Generalized Logistic Distributions
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Summary Notes for Survival Analysis
Summary Notes for Survival Analysis Instructor: Mei-Cheng Wang Department of Biostatistics Johns Hopkins University Spring, 2006 1 1 Introduction 1.1 Introduction De¯nition: A failure time (survival time, lifetime), T , is a nonnegative-valued random vari- able. For most of the applications, the value of T is the time from a certain event to a failure event. For example, a) in a clinical trial, time from start of treatment to a failure event b) time from birth to death = age at death c) to study an infectious disease, time from onset of infection to onset of disease d) to study a genetic disease, time from birth to onset of a disease = onset age 1.2 De¯nitions De¯nition. Cumulative distribution function F (t). F (t) = Pr(T · t) De¯nition. Survial function S(t). S(t) = Pr(T > t) = 1 ¡ Pr(T · t) Characteristics of S(t): a) S(t) = 1 if t < 0 b) S(1) = limt!1 S(t) = 0 c) S(t) is non-increasing in t In general, the survival function S(t) provides useful summary information, such as the me- dian survival time, t-year survival rate, etc. De¯nition. Density function f(t). 2 a) If T is a discrete random variable, f(t) = Pr(T = t) b) If T is (absolutely) continuous, the density function is Pr (Failure occurring in [t; t + ¢t)) f(t) = lim ¢t!0+ ¢t = Rate of occurrence of failure at t: Note that dF (t) dS(t) f(t) = = ¡ : dt dt De¯nition. Hazard function ¸(t). -
On the Scale Parameter of Exponential Distribution
Review of the Air Force Academy No.2 (34)/2017 ON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION Anca Ileana LUPAŞ Military Technical Academy, Bucharest, Romania ([email protected]) DOI: 10.19062/1842-9238.2017.15.2.16 Abstract: Exponential distribution is one of the widely used continuous distributions in various fields for statistical applications. In this paper we study the exact and asymptotical distribution of the scale parameter for this distribution. We will also define the confidence intervals for the studied parameter as well as the fixed length confidence intervals. 1. INTRODUCTION Exponential distribution is used in various statistical applications. Therefore, we often encounter exponential distribution in applications such as: life tables, reliability studies, extreme values analysis and others. In the following paper, we focus our attention on the exact and asymptotical repartition of the exponential distribution scale parameter estimator. 2. SCALE PARAMETER ESTIMATOR OF THE EXPONENTIAL DISTRIBUTION We will consider the random variable X with the following cumulative distribution function: x F(x ; ) 1 e ( x 0 , 0) (1) where is an unknown scale parameter Using the relationships between MXXX( ) ; 22( ) ; ( ) , we obtain ()X a theoretical variation coefficient 1. This is a useful indicator, especially if MX() you have observational data which seems to be exponential and with variation coefficient of the selection closed to 1. If we consider x12, x ,... xn as a part of a population that follows an exponential distribution, then by using the maximum likelihood estimation method we obtain the following estimate n ˆ 1 xi (2) n i1 119 On the Scale Parameter of Exponential Distribution Since M ˆ , it follows that ˆ is an unbiased estimator for . -
Estimation of Common Location and Scale Parameters in Nonregular Cases Ahmad Razmpour Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1982 Estimation of common location and scale parameters in nonregular cases Ahmad Razmpour Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Razmpour, Ahmad, "Estimation of common location and scale parameters in nonregular cases " (1982). Retrospective Theses and Dissertations. 7528. https://lib.dr.iastate.edu/rtd/7528 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. -
SOLUTION for HOMEWORK 1, STAT 3372 Welcome to Your First
SOLUTION FOR HOMEWORK 1, STAT 3372 Welcome to your first homework. Remember that you are always allowed to use Tables allowed on SOA/CAS exam 4. You can find them on my webpage. Another remark: 6 minutes per problem is your “speed”. Now you probably will not be able to solve your problems so fast — but this is the goal. Try to find mistakes (and get extra points) in my solutions. Typically they are silly arithmetic mistakes (not methodological ones). They allow me to check that you did your HW on your own. Please do not e-mail me about your findings — just mention them on the first page of your solution and count extra points. You can use them to compensate for wrongly solved problems in this homework. They cannot be counted beyond 10 maximum points for each homework. Now let us look at your problems. General Remark: It is always prudent to begin with writing down what is given and what you need to establish. This step can help you to understand/guess a possible solution. Also, your solution should be written neatly so, if you have extra time, it is easier to check your solution. 1. Problem 2.4 Given: The hazard function is h(x)=(A + e2x)I(x ≥ 0) and S(0.4) = 0.5. Find: A Solution: I need to relate h(x) and S(x) to find A. There is a formula on page 17, also discussed in class, which is helpful here. Namely, x − h(u)du S(x)= e 0 . R It allows us to find A. -
A Comparison of Unbiased and Plottingposition Estimators of L
WATER RESOURCES RESEARCH, VOL. 31, NO. 8, PAGES 2019-2025, AUGUST 1995 A comparison of unbiased and plotting-position estimators of L moments J. R. M. Hosking and J. R. Wallis IBM ResearchDivision, T. J. Watson ResearchCenter, Yorktown Heights, New York Abstract. Plotting-positionestimators of L momentsand L moment ratios have several disadvantagescompared with the "unbiased"estimators. For generaluse, the "unbiased'? estimatorsshould be preferred. Plotting-positionestimators may still be usefulfor estimatingextreme upper tail quantilesin regional frequencyanalysis. Probability-Weighted Moments and L Moments •r+l-" (--1)r • P*r,k Olk '- E p *r,!•[J!•. Probability-weightedmoments of a randomvariable X with k=0 k=0 cumulativedistribution function F( ) and quantile function It is convenient to define dimensionless versions of L mo- x( ) were definedby Greenwoodet al. [1979]to be the quan- tities ments;this is achievedby dividingthe higher-orderL moments by the scale measure h2. The L moment ratios •'r, r = 3, Mp,ra= E[XP{F(X)}r{1- F(X)} s] 4, '", are definedby ßr-" •r/•2 ß {X(u)}PUr(1 -- U)s du. L momentratios measure the shapeof a distributionindepen- dently of its scaleof measurement.The ratios *3 ("L skew- ness")and *4 ("L kurtosis")are nowwidely used as measures Particularlyuseful specialcases are the probability-weighted of skewnessand kurtosis,respectively [e.g., Schaefer,1990; moments Pilon and Adamowski,1992; Royston,1992; Stedingeret al., 1992; Vogeland Fennessey,1993]. 12•r= M1,0, r = •01 (1 - u)rx(u) du, Estimators Given an ordered sample of size n, Xl: n • X2:n • ''' • urx(u) du. X.... there are two establishedways of estimatingthe proba- /3r--- Ml,r, 0 =f01 bility-weightedmoments and L moments of the distribution from whichthe samplewas drawn. -
Study of Generalized Lomax Distribution and Change Point Problem
STUDY OF GENERALIZED LOMAX DISTRIBUTION AND CHANGE POINT PROBLEM Amani Alghamdi A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2018 Committee: Arjun K. Gupta, Committee Co-Chair Wei Ning, Committee Co-Chair Jane Chang, Graduate Faculty Representative John Chen Copyright c 2018 Amani Alghamdi All rights reserved iii ABSTRACT Arjun K. Gupta and Wei Ning, Committee Co-Chair Generalizations of univariate distributions are often of interest to serve for real life phenomena. These generalized distributions are very useful in many fields such as medicine, physics, engineer- ing and biology. Lomax distribution (Pareto-II) is one of the well known univariate distributions that is considered as an alternative to the exponential, gamma, and Weibull distributions for heavy tailed data. However, this distribution does not grant great flexibility in modeling data. In this dissertation, we introduce a generalization of the Lomax distribution called Rayleigh Lo- max (RL) distribution using the form obtained by El-Bassiouny et al. (2015). This distribution provides great fit in modeling wide range of real data sets. It is a very flexible distribution that is related to some of the useful univariate distributions such as exponential, Weibull and Rayleigh dis- tributions. Moreover, this new distribution can also be transformed to a lifetime distribution which is applicable in many situations. For example, we obtain the inverse estimation and confidence intervals in the case of progressively Type-II right censored situation. We also apply Schwartz information approach (SIC) and modified information approach (MIC) to detect the changes in parameters of the RL distribution. -
The Asymmetric T-Copula with Individual Degrees of Freedom
The Asymmetric t-Copula with Individual Degrees of Freedom d-fine GmbH Christ Church University of Oxford A thesis submitted for the degree of MSc in Mathematical Finance Michaelmas 2012 Abstract This thesis investigates asymmetric dependence structures of multivariate asset returns. Evidence of such asymmetry for equity returns has been reported in the literature. In order to model the dependence structure, a new t-copula approach is proposed called the skewed t-copula with individ- ual degrees of freedom (SID t-copula). This copula provides the flexibility to assign an individual degree-of-freedom parameter and an individual skewness parameter to each asset in a multivariate setting. Applying this approach to GARCH residuals of bivariate equity index return data and using maximum likelihood estimation, we find significant asymmetry. By means of the Akaike information criterion, it is demonstrated that the SID t-copula provides the best model for the market data compared to other copula approaches without explicit asymmetry parameters. In addition, it yields a better fit than the conventional skewed t-copula with a single degree-of-freedom parameter. In a model impact study, we analyse the errors which can occur when mod- elling asymmetric multivariate SID-t returns with the symmetric multi- variate Gauss or standard t-distribution. The comparison is done in terms of the risk measures value-at-risk and expected shortfall. We find large deviations between the modelled and the true VaR/ES of a spread posi- tion composed of asymmetrically distributed risk factors. Going from the bivariate case to a larger number of risk factors, the model errors increase. -
A Tail Quantile Approximation Formula for the Student T and the Symmetric Generalized Hyperbolic Distribution
A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schlüter, Stephan; Fischer, Matthias J. Working Paper A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution IWQW Discussion Papers, No. 05/2009 Provided in Cooperation with: Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics Suggested Citation: Schlüter, Stephan; Fischer, Matthias J. (2009) : A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution, IWQW Discussion Papers, No. 05/2009, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW), Nürnberg This Version is available at: http://hdl.handle.net/10419/29554 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No. -
Review of Multivariate Survival Data
Review of Multivariate Survival Data Guadalupe G´omez(1), M.Luz Calle(2), Anna Espinal(3) and Carles Serrat(1) (1) Universitat Polit`ecnica de Catalunya (2) Universitat de Vic (3) Universitat Aut`onoma de Barcelona DR2004/15 Review of Multivariate Survival Data ∗ Guadalupe G´omez [email protected] Departament d'Estad´ıstica i I.O. Universitat Polit`ecnica de Catalunya M.Luz Calle [email protected] Departament de Matem`atica i Inform`atica Universitat de Vic Carles Serrat [email protected] Departament de Matem`atica Aplicada I Universitat Polit`ecnica de Catalunya Anna Espinal [email protected] Departament de Matem`atiques Universitat Aut`onoma de Barcelona September 2004 ∗Partially supported by grant BFM2002-0027 PB98{0919 from the Ministerio de Cien- cia y Tecnologia Contents 1 Introduction 1 2 Review of bivariate nonparametric approaches 3 2.1 Introduction and notation . 3 2.2 Nonparametric estimation of the bivariate distribution func- tion under independent censoring . 5 2.3 Nonparametric estimation of the bivariate distribution func- tion under dependent right-censoring . 9 2.4 Average relative risk dependence measure . 18 3 Extensions of the bivariate survival estimator 20 3.1 Estimation of the bivariate survival function for two not con- secutive gap times. A discrete approach . 20 3.2 Burke's extension to two consecutive gap times . 25 4 Multivariate Survival Data 27 4.1 Notation . 27 4.2 Likelihood Function . 29 4.3 Modelling the time dependencies via Partial Likelihood . 31 5 Bayesian approach for modelling trends and time dependen- cies 35 5.1 Modelling time dependencies via indicators . -
Asymmetric Generalizations of Symmetric Univariate Probability Distributions Obtained Through Quantile Splicing
Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing a* a b Brenda V. Mac’Oduol , Paul J. van Staden and Robert A. R. King aDepartment of Statistics, University of Pretoria, Pretoria, South Africa; bSchool of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia *Correspondence to: Brenda V. Mac’Oduol; Department of Statistics, University of Pretoria, Pretoria, South Africa. email: [email protected] Abstract Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered. Keywords Two-piece; Half distributions; Quantile functions; L-moments 1. Introduction Balakrishnan, Dai, and Liu (2017) introduced a skew logistic distribution as an alterna- tive to the model proposed by van Staden and King (2015). It proposed taking the half distribution to the right of the location parameter and joining it to the half distribution to the left of the location parameter which has an inclusion of a single shape parameter, a > 0. The methodology made use of the cumulative distribution functions (CDFs) of the parent distributions to obtain the two-piece distribution. -
Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications
S S symmetry Article Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications Perla Celis 1,*, Rolando de la Cruz 1 , Claudio Fuentes 2 and Héctor W. Gómez 3 1 Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Diagonal Las Torres 2640, Peñalolén, Santiago 7941169, Chile; [email protected] 2 Department of Statistics, Oregon State University, 217 Weniger Hall, Corvallis, OR 97331, USA; [email protected] 3 Departamento de Matemática, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta 1240000, Chile; [email protected] * Correspondence: [email protected] Abstract: We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon– positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. -
A Comparison Between Skew-Logistic and Skew-Normal Distributions 1
MATEMATIKA, 2015, Volume 31, Number 1, 15–24 c UTM Centre for Industrial and Applied Mathematics A Comparison Between Skew-logistic and Skew-normal Distributions 1Ramin Kazemi and 2Monireh Noorizadeh 1,2Department of Statistics, Imam Khomeini International University Qazvin, Iran e-mail: [email protected] Abstract Skew distributions are reasonable models for describing claims in property- liability insurance. We consider two well-known datasets from actuarial science and fit skew-normal and skew-logistic distributions to these dataset. We find that the skew-logistic distribution is reasonably competitive compared to skew-normal in the literature when describing insurance data. The value at risk and tail value at risk are estimated for the dataset under consideration. Also, we compare the skew distributions via Kolmogorov-Smirnov goodness-of-fit test, log-likelihood criteria and AIC. Keywords Skew-logistic distribution; skew-normal distribution; value at risk; tail value at risk; log-likelihood criteria; AIC; Kolmogorov-Smirnov goodness-of-fit test. 2010 Mathematics Subject Classification 62P05, 91B30 1 Introduction Fitting an adequate distribution to real data sets is a relevant problem and not an easy task in actuarial literature, mainly due to the nature of the data, which shows several fea- tures to be accounted for. Eling [1] showed that the skew-normal and the skew-Student t distributions are reasonably competitive compared to some models when describing in- surance data. Bolance et al. [2] provided strong empirical evidence in favor of the use of the skew-normal, and log-skew-normal distributions to model bivariate claims data from the Spanish motor insurance industry.