Thread: 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?

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).