When there are actual data, the estimate takes a particular numerical value, which will be the maximum likelihood estimator. Namely, the random sample is p2. Example. Given the iid uniform random variables {X i} the likelihood (it is easier to study the likelihood rather than the log-likelihood) is L n(X n; )= 1 n Yn i=1 I [0, ](X i). Assume X 1; ;X n ˘Uni[0; ]. Then having observed n independent observations, we can write the likelihood as, L ( A) = 1 A n ∏ i = 1 n I ( c < X i < c + A) = 1 A n I ( min X i ≥ c) I ( max X i ≤ c + A). Example. 1,L(µ) is deflned as a product ofnterms, which … Using L n(X n; ), the maximum likelihood estimator of is b n =max Example 2.2.1 (The uniform distribution) Consider the uniform distribution, which has the density f(x; )= 1I [0, ](x). θ ⋅ n n + 1. Then the Fisher information can be computed as I(p) = −E 2. log f(X p) = EX + 1 − EX = p + 1 − p = 1 . In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. Copy link. The answer is. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. If X ∼ U ( c, c + A). (Uniform distribution) Here is a case where we cannot use the score function to obtain the MLE but still we can directly nd the MLE. Since 1 / A n is a decreasing function of A, the MLE will be the smallest value possible such that c + A ≥ max X i. This follows from the fact that the order statistics from a uniform (0,1) follow a beta distribution (and the max is the n 'th order statistic), and uniform (0, θ) is just a scaled version of a uniform (0,1). From Eqn. | p2(1 − p)2p2(1 − p)2p(1 − p) The MLE of p is pˆ = X¯ and the asymptotic normality result states that ≥ n(pˆ − p0) N(0,p0(1 − p0)) which, of course, also follows directly from the CLT. Namely, the MLE is the inverse of the sample average. Share a link to this answer. MLE requires us to maximum the likelihood functionL(µ) with respect to the unknown parameterµ. share.