Is the MLE of the mean always the sample mean?

The maximum likelihood estimator (MLE) for the mean is the value of that maximizes the joint distribution . It is easy to find using calculus. The sample mean is simply . It turns out that for Gaussian, Poisson, and Bernoulli distributions, the MLE estimator for the mean equals the sample mean.

Is the MLE The mean?

The solution also says that MLE estimate is the average of the data.

What is the MLE of a normal distribution?

“A method of estimating the parameters of a distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable.” MLE tells us which curve has the highest likelihood of fitting our data.

What is the sampling distribution of the MLE?

The distribution of the MLE means the distribution of these ˆθj values. Essentially it tells us what a histogram of the ˆθj values would look like. This distribution is often called the “sampling distribution” of the MLE to emphasise that it is the distribution one would get when sampling many different data sets.

Is MLE always efficient?

It is easy to check that the MLE is an unbiased estimator (E[̂θMLE(y)] = θ). To determine the CRLB, we need to calculate the Fisher information of the model. Yk) = σ2 n . (6) So CRLB equality is achieved, thus the MLE is efficient.

Is MLE of normal distribution unbiased?

But this MLE of σ2 is biased. A point estimateor ^θ is said to be an unbiased estimator of θ is E(^θ)=θ E ( θ ^ ) = θ for every possible value of θ . If ^θ is not unbiased, the difference E(^θ)−θ E ( θ ^ ) − θ is called the bias of ^θ .

What are the assumptions of a normal distribution?

If your data comes from a normal distribution, the box will be symmetrical with the mean and median in the center. If the data meets the assumption of normality, there should also be few outliers. A normal probability plot showing data that’s approximately normal.

What does it mean when we say that the normal distribution is asymptotic?

“Asymptotic” refers to how an estimator behaves as the sample size gets larger (i.e. tends to infinity). “Normality” refers to the normal distribution, so an estimator that is asymptotically normal will have an approximately normal distribution as the sample size gets infinitely large.

Is MLE always asymptotically efficient?

It is consistent and asymptotically efficient (as N→∞ we are doing as well as MVUE). When an efficient estimator exists, it is the MLE. The MLE is invariant to reparameterization.

Is a normal distribution asymptotic?

Perhaps the most common distribution to arise as an asymptotic distribution is the normal distribution. In particular, the central limit theorem provides an example where the asymptotic distribution is the normal distribution.