1. May 7, 2010

### mike1111

1. The problem statement, all variables and given/known data

I got a stats problem which I don't know how to approach. It concerns a method of predicting election outcomes based solely on the length of incumbency of an existing Government. For Australian National or State governments elected the lengths of holding office are summarised in the Table below. The count mj is the number of Governments who win j elections before finally losing.

j 1 2 3 4 5
mj 3 9 7 3 1

This Table does not include any of the present State or National Governments, whose current lengths of office are given below.

Government term number
national 1
NSW 4
Queensland 5
SA 3
Tasmania 4
Victoria 3
WA 1

(a) Let X » Bi(n; p), and let Xn = X/n be the corresponding binomial proportion. Use the delta method to express

log(Xn) = a +b*Z/(n^0.5)

where Z ~ N(0,1) approximately, for large n, and the constants a, b.

(b) Let N be the total number of elections won by a 'random Government', past or future, and define a 'success' as winning an election. Consider the success probabilities

pj = P(N > j|N >= j) for j = 1, 2,...

the success probability for a Government attempting to win a (j +1)th election. By considering election outcomes as Bernouilli trials, identify the binomial proportions which yield independent estimates pj^ of pj for j = 1; 2; 3; 4.
(Do not consider j = 5, for in that case there is only one observation).

(c) A plausible model is

pj = theta*(2/3)(j+1) for j = 1, 2, ...

describing how re-election chances decrease as the length of time in office increases. For each numerical pj^ value in part (b), write down a corresponding algebraic expression for
log (pj^) , in terms of theta and j , using part (a). For this
purpose you may regard n = 4 as 'large'.

(d) Average the representations in part (c), and hence obtain an estimate log (theta)^ of
log (theta). Also, derive an expression for and evaluate the standard error.

(e) Use the delta-method to find an estimate of theta, along with its standard error.

(f) Finally, estimate the probabilities for each of the National Government, the NSW
Government and the Victorian Government being re-elected at forthcoming elections.

3. The attempt at a solution
(a)For Binomial
/mu = p
\sigma2 = p(1-p)
g(x) = log x
g'(x) = 1/x
using delta method
then Xn = g(\mu) +\sigma *Z* g'(\mu)
=log (p) +(1-p)Z

(b) total number of election is 58 ( not counting j=5 data)

pj^ =mj / $$\sum mj$$
I think this is wrong!

anyone know how to approach the rest of this question and if the start is right?