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Is the sample mean a consistent estimator?
The sample mean is a consistent estimator for the population mean. In other words, the more data you collect, a consistent estimator will be close to the real population parameter you’re trying to measure. The sample mean and sample variance are two well-known consistent estimators.
What does it mean when we say that the sample mean is an unbiased estimator of μ?
When a statistic like the sample mean X is aimed at a population parameter like μ, we call X an estimator of μ. An estimator is unbiased if its mean over all samples is equal to the population parameter that it is estimating. For example, E(X) = μ.
Which is an unbiased estimator of the parameter θ?
If the following holds: then the statistic u ( X 1, X 2, …, X n) is an unbiased estimator of the parameter θ. Otherwise, u ( X 1, X 2, …, X n) is a biased estimator of θ. If X i is a Bernoulli random variable with parameter p, then: is the maximum likelihood estimator (MLE) of p.
Can a unbiased but not consistent estimator be used?
“An estimator can be unbiased but not consistent. For example, for an iid sample ${x _1,…, x_n}$ one can use $T(X) = x_1$ as the estimator of the mean $E[x]$. This estimator is obviously unbiased, and obviously inconsistent.”.
Is the maximum likelihood estimator of μ unbiased?
Therefore, the maximum likelihood estimator of μ is unbiased. Now, let’s check the maximum likelihood estimator of σ 2. First, note that we can rewrite the formula for the MLE as: σ ^ 2 = ( 1 n ∑ i = 1 n X i 2) − X ¯ 2. because: Then, taking the expectation of the MLE, we get: E ( σ ^ 2) = ( n − 1) σ 2 n. as illustrated here:
Which is an example of an inconsistent estimator?
Your estimator is on the other hand inconsistent, since x ~ is fixed at x 1 and will not change with the changing sample size, i.e. will not converge in probability to μ. Perhaps an easier example would be the following. Let β n be an estimator of the parameter β.