A Bivariate Weibull Regression Model
|
|
- Earl Tyler
- 5 years ago
- Views:
Transcription
1 c Heldermann Verlag Economic Quality Control ISSN Vol 20 (2005), No. 1, 1 A Bivariate Weibull Regression Model David D. Hanagal Abstract: In this paper, we propose a new bivariate Weibull regression model based on censored samples with common covariates. There are some interesting biometrical situations which motivate the study of a bivariate Weibull regression model of the proposed type. A procedure for obtaining the maximum likelihood estimators for the parameters in the model is derived and a test of significance for the regression parameters is sketched. Key words: Bivariate Weibull model, parametric regression, Ssurvival times. 1 Introduction We introduce a new bivariate Weibull regression model based on censored samples with common covariates. Freund [1] proposed a bivariate exponential model (BVE) and Proschan- Sullo [7] modified Freund s BVE by allowing simultaneous failures. A modified bivariate Weibull (BVW) model is obtained by taking simple transformation of the bivariate exponential model (BVE) of Proschan-Sullo [7]. Hanagal [4] proposed a multivariate Weibull distribution which is a generalization of the multivariate exponential model of Marshall-Olkin [6]. We choose the BVW model because it is superior compared with the BVE model of Proschan-Sullo [7]. There are some situations arising in biometry which motivate the study of a BVW regression model of a particular type. For example, paired organs like Kidneys, Eyes, Ears or any other paired organs of an individual (or patient) may be looked at as a two component system. Failure of an organ increases the risk of other organ. We assume here that the lifetime of an individual is independent of the lifetimes of the paired organs an use the univariate censoring given by Hanagal [2, 3], since the death of an individual will censor both lifetimes of organs. Hanagal [5] proposed a bivariate Weibull regression model for the situation described above by extending Marshall-Olkin s bivariate exponential distribution. The covariates may be age of the patient, sex of the patient, smoking or alcoholic habits, diabetic or non-diabetic conditions, some specific diseases of the patient etc. Treating in such situations the lifetimes of paired organs of each patient as identically distributed BVW violates reality and, therefore, it is not advisable. Each patient features certain characteristics and, hence, it is necessary to incorporate covariates on which the
2 2 David D. Hanagal lifetimes of the paired organs may depend. These covariates represent individual properties of a patient relevant to the lifetimes and, thus, can be assumed to be the same for each considered pair of organs, i.e., we may assume common covariates. Unfortunately, at this stage of the investigations there were no real data available for evaluating our proposed model. In Section 2, we introduce the BVW regression model and in Section 3, we derive estimators for the parameters of the proposed model. In Section 4, we present a test procedures for checking the significance of the regression parameters. 2 Bivariate Weibull Regression Model The BVE of Freund [1] is given by the following joint probability density function (pdf): 1 22 e 22x 2 ( 22 )x 1 for 0 <x 1 <x 2 < f(x 1,x 2 ) = 2 11 e 11x 1 ( 11 )x 2 for 0 <x 2 <x 1 < (1) 3 e for 0 <x 1 = x 2 = x< where =( )and 1, 2, 3, 11, 22 > 0. By means of the transformations T 1 = X1 σ and T 2 = X2 σ, σ>0we get a bivariate Weibull model with joint pdf as follows: 1 22 σ 2 (t 1 t 2 ) σ 1 e 22t σ 2 ( 22)t σ 1 for 0 <t 1 <t 2 < f(t 1,t 2 ) = 2 11 σ 2 (t 1 t 2 ) σ 1 e 11t σ 1 ( 11)t σ 2 for 0 <t 2 <t 1 < (2) 3 σt σ 1 e for 0 <t 1 = t 2 = t< As it is well known for the BVE of Proschan-Sullo [7], the marginals are weighted combinations of two exponential distributions. Also in the BVW case, the marginals are weighted combinations of two Weibull distributions with same weights. The minimum of the two lifetimes min(t 1,T 2 ) is Weibull distributed with scale parameter ( ) and shape parameter σ. The lifetimes T 1 and T 2 are independent. whenever 11 = 1 and 22 = 2 holds. The probabilities in the two regions are given by P [T 1 <T 2 ] = 1 P [T 1 >T 2 ] = 2 P [T 1 = T 2 ] = 3 (3) Taking logarithm of the BVE variates in (1), i.e., Y 1 =logx 1 and Y 2 =logx 2,wegeta bivariate extreme value (BVEV) distribution with pdf given by:
3 A Bivariate Weibull Regression Model 3 f(y 1,y 2 ) = e y 1+log 1 +y 2 +log 22 e y 1 +log( 22 ) e y 2 +log 22 e y 2+log 2 +y 1 +log 11 e y 2 +log( 11 ) e y 1 +log 11 e y+log 3 e y+log for <y 1 <y 2 < for <y 2 <y 1 < (4) for <y 1 =y 2 =y< The two joint density function (1) and (4) show that the parameters ( 1, 2, 3, 11, 22 ) are not proper scale parameters for BVE of Proschan-Sullo [7]. It follows that BVE of Proschan-Sullo and also the corresponding BVEV do not belong to the location-scale family. The regression model for the two component system of interest is given by: ( ) ( ) Y1 β = 1X ( ) U1 Y 2 β 2X 2 σ U 2 (5) where X 1 and X 2 are m-dimensional vectors of regressor variables or covariates, β 1 and β 2 are m-dimensional vectors of regression coefficients and (U 1,U 2 ) are random variables having density functions as given in (4). For Y 1 =logt 1 and Y 2 =logt 2,weget ( ) ( ) T1 e = β 1 X 1 V 1 σ 1 T 2 e β 2 X 2 V 1 σ 2 (6) where (V 1 = e U 1,V 2 = e U 2 ) is BVE of Proschan-Sullo [7]. Alternatively we can write ( ) ( ) V1 e = σβ 1 X 1 T1 σ V 2 e σβ 2 X 2 T2 σ Assuming additionally that both components have not only common covariates, but also common regression parameters, i.e., X 1 = X 2 and β 1 = β 2 then (7) simplifies to: ( ) ( ) V1 T σ = 1 e σβ X V 2 T σ 2 3 Estimation of the Parameters Suppose that the study shall be based on an independent sample of size n and let the ith pair of the components have lifetimes (T 1i,T 2i ) and censoring time (Z i ). We assume the censoring time Z to be independent of the lifetimes (T 1,T 2 ). The lifetimes associated with the ith pair of the organs are given by (T 1i,T 2i ) = (T 1i,T 2i ) for max(t 1i,T 2i ) <Z i (T 1i,Z i ) for T 1i <Z i <T 2i (Z i,t 2i ) for T 2i <Z i <T 1i (Z i,z i ) for Z i < min(tt 1i,T 2i ) (7)
4 4 David D. Hanagal The likelihood function of the sample of size n is given by n 1 n 2 n 3 n 4 n 5 n 6 L = ( f 1,i )( f 2,i )( f 3,i )( f 4,i )( f 5,i )( F i ) (8) where f 1,i (t 1i t 2i ) = σ (t 1i t 2i ) σ 1 e 2σX β [( )t σ 1i + 22t σ 2i ]e σx β for 0 <t 1i <t 2i <z i f 2,i (t 1i t 2i ) = σ (t 1i t 2i ) σ 1 e 2σX β [( )t σ 2i + 11t σ 1i ]e σx β (9) (10) for 0 <t 2i <t 1i <z i f 3,i (t 1i t 2i ) = σ 3 t σ 1 i e σx β [( )t σ i e σx β (11) for 0 <t 1i = t 2i = t i <z i f 4,i (t 1i t 2i ) = lim δt i 0 f 5,i (t 1i t 2i ) = lim δt i 0 P [t 1i <T 1i <t 1i + δt i T 2i >z i ]P [T 2i >z i ] δt i β (12) = σ 1 t σ 1 1i e σx β [( )t σ 1i + 22zi σ]e σx for 0 <t 1i <z i <t 2i P [t 2i <T 2i <t 2i + δt i T 1i >z i ]P [T 1i >z i ] δt i β (13) = σ 2 t σ 1 1i e σx β [( )t σ 2i + 11zi σ]e σx for 0 <t 2i <z i <t 1i F i (z i ) = P [T 1i >z i,t 2i >z i ] = e ( )zi σe σx β (14) X = (X 1,..., X m ) (15) β = (β 1,..., β m ) (16) The integers n 1, n 2, n 3, n 4, n 5 and n 6 represent the number of observations falling in the range corresponding to f 1, f 2, f 3, f 4, f 5 and F, respectively. As can be seen from the above formulas, the densities f 1 and f 2 refer to Lebesque measures in R 2, while f 3, f 4 and f 5 refer Lebesque measures in R 1. The logarithm of the likelihood function is given by: log L = (2n 1 +2n 2 + n 3 + n 4 + n 5 )logσ +(n 1 + n 4 )log 1 +(n 2 + n 5 )log 2 +n 3 log 3 + n 1 log 22 + n 2 log 11 +(σ 1) log t 1i +(σ 1) iεa iεb log t 2i σ iεc X iβ σ iεd X iβ ( ) i exp( σx iβ) 11 (T1i σ T2i)exp( σx σ iβ) 22 (T2i σ T1i)exp( σx σ iβ) (17) where
5 A Bivariate Weibull Regression Model 5 A = {t 1i t 1i <z i } B = {t 2i t 2i <z i } C = {(t 1i,t 2i ) 0 <t 1i,t 2i <z i } D = {(t 1i,t 2i ) t 1i <z i or t 2i <z i } F = {(T 1i,T 2i ) T 2i <T 1i } G = {(T 1i,T 2i ) T 1i <T 2i } Wi σ = Min(T 1i,T 2i ) (18) From (17) the likelihood equations are obtained: (n 1 + n 4 ) Wi σ exp σx i β = 0 (19) 1 (n 2 + n 5 ) n n 2 i exp σx i β = 0 (20) i exp σx i β = 0 (21) (T1i σ T 2i)exp σ σx i β = 0 (22) 11 n 1 (T2i σ T 1i)exp σ σx i β = 0 (23) 22 (2n 1 +2n 2 + n 3 + n 4 + n 5 ) + log t 1i + log t 2i X σ iβ iεa iεb iεc X iβ ( ) Wi σ exp σx i β [log W i X iβ] iεd 11 [T1i[log σ T 1i X iβ] T2i[log σ T 2i X iβ]] exp σx i β 22 [T2i[log σ T 2i X iβ] T1i[log σ T 1i X iβ]] exp σx i β = 0 (24) σ X ji σ X ji + σ( ) iεc iεd +σ 11 X ji (T1i σ T2i)exp σ σx i β +σ 22 X ji (T2i σ T1i)exp σ σx i β = 0 X ji i for j =1,..., m exp σx i β The above likelihood equations cannot be solved analytically for obtaining explicit expressions for the maximum likelihood estimators(mles). However, they can be solved numerically, for example by the Newton-Raphson procedure. The second order partial derivatives of the log-likelihood function are as follows: 2 1 = (n 1 + n 4 ) 2 1 (25)
6 6 David D. Hanagal = (n 2 + n 5 ) 2 2 = n = n = n = (2n 1 +2n 2 + n 3 + n 4 + n 5 ) σ 2 σ 2 ( ) i exp σx i β (log W 1i X iβ) 2 (26) (27) (28) (29) 11 exp σx i β [T1i(log σ T 1i X iβ) 2 T2i(log σ T 2i X iβ) 2 ] 22 exp σx i β [T2i(log σ T 2i X iβ) 2 T1i(log σ T 1i X iβ) 2 ] (30) β j β k = ( )σ 2 11 σ 2 i X ji X ki exp σx i β (T σ 1i T σ 2i)X ji X ki exp σx i β 22 σ 2 (T σ 2i T σ 1i)X ji X ki exp σx i β for j, k =1,..., m (31) = 0 2 log L = 2 log L i j ii jj i jj for i, j =1, 2, 3; ii, jj =1, 2 (32) σ j = σ 11 = σ 22 = = σ β j k = σ β j 11 i exp σx i β (log W i X iβ) for j =1,.., m (33) exp σx i β [T σ 1i(log T 1i X iβ) T σ 2i(log T 2i X iβ)] (34) exp σx i β [T σ 2i(log T 2i X iβ) T σ 1i(log T 1i X iβ)] (35) i X ji exp σx i β for k =1, 2; j =1,.., m (36) X ji exp σx i β [T σ 1i T σ 2i] for j =1,..., m (37) β j 22 = σ X ji exp σx i β [T2i σ T1i] σ for j =1,..., m (38) β j σ = ( ) i X ji exp σx i β [1 + σ(log W i X iβ)]
7 A Bivariate Weibull Regression Model X ji exp σx i β [T1i[1 σ + σ(log T 1i X iβ)] T 2i[1+σ(log σ T 2i X iβ)]] + 22 X ji exp σx i β [T2i[1 σ + σ(log T 2i X iβ)] T 1i[1+σ(log σ T 1i X iβ)]] X ji X ji for j =1,.., m (39) iεc iεd The Fisher information matrix I is of (m +6) (m + 6) type and has the following form: I = i j 0 0 log L ii jj j σ jj σ j β l jj β l i σ ii σ σ 2 σ β l i β k ii β k σ β k β l β k with the second order partial derivatives given above. The inverse of the Fisher information matrix is the variance-covariance matrix (Σ = I 1 ) of the maximum likelihood estimators ˆ = (ˆ 1, ˆ 2, ˆ 3, ˆ 11, ˆ 22, ˆσ, ˆβ 1,..., ˆβ m ) of the distribution parameter =( 1, 2, 3, 11, 22,σ,β 1,...., β m ). Thus, the sample statistics n(ˆ ) (41) (40) follows asymptotically a multivariate normal distribution with mean vector zero and variance-covariance matrix Σ. 4 Test for Regression Coefficients In order to confirm that certain covariates are relevant, the hypotheses about β is put in the form H 0 : β 1 =0,withβ partitioned as β =(β 1,β 2 ) where β 1 is a k-dimensional vector. To test H 0 with significance level α one can use Λ 1 = ˆβ 1Σ 1 ˆβ 11 1 (42) where Σ 11 is the k k empirical variance-covariance matrix referring to ˆβ 1. Under H 0, Λ 1 follows asymptotically a χ 2 -distribution with k degrees of freedom. If the value of Λ 1 exceeds the corresponding (1 α)-quantile, the null-hypothesis H 0 can be rejected.
8 8 David D. Hanagal 5 Conclusions Statistical analysis is often reduced to one dimension because multidimensional models and procedures are hardly available. In this paper a model and a method is proposed which can be applied for analysing phenomena when it makes no sense to investigate the aspects of interest one by one using an one-dimensional model. This is the case for paired organs in biometry or systems with a number of identical components operating in parallel as hot redundancy in technical equipments. References [1] Freund, J.E. (1961): A bivariate extension of the exponential distribution. Journal of Amer. Statist. Assoc. 56, [2] Hanagal, D.D.(1992): Some inference results in bivariate exponential distributions basedoncensoredsamples. Comm. Statistics, Theory and Methods 21, [3] Hanagal, D.D. (1992).: Some inference results in modified Freund s bivariate exponential distribution. Biometrical Journal 34(6), [4] Hanagal, D.D.(1996). A multivariate Weibull distribution. Economic Quality Control 11, [5] Hanagal, D.D. (2004). Parametric bivariate regression analysis based on censored samples. Economic Quality Control 19, [6] Marshall, A.W. and Olkin, I.(1967). A multivariate exponential distribution. Journal of Amer. Statist. Assoc. 62, [7] Proschan, F. and Sullo, P. (1974): Estimating the parameters of bivariate exponential distributions in several sampling situations. In Reliability and Biometry. Eds. F. Proschan and R.J. Serfling, Philadelphia: SIAM, David D. Hanagal Department of Statistics University of Pune Pune , India
Estimation of the Bivariate Generalized. Lomax Distribution Parameters. Based on Censored Samples
Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 6, 257-267 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4329 Estimation of the Bivariate Generalized Lomax Distribution Parameters
More informationOptimizing Combination Therapy under a Bivariate Weibull Distribution, with Application to Toxicity and Efficacy Responses
Optimizing Combination Therapy under a Bivariate Weibull Distribution, with Application to Toxicity and Efficacy Responses Kim, Sungwook (Peter) University of Missouri Issac Newton Institute for Mathematical
More informationSTT 843 Key to Homework 1 Spring 2018
STT 843 Key to Homework Spring 208 Due date: Feb 4, 208 42 (a Because σ = 2, σ 22 = and ρ 2 = 05, we have σ 2 = ρ 2 σ σ22 = 2/2 Then, the mean and covariance of the bivariate normal is µ = ( 0 2 and Σ
More informationTMA 4275 Lifetime Analysis June 2004 Solution
TMA 4275 Lifetime Analysis June 2004 Solution Problem 1 a) Observation of the outcome is censored, if the time of the outcome is not known exactly and only the last time when it was observed being intact,
More informationTests of independence for censored bivariate failure time data
Tests of independence for censored bivariate failure time data Abstract Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of χ
More informationGOODNESS-OF-FIT TESTS FOR ARCHIMEDEAN COPULA MODELS
Statistica Sinica 20 (2010), 441-453 GOODNESS-OF-FIT TESTS FOR ARCHIMEDEAN COPULA MODELS Antai Wang Georgetown University Medical Center Abstract: In this paper, we propose two tests for parametric models
More informationA New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto
International Mathematical Forum, 2, 27, no. 26, 1259-1273 A New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto A. S. Al-Ruzaiza and Awad El-Gohary 1 Department of Statistics
More informationMarshall-Olkin Bivariate Exponential Distribution: Generalisations and Applications
CHAPTER 6 Marshall-Olkin Bivariate Exponential Distribution: Generalisations and Applications 6.1 Introduction Exponential distributions have been introduced as a simple model for statistical analysis
More informationAnalysis of Time-to-Event Data: Chapter 4 - Parametric regression models
Analysis of Time-to-Event Data: Chapter 4 - Parametric regression models Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/25 Right censored
More informationPh.D. Qualifying Exam Friday Saturday, January 6 7, 2017
Ph.D. Qualifying Exam Friday Saturday, January 6 7, 2017 Put your solution to each problem on a separate sheet of paper. Problem 1. (5106) Let X 1, X 2,, X n be a sequence of i.i.d. observations from a
More informationSTAT 6350 Analysis of Lifetime Data. Failure-time Regression Analysis
STAT 6350 Analysis of Lifetime Data Failure-time Regression Analysis Explanatory Variables for Failure Times Usually explanatory variables explain/predict why some units fail quickly and some units survive
More informationInterval Estimation for Parameters of a Bivariate Time Varying Covariate Model
Pertanika J. Sci. & Technol. 17 (2): 313 323 (2009) ISSN: 0128-7680 Universiti Putra Malaysia Press Interval Estimation for Parameters of a Bivariate Time Varying Covariate Model Jayanthi Arasan Department
More informationPart III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data
1 Part III. Hypothesis Testing III.1. Log-rank Test for Right-censored Failure Time Data Consider a survival study consisting of n independent subjects from p different populations with survival functions
More informationMultivariate Survival Data With Censoring.
1 Multivariate Survival Data With Censoring. Shulamith Gross and Catherine Huber-Carol Baruch College of the City University of New York, Dept of Statistics and CIS, Box 11-220, 1 Baruch way, 10010 NY.
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More informationCopula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011
Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models
More informationCOMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES
COMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES Jerzy Filus Department of Mathematics and Computer Science, Oakton Community
More information[y i α βx i ] 2 (2) Q = i=1
Least squares fits This section has no probability in it. There are no random variables. We are given n points (x i, y i ) and want to find the equation of the line that best fits them. We take the equation
More informationHypothesis Testing Based on the Maximum of Two Statistics from Weighted and Unweighted Estimating Equations
Hypothesis Testing Based on the Maximum of Two Statistics from Weighted and Unweighted Estimating Equations Takeshi Emura and Hisayuki Tsukuma Abstract For testing the regression parameter in multivariate
More informationAn Extension of the Freund s Bivariate Distribution to Model Load Sharing Systems
An Extension of the Freund s Bivariate Distribution to Model Load Sharing Systems G Asha Department of Statistics Cochin University of Science and Technology Cochin, Kerala, e-mail:asha@cusat.ac.in Jagathnath
More informationFULL LIKELIHOOD INFERENCES IN THE COX MODEL
October 20, 2007 FULL LIKELIHOOD INFERENCES IN THE COX MODEL BY JIAN-JIAN REN 1 AND MAI ZHOU 2 University of Central Florida and University of Kentucky Abstract We use the empirical likelihood approach
More informationChapter 17. Failure-Time Regression Analysis. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University
Chapter 17 Failure-Time Regression Analysis William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University Copyright 1998-2008 W. Q. Meeker and L. A. Escobar. Based on the authors
More informationCOPYRIGHTED MATERIAL CONTENTS. Preface Preface to the First Edition
Preface Preface to the First Edition xi xiii 1 Basic Probability Theory 1 1.1 Introduction 1 1.2 Sample Spaces and Events 3 1.3 The Axioms of Probability 7 1.4 Finite Sample Spaces and Combinatorics 15
More informationApproximation of Survival Function by Taylor Series for General Partly Interval Censored Data
Malaysian Journal of Mathematical Sciences 11(3): 33 315 (217) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal Approximation of Survival Function by Taylor
More informationLecture 3. Truncation, length-bias and prevalence sampling
Lecture 3. Truncation, length-bias and prevalence sampling 3.1 Prevalent sampling Statistical techniques for truncated data have been integrated into survival analysis in last two decades. Truncation in
More informationSurvival Analysis Math 434 Fall 2011
Survival Analysis Math 434 Fall 2011 Part IV: Chap. 8,9.2,9.3,11: Semiparametric Proportional Hazards Regression Jimin Ding Math Dept. www.math.wustl.edu/ jmding/math434/fall09/index.html Basic Model Setup
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO
UNIVERSITY OF CALIFORNIA, SAN DIEGO Estimation of the primary hazard ratio in the presence of a secondary covariate with non-proportional hazards An undergraduate honors thesis submitted to the Department
More informationSurvival Analysis. Lu Tian and Richard Olshen Stanford University
1 Survival Analysis Lu Tian and Richard Olshen Stanford University 2 Survival Time/ Failure Time/Event Time We will introduce various statistical methods for analyzing survival outcomes What is the survival
More informationStatistical Analysis of Competing Risks With Missing Causes of Failure
Proceedings 59th ISI World Statistics Congress, 25-3 August 213, Hong Kong (Session STS9) p.1223 Statistical Analysis of Competing Risks With Missing Causes of Failure Isha Dewan 1,3 and Uttara V. Naik-Nimbalkar
More informationThe Log-generalized inverse Weibull Regression Model
The Log-generalized inverse Weibull Regression Model Felipe R. S. de Gusmão Universidade Federal Rural de Pernambuco Cintia M. L. Ferreira Universidade Federal Rural de Pernambuco Sílvio F. A. X. Júnior
More informationCOMPARISON OF FIVE TESTS FOR THE COMMON MEAN OF SEVERAL MULTIVARIATE NORMAL POPULATIONS
Communications in Statistics - Simulation and Computation 33 (2004) 431-446 COMPARISON OF FIVE TESTS FOR THE COMMON MEAN OF SEVERAL MULTIVARIATE NORMAL POPULATIONS K. Krishnamoorthy and Yong Lu Department
More informationChapter 2 Inference on Mean Residual Life-Overview
Chapter 2 Inference on Mean Residual Life-Overview Statistical inference based on the remaining lifetimes would be intuitively more appealing than the popular hazard function defined as the risk of immediate
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationNotes on the Multivariate Normal and Related Topics
Version: July 10, 2013 Notes on the Multivariate Normal and Related Topics Let me refresh your memory about the distinctions between population and sample; parameters and statistics; population distributions
More informationPower and Sample Size Calculations with the Additive Hazards Model
Journal of Data Science 10(2012), 143-155 Power and Sample Size Calculations with the Additive Hazards Model Ling Chen, Chengjie Xiong, J. Philip Miller and Feng Gao Washington University School of Medicine
More informationLinear Models and Estimation by Least Squares
Linear Models and Estimation by Least Squares Jin-Lung Lin 1 Introduction Causal relation investigation lies in the heart of economics. Effect (Dependent variable) cause (Independent variable) Example:
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationStatistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach
Statistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach Jae-Kwang Kim Department of Statistics, Iowa State University Outline 1 Introduction 2 Observed likelihood 3 Mean Score
More informationExercises. (a) Prove that m(t) =
Exercises 1. Lack of memory. Verify that the exponential distribution has the lack of memory property, that is, if T is exponentially distributed with parameter λ > then so is T t given that T > t for
More informationMultivariate Regression
Multivariate Regression The so-called supervised learning problem is the following: we want to approximate the random variable Y with an appropriate function of the random variables X 1,..., X p with the
More informationThe regression model with one fixed regressor cont d
The regression model with one fixed regressor cont d 3150/4150 Lecture 4 Ragnar Nymoen 27 January 2012 The model with transformed variables Regression with transformed variables I References HGL Ch 2.8
More informationQuantile Regression for Residual Life and Empirical Likelihood
Quantile Regression for Residual Life and Empirical Likelihood Mai Zhou email: mai@ms.uky.edu Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA Jong-Hyeon Jeong email: jeong@nsabp.pitt.edu
More informationExact Inference for the Two-Parameter Exponential Distribution Under Type-II Hybrid Censoring
Exact Inference for the Two-Parameter Exponential Distribution Under Type-II Hybrid Censoring A. Ganguly, S. Mitra, D. Samanta, D. Kundu,2 Abstract Epstein [9] introduced the Type-I hybrid censoring scheme
More informationEstimation of Conditional Kendall s Tau for Bivariate Interval Censored Data
Communications for Statistical Applications and Methods 2015, Vol. 22, No. 6, 599 604 DOI: http://dx.doi.org/10.5351/csam.2015.22.6.599 Print ISSN 2287-7843 / Online ISSN 2383-4757 Estimation of Conditional
More informationAFT Models and Empirical Likelihood
AFT Models and Empirical Likelihood Mai Zhou Department of Statistics, University of Kentucky Collaborators: Gang Li (UCLA); A. Bathke; M. Kim (Kentucky) Accelerated Failure Time (AFT) models: Y = log(t
More informationStatistics 3858 : Maximum Likelihood Estimators
Statistics 3858 : Maximum Likelihood Estimators 1 Method of Maximum Likelihood In this method we construct the so called likelihood function, that is L(θ) = L(θ; X 1, X 2,..., X n ) = f n (X 1, X 2,...,
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 00 MODULE : Statistical Inference Time Allowed: Three Hours Candidates should answer FIVE questions. All questions carry equal marks. The
More informationFinal Exam. 1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given.
1. (6 points) True/False. Please read the statements carefully, as no partial credit will be given. (a) If X and Y are independent, Corr(X, Y ) = 0. (b) (c) (d) (e) A consistent estimator must be asymptotically
More informationSPRING 2007 EXAM C SOLUTIONS
SPRING 007 EXAM C SOLUTIONS Question #1 The data are already shifted (have had the policy limit and the deductible of 50 applied). The two 350 payments are censored. Thus the likelihood function is L =
More informationUniversity of California, Berkeley
University of California, Berkeley U.C. Berkeley Division of Biostatistics Working Paper Series Year 24 Paper 153 A Note on Empirical Likelihood Inference of Residual Life Regression Ying Qing Chen Yichuan
More informationParameters Estimation for a Linear Exponential Distribution Based on Grouped Data
International Mathematical Forum, 3, 2008, no. 33, 1643-1654 Parameters Estimation for a Linear Exponential Distribution Based on Grouped Data A. Al-khedhairi Department of Statistics and O.R. Faculty
More informationMultivariate Statistics
Multivariate Statistics Chapter 2: Multivariate distributions and inference Pedro Galeano Departamento de Estadística Universidad Carlos III de Madrid pedro.galeano@uc3m.es Course 2016/2017 Master in Mathematical
More informationQuasi-likelihood Scan Statistics for Detection of
for Quasi-likelihood for Division of Biostatistics and Bioinformatics, National Health Research Institutes & Department of Mathematics, National Chung Cheng University 17 December 2011 1 / 25 Outline for
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationRELATIVE ERRORS IN RELIABILITY MEASURES. University of Maine and University of New Brunswick
RELATIVE ERRORS IN RELIABILITY MEASURES BY PUSHPA L. GUPTA AND R.D. GUPTA 1 University of Maine and University of New Brunswick A common assumption, in reliability and lifetesting situations when the components
More informationROBUSTNESS OF TWO-PHASE REGRESSION TESTS
REVSTAT Statistical Journal Volume 3, Number 1, June 2005, 1 18 ROBUSTNESS OF TWO-PHASE REGRESSION TESTS Authors: Carlos A.R. Diniz Departamento de Estatística, Universidade Federal de São Carlos, São
More informationMAS3301 / MAS8311 Biostatistics Part II: Survival
MAS3301 / MAS8311 Biostatistics Part II: Survival M. Farrow School of Mathematics and Statistics Newcastle University Semester 2, 2009-10 1 13 The Cox proportional hazards model 13.1 Introduction In the
More informationIDENTIFIABILITY OF THE MULTIVARIATE NORMAL BY THE MAXIMUM AND THE MINIMUM
Surveys in Mathematics and its Applications ISSN 842-6298 (electronic), 843-7265 (print) Volume 5 (200), 3 320 IDENTIFIABILITY OF THE MULTIVARIATE NORMAL BY THE MAXIMUM AND THE MINIMUM Arunava Mukherjea
More informationLecture 3. Inference about multivariate normal distribution
Lecture 3. Inference about multivariate normal distribution 3.1 Point and Interval Estimation Let X 1,..., X n be i.i.d. N p (µ, Σ). We are interested in evaluation of the maximum likelihood estimates
More informationStep-Stress Models and Associated Inference
Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated
More informationAnalysis of 2 n Factorial Experiments with Exponentially Distributed Response Variable
Applied Mathematical Sciences, Vol. 5, 2011, no. 10, 459-476 Analysis of 2 n Factorial Experiments with Exponentially Distributed Response Variable S. C. Patil (Birajdar) Department of Statistics, Padmashree
More informationIntroduction to Normal Distribution
Introduction to Normal Distribution Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 17-Jan-2017 Nathaniel E. Helwig (U of Minnesota) Introduction
More information3. Linear Regression With a Single Regressor
3. Linear Regression With a Single Regressor Econometrics: (I) Application of statistical methods in empirical research Testing economic theory with real-world data (data analysis) 56 Econometrics: (II)
More informationDistribution Theory. Comparison Between Two Quantiles: The Normal and Exponential Cases
Communications in Statistics Simulation and Computation, 34: 43 5, 005 Copyright Taylor & Francis, Inc. ISSN: 0361-0918 print/153-4141 online DOI: 10.1081/SAC-00055639 Distribution Theory Comparison Between
More informationHomoskedasticity. Var (u X) = σ 2. (23)
Homoskedasticity How big is the difference between the OLS estimator and the true parameter? To answer this question, we make an additional assumption called homoskedasticity: Var (u X) = σ 2. (23) This
More informationOptimum Test Plan for 3-Step, Step-Stress Accelerated Life Tests
International Journal of Performability Engineering, Vol., No., January 24, pp.3-4. RAMS Consultants Printed in India Optimum Test Plan for 3-Step, Step-Stress Accelerated Life Tests N. CHANDRA *, MASHROOR
More informationSemiparametric Regression
Semiparametric Regression Patrick Breheny October 22 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/23 Introduction Over the past few weeks, we ve introduced a variety of regression models under
More informationECON 4160, Autumn term Lecture 1
ECON 4160, Autumn term 2017. Lecture 1 a) Maximum Likelihood based inference. b) The bivariate normal model Ragnar Nymoen University of Oslo 24 August 2017 1 / 54 Principles of inference I Ordinary least
More informationLecture 6 Multiple Linear Regression, cont.
Lecture 6 Multiple Linear Regression, cont. BIOST 515 January 22, 2004 BIOST 515, Lecture 6 Testing general linear hypotheses Suppose we are interested in testing linear combinations of the regression
More informationEstimation Under Multivariate Inverse Weibull Distribution
Global Journal of Pure and Applied Mathematics. ISSN 097-768 Volume, Number 8 (07), pp. 4-4 Research India Publications http://www.ripublication.com Estimation Under Multivariate Inverse Weibull Distribution
More informationMULTIVARIATE DISCRETE PHASE-TYPE DISTRIBUTIONS
MULTIVARIATE DISCRETE PHASE-TYPE DISTRIBUTIONS By MATTHEW GOFF A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department
More informationAkaike Information Criterion
Akaike Information Criterion Shuhua Hu Center for Research in Scientific Computation North Carolina State University Raleigh, NC February 7, 2012-1- background Background Model statistical model: Y j =
More information4. Comparison of Two (K) Samples
4. Comparison of Two (K) Samples K=2 Problem: compare the survival distributions between two groups. E: comparing treatments on patients with a particular disease. Z: Treatment indicator, i.e. Z = 1 for
More informationNegative Multinomial Model and Cancer. Incidence
Generalized Linear Model under the Extended Negative Multinomial Model and Cancer Incidence S. Lahiri & Sunil K. Dhar Department of Mathematical Sciences, CAMS New Jersey Institute of Technology, Newar,
More informationDiscriminating Between the Bivariate Generalized Exponential and Bivariate Weibull Distributions
Discriminating Between the Bivariate Generalized Exponential and Bivariate Weibull Distributions Arabin Kumar Dey & Debasis Kundu Abstract Recently Kundu and Gupta ( Bivariate generalized exponential distribution,
More informationAnalysis of Time-to-Event Data: Chapter 6 - Regression diagnostics
Analysis of Time-to-Event Data: Chapter 6 - Regression diagnostics Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/25 Residuals for the
More informationIn contrast, parametric techniques (fitting exponential or Weibull, for example) are more focussed, can handle general covariates, but require
