Construction and Selection of Parametric Models

Question 1

Maximum Likelihood Estimators

Answer 1

1=L(theta)=product of f(xi)
2=l(theta)= ln(L(theta))
3= derivative of l(theta)
4=Set 3=0 and isolate

Question 2

Incomplete Data

Answer 2

1: left truncated at d = f(x)/S(d)
2: Right-censored at u = S(u)
3: Grouped data on interval (a,b) = Pr( a

Question 3

Special Cases for MLE

Answer 3

Gamma, fixed alpha : theta= average/alpha

Normal : û = average
sigma^2 = sum of xi^2 / n - û^2

LogNormal : û=sum of ln(xi) /n
sigma^2 = sum of ln(xi)^2 /n - û^2

Poisson : lambda= average

Binomial fixed m : q= average/m

Negative Binomial fixed r : Beta = average /r

Question 4

Zero-Truncated Distribution

Answer 4

Match E(X^t) to average

Question 5

Zero-Modified Distribution

Answer 5

Match po^m to the proportion of zero observations 
Match E(X^m) to average

Question 6

MLE Uniform on (0, theta)

Answer 6

theta = max(x1,x2,…)

Question 7

Choosing from (a,b,0) class

Answer 7

1: compare u and sigma^2 ****
2: observe the slope of knk/nk-1
Poisson average=sigma^2 : slope=0
Binomial average > sigma^2 : negative slope
Negative Binomial average < sigma^2 : positive slope

Question 8

Variance of MLE : Fisher Information

Answer 8

One parameter

I(theta)=-Ex(I’‘(theta))

Var(theta) = I(theta)^-1

Two Parameters

information matrix = I(alpha, theta)

information matrix ^-1 = Var(alpha), Var(theta)
diagonale= cov(alpha, theta)

Question 9

Delta Approximation

Answer 9

one variable
Var(g(theta))= (d/dtheta g(theta))^2 Var(theta)

two variable
Var(g(alpha, theta)) = (g(alpha)’ )^2 Var(alpha) + 2galpha’gtheta’* Cov(alpha, theta) + (g(theta)’)^2 Var(theta)

Question 10

Confidence Interval

Answer 10

Theta +- Z(1+p)/2 Var(theta)^0.5

Question 11

Kolmogorov-Smirnov

Answer 11

Test Statistic : D = max(Dj) where

Dj=max(abs(Fn(xj)-F(xj)), abs(Fn(xj-1)-F(xj)))

If data is truncated at d, then

F*(x)= (F(x)-F(d))/(1-F(d)) for x > d

Question 12

Kolmogorov-Smirnov Properties

Answer 12

Individual data only
Continuous fit only
Lower critical value for censored data
If parameters are estimated , critical value should be adjusted
Lower critical value if sample size is larger
no discretion
Uniform weight on all parts of distribution

Question 13

D(x) Plot

Answer 13

Graph the difference between empirical CDF and fitter CDF

Peak = Fn(xj)-F*(xj)
Valler = Fn(xj-1)-F*(xj)

Question 14

P-p Plot

Answer 14

Coordinate : (Fn(xj), F*(xj))) where Fn(xj)= j/(n+1)

Question 15

Hypothesis Tests: Chi-Square Goodness-of-Fit

Answer 15

Test Statistic : X^2 = sum (Ej-Oj)^2 / Ej or sum Oj^2/Ej - n

Degrees of freedom : k(number of group)-1-r(number of estimated parameters)

Question 16

Chi-Square Properties

Answer 16

Individual and grouped data
continuous and discrete fit
no adjustments to critical value for censored data
If parameters are estimated , critical value is automatically adjusted via degrees of freedom
no change for critical value is sample size is large
more weights on intervals with poor fit

Question 17

Hypothesis Test: Likelihood Ratio

Answer 17

Test statistic : T=2(l(theta1)-l(theta0))

Degrees of freedom = number of free parameters in H1 - number of free parameters in H0

Question 18

Score-Based Approaches

Answer 18

SBC : Schwarz Bayesian Criterion = BIC : Bayesian Information Criterion = l-r/2*ln(n)

Akaike Information Criterion : AIC = l-r

l: log-likelihood
r: number of parameters
n: sample size

Select model with the highest AIC or BIC value