Question

List all basic distributions for which:

a) MLE is unbiased, but Method of Moments (MM) estimator is biased

b) MLE is biased, but MM estimator is unbiased

Answer 1

Solution:

a),Maximum likelihood estimation (MLE) is a technique used for estimating the parameters of a given distribution, using some observed data. For example, if a population is known to follow a normal distribution but the mean and variance are unknown, MLE can be used to estimate them using a limited sample of the population, by finding particular values of the mean and variance so that the observation is the most likely result to have occurred.

.MLE is helpful in an assortment of settings, running from econometrics to MRIs to satellite imaging. It is additionally identified with Bayesian measurements.

."In the event that a most productive unprejudiced estimator ˆθ of θ exists (i.e. ˆθ is unprejudiced and its change is equivalent to the CRLB) at that point the most extreme probability strategy for estimation will create it."

.The method of moments is a method of estimation of population parameters. One starts with deriving equations that relate the population moments(i.e., the expected values of powers of the random variable under consideration) to the parameters of interest.

.Factual predisposition is a methodical inclination during the time spent information gathering, which results in unbalanced, misdirecting results. ... It is a property of a factual procedure or of its outcomes whereby the normal estimation of the outcomes contrasts from the genuine hidden quantitative parameter being assessed.

b).In statistics, we evaluate the “goodness” of the estimation by checking if the estimation is “unbiased”. By saying “unbiased”, it means the expectation of the estimator equals to the true value, e.g. if E[x] = µ then the mean estimator is unbiased. Now we will show that the equation actually holds for mean estimator. E[x] = E[ 1 N X N i=1 xi ] = 1 N X N i=1 E[x] = 1 N · N · E[x] = E[x] = µ The first line makes use of the assumption that the samples are drawn i.i.d from the true distribution, thus E[xi ] is actually E[x]. From the proof above, it is shown that the mean estimator is unbiased. Now we move to the variance estimator. At the first glance, the variance estimator s 2 = 1 N PN i=1(xi − x) 2 should follow because mean estimator x bar is unbiased.

.The strategy for minutes is a technique for estimation of populace parameters. ... At that point an example is drawn and the populace minutes are assessed from the example. The conditions are then tackled for the parameters of enthusiasm, utilizing the example minutes instead of the (obscure) populace minutes.