M4: Binomial and Poisson Models

Question 1

Binomial Model has the RV D_x dist via

Answer 1

~ Binomial(N_x,q_x)

N is the number of independent lives aged x

Question 2

Under MLE, we can estimate q_x^hat by_z

Answer 2

q_x=d_x / N_x

Question 3

Properties of the Binomial MLE est of q_x (3)

Answer 3

1. Unbiased
2. Minimum variance of all estimators (efficient)
3. Asympotically, our estimate for q_x is distributed ~Normal(q, q(1-q)/N)

Question 4

When introducing censoring into the binomial model, a_I and b_i represent

Answer 4

x+a is the entrance time

x+b is the exit time

Question 5

To help with censored binomial model we introduce the RV D_i which has values:

Answer 5

0 if the life survives thru the obvs period.

1 if the life dies.

Question 6

Likelihood equation for censored binomial model

Answer 6

L() = Π q^d • (1-q)^1-d

• q is chance of death, so 1-q is survival
• d the power is used as an indicator
• we take product across all observations

Question 7

Problem with MLE eqn for censored binomial (and solution)

Answer 7

• we may get as many solutions are there are observations
  
  • Fix: use a simplifying assumption about the form of q

Question 8

3 possible simplifying assumptions to use in the censored Binomial MLE eqn

Answer 8

1. Uniform dist of deaths
2. Balducci assumptions (leads to actuarial estimate)
3. Constant force of mortality

Question 9

How to we find E() of number of deaths under the actuarial estimte?

Answer 9

E[D] = Σ_b-aq_x+a

Sum of the chances of each indivdual life dieing

Question 10

What does actuarial estimate allow us to do

Answer 10

1. Simplify the E[D] formula
2. Re-arrange the E[D] formula in terms of q to get an estimate for mortality.

Question 11

What is initial exposed to risk and what can we do with it

Answer 11

• The amount of time people could die
  
  • Allows us to estimate q hat easily = deaths/exposed time

Question 12

When can we use central exposed to risk

Answer 12

When we know when deaths occur.

(with intial we only have the entry/exit times, and the info of death/survive during the period)

Question 13

How we will incorporate the extra knowledge of central exposed to risk

Answer 13

Modify intial exposed to risk to become more accurate.

Survivors add (b-a)

Deaths at (t-a)

Question 14

When we dont know time of deaths, but still must use central exposed to risk, what will we do?

Answer 14

• Assume deaths occur (on average) at x+0.5
• This leads to E_x= E_x^Central + d/2
  
  • d is the number of deaths

Question 15

The parameter of the poisson dist is

Answer 15

Poisson(uE^c_x)

Question 16

MLE to find u hat for poisson parameter leads to…

Answer 16

u hat = D_x / E^c_x

Question 17

Properties of the Possion model estimator

Answer 17

1. Unbiased
2. u hat is asmpytopically normal