Chapter 5 modelling Semi parametric We have considered parametric and nonparametric techniques for comparing survival distributions between different treatment groups. Nonparametric techniques, such as
More informationThe comparative studies on reliability for Rayleigh models
Journal of the Korean Data & Information Science Society 018, 9, 533 545 http://dx.doi.org/10.7465/jkdi.018.9..533 한국데이터정보과학회지 The comparative studies on reliability for Rayleigh models Ji Eun Oh 1 Joong
More informationChap 2. Linear Classifiers (FTH, ) Yongdai Kim Seoul National University
Chap 2. Linear Classifiers (FTH, 4.1-4.4) Yongdai Kim Seoul National University Linear methods for classification 1. Linear classifiers For simplicity, we only consider two-class classification problems
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationReliability Modelling Incorporating Load Share and Frailty
Reliability Modelling Incorporating Load Share and Frailty Vincent Raja Anthonisamy Department of Mathematics, Physics and Statistics Faculty of Natural Sciences, University of Guyana Georgetown, Guyana,
More informationCTDL-Positive Stable Frailty Model
CTDL-Positive Stable Frailty Model M. Blagojevic 1, G. MacKenzie 2 1 Department of Mathematics, Keele University, Staffordshire ST5 5BG,UK and 2 Centre of Biostatistics, University of Limerick, Ireland
More informationDepartment of Statistical Science FIRST YEAR EXAM - SPRING 2017
Department of Statistical Science Duke University FIRST YEAR EXAM - SPRING 017 Monday May 8th 017, 9:00 AM 1:00 PM NOTES: PLEASE READ CAREFULLY BEFORE BEGINNING EXAM! 1. Do not write solutions on the exam;
More informationStatistical Process Control Methods from the Viewpoint of Industrial Application
c Heldermann Verlag Economic Quality Control ISSN 0940-5151 Vol 16 (2001), No. 1, 49 63 Statistical Process Control Methods from the Viewpoint of Industrial Application Constantin Anghel Abstract: Statistical
More informationThe linear model is the most fundamental of all serious statistical models encompassing:
Linear Regression Models: A Bayesian perspective Ingredients of a linear model include an n 1 response vector y = (y 1,..., y n ) T and an n p design matrix (e.g. including regressors) X = [x 1,..., x
More informationA STATISTICAL TEST FOR MONOTONIC AND NON-MONOTONIC TREND IN REPAIRABLE SYSTEMS
A STATISTICAL TEST FOR MONOTONIC AND NON-MONOTONIC TREND IN REPAIRABLE SYSTEMS Jan Terje Kvaløy Department of Mathematics and Science, Stavanger University College, P.O. Box 2557 Ullandhaug, N-491 Stavanger,
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
More informationProblems. Suppose both models are fitted to the same data. Show that SS Res, A SS Res, B
Simple Linear Regression 35 Problems 1 Consider a set of data (x i, y i ), i =1, 2,,n, and the following two regression models: y i = β 0 + β 1 x i + ε, (i =1, 2,,n), Model A y i = γ 0 + γ 1 x i + γ 2
More informationProbability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More informationMultivariate Regression Generalized Likelihood Ratio Tests for FMRI Activation
Multivariate Regression Generalized Likelihood Ratio Tests for FMRI Activation Daniel B Rowe Division of Biostatistics Medical College of Wisconsin Technical Report 40 November 00 Division of Biostatistics
More informationProblem 1 (20) Log-normal. f(x) Cauchy
ORF 245. Rigollet Date: 11/21/2008 Problem 1 (20) f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 4 2 0 2 4 Normal (with mean -1) 4 2 0 2 4 Negative-exponential x x f(x) f(x) 0.0 0.1 0.2 0.3 0.4 0.5
More informationTest Problems for Probability Theory ,
1 Test Problems for Probability Theory 01-06-16, 010-1-14 1. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30
More informationPoint and Interval Estimation for Gaussian Distribution, Based on Progressively Type-II Censored Samples
90 IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 Point and Interval Estimation for Gaussian Distribution, Based on Progressively Type-II Censored Samples N. Balakrishnan, N. Kannan, C. T.
More informationSTAT331. Cox s Proportional Hazards Model
STAT331 Cox s Proportional Hazards Model In this unit we introduce Cox s proportional hazards (Cox s PH) model, give a heuristic development of the partial likelihood function, and discuss adaptations
More informationLocation-Scale Bivariate Weibull Distributions For Bivariate. Lifetime Modeling. Yi Han
Location-Scale Bivariate Weibull Distributions For Bivariate Lifetime Modeling Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration
More informationMarshall-Olkin Univariate and Bivariate Logistic Processes
CHAPTER 5 Marshall-Olkin Univariate and Bivariate Logistic Processes 5. Introduction The logistic distribution is the most commonly used probability model for modelling data on population growth, bioassay,
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More informationUNIVERSITÄT POTSDAM Institut für Mathematik
UNIVERSITÄT POTSDAM Institut für Mathematik Testing the Acceleration Function in Life Time Models Hannelore Liero Matthias Liero Mathematische Statistik und Wahrscheinlichkeitstheorie Universität Potsdam
More information