Resampling methods: Concepts, Applications, and Justification," Practical Assessment, Research, and Evaluation: Vol. 8 , Article 19. DOI: https://doi.org/10.7275/9cms-my97 Available at: https://scholarworks.umass.edu/pare/vol8/iss1/19 This Article is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Practical Assessment, Research, and Evaluation by an authorized editor of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Yu: Resampling methods: Concepts, Applications, and Justification A peer-reviewed electronic journal. Copyright is retained by the first or sole author, who grants right of first publication to Practical Assessment, Research & Evaluation. Permission is granted to distribute this article for nonprofit, educational purposes if it is copied in its entirety and the journal is credited. PARE has the right to authorize third party reproduction of this article in print, electronic and database forms. Volume 8, Number 19, September, 2003 ISSN=1531-7714 Resampling methods: Concepts, Applications, and Justification Chong Ho Yu Aries Technology/Cisco Systems Introduction In recent years many emerging statistical analytical tools, such as exploratory data analysis (EDA), data visualization, robust procedures, and resampling methods, have been gaining attention among psychological and educational researchers. However, many researchers tend to embrace traditional statistical methods rather than experimenting with these new techniques, even though the data structure does not meet certain parametric assumptions. -
Monte Carlo Simulation and Resampling
Monte Carlo Simulation and Resampling Tom Carsey (Instructor) Jeff Harden (TA) ICPSR Summer Course Summer, 2011 | Monte Carlo Simulation and Resampling 1/68 Resampling Resampling methods share many similarities to Monte Carlo simulations { in fact, some refer to resampling methods as a type of Monte Carlo simulation. Resampling methods use a computer to generate a large number of simulated samples. Patterns in these samples are then summarized and analyzed. However, in resampling methods, the simulated samples are drawn from the existing sample of data you have in your hands and NOT from a theoretically defined (researcher defined) DGP. Thus, in resampling methods, the researcher DOES NOT know or control the DGP, but the goal of learning about the DGP remains. | Monte Carlo Simulation and Resampling 2/68 Resampling Principles Begin with the assumption that there is some population DGP that remains unobserved. That DGP produced the one sample of data you have in your hands. Now, draw a new \sample" of data that consists of a different mix of the cases in your original sample. Repeat that many times so you have a lot of new simulated \samples." The fundamental assumption is that all information about the DGP contained in the original sample of data is also contained in the distribution of these simulated samples. If so, then resampling from the one sample you have is equivalent to generating completely new random samples from the population DGP. | Monte Carlo Simulation and Resampling 3/68 Resampling Principles (2) Another way to think about this is that if the sample of data you have in your hands is a reasonable representation of the population, then the distribution of parameter estimates produced from running a model on a series of resampled data sets will provide a good approximation of the distribution of that statistics in the population. -
Tests of Hypotheses Using Statistics
Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables .