Theorem on Asymptotic Validity of Maximum Likelihood Method Conditions---First Assertion

• (a) If we fix n and conduct many such series of n observations, the mean of satisfies:

  

• Note is not exactly ( is true theoretical parameter).

• If for any estimate , then we say estimate is unbiased:

• if then we have a biased estimate with bias .

• So maximum likelihood estimate is biased for finite n and is only asymptotically unbiased.

• However, bias is smaller than Standard Deviation and so it is of no real importance.

• Bias is discussed in detail by Cramer.