BA3202-L10 - BA3202 Actuarial Statistics Lecture 9 Summary...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
BA3203 L10 BA3202 Actuarial Statistics Lecture 9 Summary: - Generalized linear models (GLM)
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L10 Exponential families A distribution for a random variable Y belongs to an exponential family if its density has the following form 𝑓 𝑌 𝑦 ; 𝜃 , 𝜑 = exp 𝑦𝜃 − 𝑏 𝜃 𝑎 𝜑 + 𝑐 𝑦 , 𝜑 where 𝑎 , 𝑏 , 𝑐 are functions, 𝜃 and 𝜑 are parameters 𝜃 : natural parameter 𝜑 : scale/dispersion parameter 𝐸 𝑌 = 𝑏 𝜃 note that 𝐸 𝑌 only depend on 𝜃 When predicting 𝑌 , only 𝜃 is of importance 𝑣𝑎𝑣 𝑌 = 𝑎 𝜑 𝑏 ′′ 𝜃 𝑏 ′′ 𝜃 : defines the way that variance related to the mean 𝑎 𝜑 : function involving the scale parameter Examples of exponential family distributions: Exponential, Normal, Poisson, Binomial, Gamma Actuarial Science NTU Jade Nie [email protected] 2
Image of page 2
BA3203 L10 Summary Note: 𝐸 𝑌 = 𝜇 𝜃 are functions of 𝜇 We denote these functions as 𝑔 𝜇 𝜃 = 𝑔 𝜇 Actuarial Science NTU Jade Nie [email protected] 3 𝑵 ( 𝝁 , 𝝈 ) 𝑃𝑃𝑃 ( 𝝁 ) 𝒁 ~ 𝑩𝑩 𝒏 , 𝝁 𝒀 = 𝒁 / 𝒏 𝚪 𝜶 , 𝜶 𝝁 𝜃 𝜇 ln 𝜇 ln 𝜇 1 − 𝜇 1 𝜇 𝜑 𝜎 2 1 𝑛 𝛼 𝑎 𝜑 𝜑 1 1 𝜑 1 𝜑 𝑏 𝜃 𝜃 2 2 𝑒 𝜃 ln 1 + 𝑒 𝜃 ln −𝜃 𝑏 𝜃 𝜃 𝑒 𝜃 𝑒 𝜃 1 + 𝑒 𝜃 1 𝜃 𝐸 𝑌 𝜇 𝜇 𝜇 𝜇 𝑣𝑎𝑣 𝑌 𝜎 2 𝜇 𝜇 1 − 𝜇 𝜇 2 𝛼
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L10 Link functions Link function represents the relationship between the mean of an exponential family distribution ( 𝜇 ) and the natural parameter ( 𝜃 ) Summary of natural/canonical link function Technically, it is necessary for the link-function to be Differentiable Invertible Actuarial Science NTU Jade Nie [email protected] 4 𝑵 ( 𝝁 , 𝝈 ) 𝑃𝑃𝑃 ( 𝝁 ) 𝒁 ~ 𝑩𝑩 𝒏 , 𝝁 𝒀 = 𝒁 / 𝒏 𝚪 𝜶 , 𝜶 𝝁 Link function 𝑔 𝜇 𝜇 ln 𝜇 ln 𝜇 1 − 𝜇 1 𝜇 The negative sign absorbed in the linear predictor and is dropped in the canonical link- function;
Image of page 4
BA3203 L10 Linear predictor Interaction between two predictors E.g. age and gender of a policyholder can be two interactive predictor linear predictor 𝛼 𝑖 + 𝛽 𝑖 𝑥 𝑃 = 1 for male and 𝑃 = 2 for female 𝑥 is the age of the policyholder Effect of age is different for males and females Summarize Actuarial Science NTU Jade Nie [email protected] 5 Model Linear predictor Age 𝛽 0 + 𝛽 1 𝑥 Gender 𝛼 𝑖 Age+gender 𝛼 𝑖 + 𝛽𝑥 Age+gender+age*gender 𝛼 𝑖 + 𝛽 𝑖 𝑥 Age*gender 𝛼 𝑖 + 𝛽 𝑖 𝑥
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
BA3203 L10 Model testing Scaled deviance 𝟐 𝐥𝐥 𝑳 𝑺 − 𝐥𝐥 𝑳 𝑴 Deviance: 𝐷 𝑀 = 𝜑 ∗ Scaled deviance, where 𝜑 is the scale parameter We want to decide if model 2 (with 𝑝 parameters and 𝐷 2 deviance) is a significant improvement over model 1 (with 𝑞 parameters and 𝐷 1 deviance) Model 1 is nested in model 2
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern