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Chapter 6 Continuous Random Variables and Probability
EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2019 Chapter 6 Continuous Random Variables and Probability Distributions Chap 6-1 Probability Distributions Probability Distributions Ch. 5 Discrete Continuous Ch. 6 Probability Probability Distributions Distributions Binomial Uniform Hypergeometric Normal Poisson Exponential Chap 6-2/62 Continuous Probability Distributions § A continuous random variable is a variable that can assume any value in an interval § thickness of an item § time required to complete a task § temperature of a solution § height in inches § These can potentially take on any value, depending only on the ability to measure accurately. Chap 6-3/62 Cumulative Distribution Function § The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x F(x) = P(X £ x) § Let a and b be two possible values of X, with a < b. The probability that X lies between a and b is P(a < X < b) = F(b) -F(a) Chap 6-4/62 Probability Density Function The probability density function, f(x), of random variable X has the following properties: 1. f(x) > 0 for all values of x 2. The area under the probability density function f(x) over all values of the random variable X is equal to 1.0 3. The probability that X lies between two values is the area under the density function graph between the two values 4. The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 x0 f(x ) = f(x)dx 0 ò xm where -
Skewed Double Exponential Distribution and Its Stochastic Rep- Resentation
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 1, 2009, (1-20) ISSN 1307-5543 – www.ejpam.com Skewed Double Exponential Distribution and Its Stochastic Rep- resentation 12 2 2 Keshav Jagannathan , Arjun K. Gupta ∗, and Truc T. Nguyen 1 Coastal Carolina University Conway, South Carolina, U.S.A 2 Bowling Green State University Bowling Green, Ohio, U.S.A Abstract. Definitions of the skewed double exponential (SDE) distribution in terms of a mixture of double exponential distributions as well as in terms of a scaled product of a c.d.f. and a p.d.f. of double exponential random variable are proposed. Its basic properties are studied. Multi-parameter versions of the skewed double exponential distribution are also given. Characterization of the SDE family of distributions and stochastic representation of the SDE distribution are derived. AMS subject classifications: Primary 62E10, Secondary 62E15. Key words: Symmetric distributions, Skew distributions, Stochastic representation, Linear combina- tion of random variables, Characterizations, Skew Normal distribution. 1. Introduction The double exponential distribution was first published as Laplace’s first law of error in the year 1774 and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error, disregarding sign. This distribution comes up as a model in many statistical problems. It is also considered in robustness studies, which suggests that it provides a model with different characteristics ∗Corresponding author. Email address: (A. Gupta) http://www.ejpam.com 1 c 2009 EJPAM All rights reserved. K. Jagannathan, A. Gupta, and T. Nguyen / Eur. -
A New Parameter Estimator for the Generalized Pareto Distribution Under the Peaks Over Threshold Framework
mathematics Article A New Parameter Estimator for the Generalized Pareto Distribution under the Peaks over Threshold Framework Xu Zhao 1,*, Zhongxian Zhang 1, Weihu Cheng 1 and Pengyue Zhang 2 1 College of Applied Sciences, Beijing University of Technology, Beijing 100124, China; [email protected] (Z.Z.); [email protected] (W.C.) 2 Department of Biomedical Informatics, College of Medicine, The Ohio State University, Columbus, OH 43210, USA; [email protected] * Correspondence: [email protected] Received: 1 April 2019; Accepted: 30 April 2019 ; Published: 7 May 2019 Abstract: Techniques used to analyze exceedances over a high threshold are in great demand for research in economics, environmental science, and other fields. The generalized Pareto distribution (GPD) has been widely used to fit observations exceeding the tail threshold in the peaks over threshold (POT) framework. Parameter estimation and threshold selection are two critical issues for threshold-based GPD inference. In this work, we propose a new GPD-based estimation approach by combining the method of moments and likelihood moment techniques based on the least squares concept, in which the shape and scale parameters of the GPD can be simultaneously estimated. To analyze extreme data, the proposed approach estimates the parameters by minimizing the sum of squared deviations between the theoretical GPD function and its expectation. Additionally, we introduce a recently developed stopping rule to choose the suitable threshold above which the GPD asymptotically fits the exceedances. Simulation studies show that the proposed approach performs better or similar to existing approaches, in terms of bias and the mean square error, in estimating the shape parameter. -
Independent Approximates Enable Closed-Form Estimation of Heavy-Tailed Distributions
Independent Approximates enable closed-form estimation of heavy-tailed distributions Kenric P. Nelson Abstract – Independent Approximates (IAs) are proven to enable a closed-form estimation of heavy-tailed distributions with an analytical density such as the generalized Pareto and Student’s t distributions. A broader proof using convolution of the characteristic function is described for future research. (IAs) are selected from independent, identically distributed samples by partitioning the samples into groups of size n and retaining the median of the samples in those groups which have approximately equal samples. The marginal distribution along the diagonal of equal values has a density proportional to the nth power of the original density. This nth-power-density, which the IAs approximate, has faster tail decay enabling closed-form estimation of its moments and retains a functional relationship with the original density. Computational experiments with between 1000 to 100,000 Student’s t samples are reported for over a range of location, scale, and shape (inverse of degree of freedom) parameters. IA pairs are used to estimate the location, IA triplets for the scale, and the geometric mean of the original samples for the shape. With 10,000 samples the relative bias of the parameter estimates is less than 0.01 and a relative precision is less than ±0.1. The theoretical bias is zero for the location and the finite bias for the scale can be subtracted out. The theoretical precision has a finite range when the shape is less than 2 for the location estimate and less than 3/2 for the scale estimate. -
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions F (x) = P r(X ≤ x) S(x) = 1 − F (x) dF (x) f(x) = dx H(x) = − ln S(x) dH(x) f(x) h(x) = = dx S(x) Functions of random variables Z 1 Expected Value E[g(x)] = g(x)f(x)dx −∞ 0 n n-th raw moment µn = E[X ] n n-th central moment µn = E[(X − µ) ] Variance σ2 = E[(X − µ)2] = E[X2] − µ2 µ µ0 − 3µ0 µ + 2µ3 Skewness γ = 3 = 3 2 1 σ3 σ3 µ µ0 − 4µ0 µ + 6µ0 µ2 − 3µ4 Kurtosis γ = 4 = 4 3 2 2 σ4 σ4 Moment generating function M(t) = E[etX ] Probability generating function P (z) = E[zX ] More concepts • Standard deviation (σ) is positive square root of variance • Coefficient of variation is CV = σ/µ • 100p-th percentile π is any point satisfying F (π−) ≤ p and F (π) ≥ p. If F is continuous, it is the unique point satisfying F (π) = p • Median is 50-th percentile; n-th quartile is 25n-th percentile • Mode is x which maximizes f(x) (n) n (n) • MX (0) = E[X ], where M is the n-th derivative (n) PX (0) • n! = P r(X = n) (n) • PX (1) is the n-th factorial moment of X. Bayes' Theorem P r(BjA)P r(A) P r(AjB) = P r(B) fY (yjx)fX (x) fX (xjy) = fY (y) Law of total probability 2 If Bi is a set of exhaustive (in other words, P r([iBi) = 1) and mutually exclusive (in other words P r(Bi \ Bj) = 0 for i 6= j) events, then for any event A, X X P r(A) = P r(A \ Bi) = P r(Bi)P r(AjBi) i i Correspondingly, for continuous distributions, Z P r(A) = P r(Ajx)f(x)dx Conditional Expectation Formula EX [X] = EY [EX [XjY ]] 3 Lesson 2: Parametric Distributions Forms of probability -
A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess
S S symmetry Article A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess Guillermo Martínez-Flórez 1, Víctor Leiva 2,* , Emilio Gómez-Déniz 3 and Carolina Marchant 4 1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 14014, Colombia; [email protected] 2 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile 3 Facultad de Economía, Empresa y Turismo, Universidad de Las Palmas de Gran Canaria and TIDES Institute, 35001 Canarias, Spain; [email protected] 4 Facultad de Ciencias Básicas, Universidad Católica del Maule, 3466706 Talca, Chile; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 June 2020; Accepted: 19 August 2020; Published: 1 September 2020 Abstract: In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Keywords: beta distribution; centered skew-normal distribution; maximum-likelihood methods; Monte Carlo simulations; proportions; R software; rates; zero/one inflated data 1. -
Lecture 2 — September 24 2.1 Recap 2.2 Exponential Families
STATS 300A: Theory of Statistics Fall 2015 Lecture 2 | September 24 Lecturer: Lester Mackey Scribe: Stephen Bates and Andy Tsao 2.1 Recap Last time, we set out on a quest to develop optimal inference procedures and, along the way, encountered an important pair of assertions: not all data is relevant, and irrelevant data can only increase risk and hence impair performance. This led us to introduce a notion of lossless data compression (sufficiency): T is sufficient for P with X ∼ Pθ 2 P if X j T (X) is independent of θ. How far can we take this idea? At what point does compression impair performance? These are questions of optimal data reduction. While we will develop general answers to these questions in this lecture and the next, we can often say much more in the context of specific modeling choices. With this in mind, let's consider an especially important class of models known as the exponential family models. 2.2 Exponential Families Definition 1. The model fPθ : θ 2 Ωg forms an s-dimensional exponential family if each Pθ has density of the form: s ! X p(x; θ) = exp ηi(θ)Ti(x) − B(θ) h(x) i=1 • ηi(θ) 2 R are called the natural parameters. • Ti(x) 2 R are its sufficient statistics, which follows from NFFC. • B(θ) is the log-partition function because it is the logarithm of a normalization factor: s ! ! Z X B(θ) = log exp ηi(θ)Ti(x) h(x)dµ(x) 2 R i=1 • h(x) 2 R: base measure. -
Statistical Inference
GU4204: Statistical Inference Bodhisattva Sen Columbia University February 27, 2020 Contents 1 Introduction5 1.1 Statistical Inference: Motivation.....................5 1.2 Recap: Some results from probability..................5 1.3 Back to Example 1.1...........................8 1.4 Delta method...............................8 1.5 Back to Example 1.1........................... 10 2 Statistical Inference: Estimation 11 2.1 Statistical model............................. 11 2.2 Method of Moments estimators..................... 13 3 Method of Maximum Likelihood 16 3.1 Properties of MLEs............................ 20 3.1.1 Invariance............................. 20 3.1.2 Consistency............................ 21 3.2 Computational methods for approximating MLEs........... 21 3.2.1 Newton's Method......................... 21 3.2.2 The EM Algorithm........................ 22 1 4 Principles of estimation 23 4.1 Mean squared error............................ 24 4.2 Comparing estimators.......................... 25 4.3 Unbiased estimators........................... 26 4.4 Sufficient Statistics............................ 28 5 Bayesian paradigm 33 5.1 Prior distribution............................. 33 5.2 Posterior distribution........................... 34 5.3 Bayes Estimators............................. 36 5.4 Sampling from a normal distribution.................. 37 6 The sampling distribution of a statistic 39 6.1 The gamma and the χ2 distributions.................. 39 6.1.1 The gamma distribution..................... 39 6.1.2 The Chi-squared distribution.................. 41 6.2 Sampling from a normal population................... 42 6.3 The t-distribution............................. 45 7 Confidence intervals 46 8 The (Cramer-Rao) Information Inequality 51 9 Large Sample Properties of the MLE 57 10 Hypothesis Testing 61 10.1 Principles of Hypothesis Testing..................... 61 10.2 Critical regions and test statistics.................... 62 10.3 Power function and types of error.................... 64 10.4 Significance level............................ -
Procedures for Estimation of Weibull Parameters James W
United States Department of Agriculture Procedures for Estimation of Weibull Parameters James W. Evans David E. Kretschmann David W. Green Forest Forest Products General Technical Report February Service Laboratory FPL–GTR–264 2019 Abstract Contents The primary purpose of this publication is to provide an 1 Introduction .................................................................. 1 overview of the information in the statistical literature on 2 Background .................................................................. 1 the different methods developed for fitting a Weibull distribution to an uncensored set of data and on any 3 Estimation Procedures .................................................. 1 comparisons between methods that have been studied in the 4 Historical Comparisons of Individual statistics literature. This should help the person using a Estimator Types ........................................................ 8 Weibull distribution to represent a data set realize some advantages and disadvantages of some basic methods. It 5 Other Methods of Estimating Parameters of should also help both in evaluating other studies using the Weibull Distribution .......................................... 11 different methods of Weibull parameter estimation and in 6 Discussion .................................................................. 12 discussions on American Society for Testing and Materials Standard D5457, which appears to allow a choice for the 7 Conclusion ................................................................ -
A New Weibull-X Family of Distributions: Properties, Characterizations and Applications Zubair Ahmad1* , M
