# Maximum Likelihood Estimator / Best Estimator .. Related to survey.. Help needed :(

• Dec 12th 2012, 09:50 PM
drinkingwater
Maximum Likelihood Estimator / Best Estimator .. Related to survey.. Help needed :(
Some respondents give false responses to survey questions. Denote the proportion of the members of a particular population that smoked more than 50 cigarettes last week by p. A random sample of n members of this population is taken, and each person in the sample is asked “Did you smoke more than 50 cigarettes last week?” If a person really smoked more than 50 cigarettes, the probability that he will give a truthful answer to this question is 1-x1. If a person did not smoke more than 50 cigarettes, the probability that he will give a truthful answer is 1-x2. From past data, x1 and x2 are known, with 0<x1<0.5, 0<x2<0.5.
a) For a sample of size one, find the likelihood function if the answer is “yes” and find the likelihood function if the answer is “no.”
b) For a random sample of size n, find the likelihood function and sufficient statistics.
c) Find the maximum likelihood estimator for p.
d) Assume that x1=0.1, x2=0, and there is one “yes” answer in a random sample of size 10. What is your best estimate of p and why?
e) Consider the same scenario as in (d), but assume that x1 is unknown (0<x1<1). In this case, what would be your best estimate of p and why?
• Dec 13th 2012, 08:02 PM
chiro
Re: Maximum Likelihood Estimator / Best Estimator .. Related to survey.. Help needed
Hey drinkingwater.

Can you show us what you have tried?

(Hint: the likelihood for sample size 1 is just the probability of that thing happening, so in other words its just the PDF where you substitute in a given x value corresponding to the sample observation).