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-x_{1}. If a person did not smoke more than 50 cigarettes, the probability that he will give a truthful answer is 1-x_{2}. From past data, x_{1} and x_{2} are known, with 0<x_{1}<0.5, 0<x_{2}<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 x_{1}=0.1, x_{2}=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 x_{1} is unknown (0<x_{1}<1). In this case, what would be your best estimate of *p* and why?

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).