(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. 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). | 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. Using L n(X n; ), the maximum likelihood estimator of is b n =max The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. p2. The answer is. MLE requires us to maximum the likelihood functionL(µ) with respect to the unknown parameterµ. θ ⋅ n n + 1. Example. Example 2.2.1 (The uniform distribution) Consider the uniform distribution, which has the density f(x; )= 1I [0, ](x). Copy link. share. Namely, the MLE is the inverse of the sample average. Share a link to this answer. If X ∼ U ( c, c + A). 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. From Eqn. 1,L(µ) is deﬂned as a product ofnterms, which … When there are actual data, the estimate takes a particular numerical value, which will be the maximum likelihood estimator. Assume X 1; ;X n ˘Uni[0; ]. Then the Fisher information can be computed as I(p) = −E 2. log f(X p) = EX + 1 − EX = p + 1 − p = 1 . 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. 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). Namely, the random sample is 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).

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