Question

1. what is the connection between the score function and maximun likelihood function?

2. what information the CRLB provides about the class of unbaised estimator of parameter

Answer 1

1.If the log-likelihood function is continuous over the parameter space, the score will vanish at a local maximum or minimum; this fact is used in maximum likelihood estimation to find the parameter values that maximize the likelihood function.

2.CRLB tstates that the variance of any unbiased estimator is at least as high as the inverse of the Fisher information. An unbiased estimator which achieves this lower bound is said to be (fully) efficient.

Answer 2

• Connection between the score function and maximum likelihood function

• The maximum likelihood function helps us find the best possible value of a parameter that makes the observed data most likely.

• The score function is simply the derivative (slope) of the log-likelihood function. It tells us how sensitive the likelihood is to changes in the parameter.

• In simple terms, when the score function is zero, it means we have reached a peak (maximum point), which helps us find the maximum likelihood estimate (MLE).

• What information does the Cramér-Rao Lower Bound (CRLB) provide?

• The CRLB sets a minimum limit on how small the variance of an unbiased estimator can be.

• Think of it like a rule that tells us, "No matter how good your estimator is, you can't get a better (lower variance) unbiased estimator than this bound."

• If an estimator reaches this bound, it's considered efficient—meaning it’s the best unbiased estimator possible.