Let X1,..., X, be an iid random sample from the Binomial(k, p) distribution, where p is known but k is unknown. For example, this could happen in an experiment where we flip a coin that we know is...

max;=1,...,n i. (e) part (d). This description of the MLE for k was found by Feldman and Fox (1968). Show that there is only one unique integer value of k that satisfies the equation in "/>
Extracted text: Let X1,..., X, be an iid random sample from the Binomial(k, p) distribution, where p is known but k is unknown. For example, this could happen in an experiment where we flip a coin that we know is fair and observe x; heads, but we do not know how many times the coin was flipped. Suppose we observe values x1,..., xn of our random sample. In this problem, you will show that there is a unique maximum likelihood estimator of k. (a) What is the likelihood function, L(k)? Why is maximization by differentiation difficult here? (b) of k. What is the value of the likelihood when k < ¸max="" x;?="" based="" on="" this,="" what="" can="" you="" we="" have="" to="" take="" a="" different="" approach="" to="" finding="" the="" maximum="" likelihood="" estimator="" i="1,...,n" conclude="" about="" the="" range="" of="" values="" the="" maximum="" likelihood="" estimator="" can="" take?="" (c)="" at="" k="" and="" k="" +1,="" based="" on="" part="" (a),="" what="" can="" you="" conclude="" about="" the="" ratio="" of="" two="" likelihoods="" evaluated="" l(k)="" l(k="" –="" 1)'="" l(k="" +="" 1)="" l(k)="" |="" for="" k="" in="" the="" range="" found="" in="" part="" (b)?="" (d)="" using="" part="" (c),="" show="" that="" the="" maximum="" likelihood="" estimator="" is="" the="" value="" of="" k="" that="" satisfies="" n="" n="" ii="" (1="" -="" )="">< (1="" –="" p)"="">< ii(!="" xi="" k="" +1="" i="1" i="for" k=""> max;=1,...,n i. (e) part (d). This description of the MLE for k was found by Feldman and Fox (1968). Show that there is only one unique integer value of k that satisfies the equation in