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

• Homework Help
• 30

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

BA3203 L10 BA3202 Actuarial Statistics Lecture 9 Summary: - Generalized linear models (GLM)

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

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 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 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

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

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 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;
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 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

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

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
This is the end of the preview. Sign up to access the rest of the document.
• Spring '16
• Autocorrelation, Stationary process, Jade Nie, Jade Nie cynie, Actuarial Science NTU

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern