November 13th, 2020

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

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