Estimators and Mean Square Error

The video tutorial covers two key properties of statistical estimators: bias and mean square error. It begins by distinguishing between an estimate and an estimator, then delves into the concept of bias, explaining how an unbiased estimator has a zero bias. Examples are provided to illustrate unbiased estimators for different distributions. The tutorial then shifts focus to mean square error, describing it as the average of the squared errors and its role in measuring estimator efficiency. The video concludes with examples demonstrating how mean square error decreases with larger sample sizes, indicating estimator consistency.

What are the two main properties of estimators discussed in this unit?

• Standard deviation and variance

• Bias and variance

• Bias and mean square error

• Mean and median

How is an estimate different from an estimator?

• Both are the same.

• An estimate is a single value, while an estimator is a distribution.

• An estimate is a distribution, while an estimator is a single value.

• An estimate is always more accurate than an estimator.

What does it mean if an estimator is unbiased?

• It has a high variance.

• On average, it gives the true value of the parameter.

• It always underestimates the true value.

• It always overestimates the true value.

Which of the following is an example of an unbiased estimator?

• Sample mode

• Sample median

• Sample variance

• Sample mean

What is the mean square error?

• The average of the square of the error

• The square of the average error

• The sum of the errors

• The difference between the highest and lowest error

How is the mean square error related to the variance and bias of an estimator?

• It is unrelated to variance and bias.

• It is the difference between the variance and the bias.

• It is the product of the variance and the bias.

• It is the sum of the variance and the square of the bias.

What happens to the mean square error as the sample size increases?

• It increases.

• It decreases.

• It remains constant.

• It becomes negative.

In the context of estimators, what does consistency mean?

• The estimator always gives the same result.

• The estimator is always unbiased.

• The estimator's mean square error tends towards zero as sample size increases.

• The estimator has a high variance.

What is the bias of the estimator P hat equal to X over N from a binomial NP distribution?

• P

• N

• X

• 0

What is the variance of X in a binomial NP distribution?

• N

• P

• NPQ

• X