Let θ be an unknown constant. Let W1,…,Wn be independent exponential random variables each with parameter 1. Let Xi=θ+Wi.

What is the maximum likelihood estimate of θ based on a single observation X1=x1? Enter your answer in terms of x1 (enter as x_1) using standard notation .

θ^ML(x1)= - unanswered

e−x1
What is the maximum likelihood estimate of θ based on a sequence of observations (X1,…,Xn)=(x1,…,xn)?
θ^ML(x1,…,xn)=

(x1x2⋯xn)1/n x1+⋯+xnn 11x1+⋯1xn minixi maxixi None of the above

You have been asked to construct a confidence interval of the particular form [Θ^−c,Θ^], where Θ^=mini{Xi} and c is a constant that we need to choose. For n=10, how should the constant c be chosen so that we have a 95% confidence interval? (Give the smallest possible value of c.) Your answer should be accurate to 3 decimal places.