Generating Random Numbers And Estimation

Question 1

What is the inverse CDF method

Answer 1

The inverse CDF method is:
If G(x) is a strictly increasing function from [a,b] to [0,1] (a,,b can be +- infinity)
Let U ~unif(0,1) and X = G^-1(U)
Then X is an RV with CDF G(x)

Question 2

What are true random numbers

Answer 2

True random numbers are all independent where each digit is equally likely

Question 3

What are pseudo random numbers

Answer 3

Pseudo random numbers follow a pre-specified pattern unknown

Question 4

What is method of moments estimation method

Answer 4

Method of moments estimation method is:
E(Xi) = mu = h(theta) where mu = 1/n sum(xi) where xi are observed values and h is a function of how to f8nd mean, e.g exponential h(theta) = 1/lambda = 1/n sum(xi)

Question 5

What is an estimate and what can they be based on

Answer 5

An estimate is a real number computed from data, an estimate of parameter theta based on x1,…,xn can be written as theta hat = h(x1,…,xn)

Question 6

What is an estimator

Answer 6

An estimator is an RV, function of RVs X1,…,Xn that comprise data, estimator is RV h(X1,…,Xn)

Question 7

What is definition of likelihood function (discrete)

Answer 7

Definition of likelihood function is:
L(theta, x) = i = 1 to n PI px(theta ,xi) where px(theta,xi) = P(Xi =xi) and X1,,..,Xn are i.i.d discrete RVs and theta denotes a parameter of a vector of parameters
likelihood is Joint PMF

Question 8

What is likelihood function (continuous)

Answer 8

Likelihood function continuous is:
L(theta, x) = i=1 to n PI fx(theta, xi)
Likelihood is joint PDF

Question 9

What is the log likelihood

Answer 9

Log likelihood (for both discrete and continuous) is 
L(theta, x) = ln{L(theta, x)}

Question 10

What is maximum likelihood estimate of a (vector of) parameter

Answer 10

Maximum likelihood estimate (MLE) of a (vector of) parameters is:
Value of theta that maximises (log) likelihood function/for observed data

Question 11

What is Q-Q plot and when is it accurate

Answer 11

Q-Q plot is a graph of xi against F^-1x(i/n+1) where there are n observed values and Fx is CDF of a distribution proposed for RVs Xi
Accurate when points lie close to line y=x