Ahmad et al. Journal of Statistical Distributions and Applications (2018) 5:5 https://doi.org/10.1186/s40488-018-0087-6 RESEARCH Open Access A new Weibull-X family of distributions: properties, characterizations and applications Zubair Ahmad1* , M. Elgarhy2 and G. G. Hamedani3 * Correspondence: [email protected] Abstract 1Department of Statistics, Quaid-i-Azam University 45320, We propose a new family of univariate distributions generated from the Weibull random Islamabad 44000, Pakistan variable, called a new Weibull-X family of distributions. Two special sub-models of the Full list of author information is proposed family are presented and the shapes of density and hazard functions are available at the end of the article investigated. General expressions for some statistical properties are discussed. For the new family, three useful characterizations based on truncated moments are presented. Three different methods to estimate the model parameters are discussed. Monti Carlo simulation study is conducted to evaluate the performances of these estimators. Finally, the importance of the new family is illustrated empirically via two real life applications. Keywords: Weibull distribution, T-X family, Moment, Characterizations, Order statistics, Estimation 1. Introduction In the field of reliability theory, modeling of lifetime data is very crucial. A number of statistical distributions such as Weibull, Pareto, Gompertz, linear failure rate, Rayleigh, Exonential etc., are available for modeling lifetime data. However, in many practical areas, these classical distributions do not provide adequate fit in modeling data, and there is a clear need for the extended version of these classical distributions. In this re- gard, serious attempts have been made to propose new families of continuous prob- ability distributions that extend the existing well-known distributions by adding additional parameter(s) to the model of the baseline random variable. -
Likelihood-Based Inference
The score statistic Inference Exponential distribution example Likelihood-based inference Patrick Breheny September 29 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/32 The score statistic Univariate Inference Multivariate Exponential distribution example Introduction In previous lectures, we constructed and plotted likelihoods and used them informally to comment on likely values of parameters Our goal for today is to make this more rigorous, in terms of quantifying coverage and type I error rates for various likelihood-based approaches to inference With the exception of extremely simple cases such as the two-sample exponential model, exact derivation of these quantities is typically unattainable for survival models, and we must rely on asymptotic likelihood arguments Patrick Breheny Survival Data Analysis (BIOS 7210) 2/32 The score statistic Univariate Inference Multivariate Exponential distribution example The score statistic Likelihoods are typically easier to work with on the log scale (where products become sums); furthermore, since it is only relative comparisons that matter with likelihoods, it is more meaningful to work with derivatives than the likelihood itself Thus, we often work with the derivative of the log-likelihood, which is known as the score, and often denoted U: d U (θ) = `(θjX) X dθ Note that U is a random variable, as it depends on X U is a function of θ For independent observations, the score of the entire sample is the sum of the scores for the individual observations: X U = Ui i Patrick Breheny Survival -
Chapter 10 “Some Continuous Distributions”.Pdf
CHAPTER 10 Some continuous distributions 10.1. Examples of continuous random variables We look at some other continuous random variables besides normals. Uniform distribution A continuous random variable has uniform distribution if its density is f(x) = 1/(b a) − if a 6 x 6 b and 0 otherwise. For a random variable X with uniform distribution its expectation is 1 b a + b EX = x dx = . b a ˆ 2 − a Exponential distribution A continuous random variable has exponential distribution with parameter λ > 0 if its λx density is f(x) = λe− if x > 0 and 0 otherwise. Suppose X is a random variable with an exponential distribution with parameter λ. Then we have ∞ λx λa (10.1.1) P(X > a) = λe− dx = e− , ˆa λa FX (a) = 1 P(X > a) = 1 e− , − − and we can use integration by parts to see that EX = 1/λ, Var X = 1/λ2. Examples where an exponential random variable is a good model is the length of a telephone call, the length of time before someone arrives at a bank, the length of time before a light bulb burns out. Exponentials are memoryless, that is, P(X > s + t X > t) = P(X > s), | or given that the light bulb has burned 5 hours, the probability it will burn 2 more hours is the same as the probability a new light bulb will burn 2 hours. Here is how we can prove this 129 130 10. SOME CONTINUOUS DISTRIBUTIONS P(X > s + t) P(X > s + t X > t) = | P(X > t) λ(s+t) e− λs − = λt = e e− = P(X > s), where we used Equation (10.1.1) for a = t and a = s + t.