Question

If an estimator is consistent, then its distribution

a) tends towards a standard normal distribution as the sample size grows
b) becomes more loosely distributed around the true parameter value as the sample size grows
c) becomes more tightly distributed around the true parameter value as the sample size grows
d) remains unaffected as the sample size grows

Answer 1

An estimator that is consistent will show a distribution that becomes more tightly distributed around the true parameter value as the sample size grows, which is consistent with the central limit theorem.

Step-by-step explanation:

If an estimator is consistent, then as the sample size grows, its distribution becomes more tightly distributed around the true parameter value. This happens because the standard deviation of the sampling distribution of the means will decrease, leading to the distribution becoming similar to the standard deviation of X. According to the central limit theorem, the larger the sample, the closer the sampling distribution of the means becomes to a normal distribution. Furthermore, while it approaches the standard normal distribution as n gets larger, it is important to understand that with a sufficient number of samples, the sample means will approximate a normal distribution regardless of the shape of the population distribution.