## What is the variance of the MLE?

This property is called asymptotic efficiency. I(θ) = −E [ ∂2 ∂θ2 ln L(θ|X) ] . Thus, the estimate of the variance given data x ˆσ2 = −1 / ∂2 ∂θ2 ln L(ˆθ|x). the negative reciprocal of the second derivative, also known as the curvature, of the log-likelihood function evaluated at the MLE.

### What is MLE for exponential distribution?

The maximum likelihood estimator is legitimate because exponentially distributed random variables can take on only positive values (and strictly so with probability 1). Therefore, the estimator is just the reciprocal of the sample mean.

#### What is the mean and variance of exponential distribution?

The mean of the exponential distribution is 1/λ and the variance of the exponential distribution is 1/λ2.

**Is MLE of variance unbiased?**

The MLE estimator is a biased estimator of the population variance and it introduces a downward bias (underestimating the parameter). The size of the bias is proportional to population variance, and it will decrease as the sample size gets larger.

**What is the asymptotic variance of the maximum likelihood estimator?**

ASYMPTOTIC VARIANCE of the MLE Maximum likelihood estimators typically have good properties when the sample size is large. For ˜θ any unbiased estmator for θo, we have a lower bound on the variance of ˜θ V ar(˜θ) ≥ 1 nI(θo) 1 Page 2 Let’s work some examples. exp(−|x|/σ).

## Is the MLE variance of normal distribution biased?

### Is MLE always asymptotically efficient?

MLE estimation is often asymptotically normal even if the model is not true, it might be consistent for the “least false” parameter values, for instance. But in such cases it wil be difficult to show efficency or other optimality properties.

#### What is asymptotic variance?

Though there are many definitions, asymptotic variance can be defined as the variance, or how far the set of numbers is spread out, of the limit distribution of the estimator.

**How do you find the MLE of a parameter?**

Definition: Given data the maximum likelihood estimate (MLE) for the parameter p is the value of p that maximizes the likelihood P(data |p). That is, the MLE is the value of p for which the data is most likely. 100 P(55 heads|p) = ( 55 ) p55(1 − p)45. We’ll use the notation p for the MLE.

**How do you find the variance of an unbiased estimator?**

Thus, the variance itself is the mean of the random variable Y=(X−μ)2. This suggests the following estimator for the variance ˆσ2=1nn∑k=1(Xk−μ)2. By linearity of expectation, ˆσ2 is an unbiased estimator of σ2.

## What does the variance of an estimator measure?

Variance. The variance of is the expected value of the squared sampling deviations; that is, . It is used to indicate how far, on average, the collection of estimates are from the expected value of the estimates. (Note the difference between MSE and variance.)