Does MLE attain Cramer-Rao lower bound?

The ML estimate is asymptotically efficient; that is, it achieves the Cramer-Rao lower bound (Appendix A). This is the lowest value of variance, which any estimate can achieve.

How is Cramer-Rao lower bound calculated?

= p(1 − p) m . Alternatively, we can compute the Cramer-Rao lower bound as follows: ∂2 ∂p2 log f(x;p) = ∂ ∂p ( ∂ ∂p log f(x;p)) = ∂ ∂p (x p − m − x 1 − p ) = −x p2 − (m − x) (1 − p)2 .

What are the assumptions of maximum likelihood estimation?

In order to use MLE, we have to make two important assumptions, which are typically referred to together as the i.i.d. assumption. These assumptions state that: Data must be independently distributed. Data must be identically distributed.

What are the properties of maximum likelihood estimator?

Maximum Likelihood Estimation (MLE) is a widely used statistical estimation method. In this lecture, we will study its properties: efficiency, consistency and asymptotic normality. MLE is a method for estimating parameters of a statistical model.

Why is Cramer Rao lower bound important?

The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. Estimators that are close to the CLRB are more unbiased (i.e. more preferable to use) than estimators further away. If you have several estimators to choose from, this can be very useful.

How is Fisher information calculated?

Given a random variable y that is assumed to follow a probability distribution f(y;θ), where θ is the parameter (or parameter vector) of the distribution, the Fisher Information is calculated as the Variance of the partial derivative w.r.t. θ of the Log-likelihood function ℓ(θ | y).

What is the Cramer Rao lower bound of the variance of an unbiased estimator of theta?

The function 1/I(θ) is often referred to as the Cramér-Rao bound (CRB) on the variance of an unbiased estimator of θ. I(θ) = −Ep(x;θ) { ∂2 ∂θ2 logp(X;θ) } . and, by Corollary 1, X is a minimum variance unbiased (MVU) estimator of λ.

Why we use Cramer Rao lower bound?

The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. Estimators that are close to the CLRB are more unbiased (i.e. more preferable to use) than estimators further away. Creating a benchmark for a best possible measure — against which all other estimators are measured.

What is the use of maximum likelihood estimation?

Maximum likelihood estimation involves defining a likelihood function for calculating the conditional probability of observing the data sample given a probability distribution and distribution parameters. This approach can be used to search a space of possible distributions and parameters.

What is maximum likelihood estimation in statistics?

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable.