Distinguish an estimator, which is a rule, from an estimate, which is a value it produces.
The idea
An estimator is a rule, a formula applied to a random sample. The sample mean is an example, the sum of the n observations divided by n. Before the data are drawn its value is uncertain, so it has a sampling distribution with an expected value, a variance and a standard error. An estimate is the single number the rule produces once a particular sample has been observed, for example 0.03% from 250 daily returns. Properties such as bias, efficiency and consistency belong to the estimator, because they describe behaviour across many possible samples. An estimate can only be close to or far from the truth in that one sample. On the exam, one estimate missing the target never proves that the estimator is biased.
A practice question
Which statement correctly distinguishes an estimator from an estimate?
- An estimator is the number obtained from one sample; an estimate is the general formula used to compute it.Why not: Reverses the terms: the formula is the estimator and the number is the estimate.
- An estimator is the true population parameter being targeted; an estimate is the sample statistic that approximates it.Why not: The true parameter is neither: it is the target. The estimator is the rule, not the parameter.
- An estimator is a rule applied to a random sample, with a sampling distribution; an estimate is the number it gives for one sample. - correct
- An estimator is used only for means; an estimate is used for variances and other moments.Why not: Estimators exist for any parameter, including variances and correlations; the distinction is rule versus value.
The sample mean formula (X1 + ... + Xn)/n applied to random data is an estimator: its value is random before the data are drawn. Once the data are observed, the number it returns is an estimate and is fixed.
A harder one
Which of the following statements can be made about an estimator, but not about a single estimate?
- It is calculated from sample data.Why not: An estimate is also calculated from sample data, so this does not separate the two.
- It has an expected value and a variance across repeated samples. - correct
- It is expressed in the same units as the parameter.Why not: An estimate is also in the units of the parameter.
- It equals the parameter it targets.Why not: Only an estimate, being a number, can equal or miss the parameter in a given sample; an estimator is judged on average across samples.
Because an estimator is a random variable, its mean and variance over repeated samples are meaningful and define bias and efficiency. A single estimate is a number and has no sampling distribution.
The hardest tier
From 36 monthly returns an analyst obtains a sample standard deviation of 2.4% and reports the sample mean with a standard error of 0.4%. What does the 0.4% represent?
- The standard deviation of the 36 monthly returns, a measure of how dispersed individual returns are.Why not: The dispersion of individual returns is the 2.4%, not the standard error; the standard error divides by sqrt(n).
- The bias of the sample mean, that is, its expected error in the sample.Why not: Standard error measures spread across samples, not systematic error; the sample mean is unbiased.
- An estimate of the standard deviation of the sampling distribution of the sample mean, that is, how much the mean estimate varies across samples. - correct
- The probability that the true mean differs from the estimate.Why not: Standard error is a standard deviation in return units, not a probability.
The standard error is s/sqrt(n) = 2.4/6 = 0.4%. It estimates the standard deviation of the estimator (the sample mean), and so describes the estimator's precision.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Sample Moments
- Use sample data to estimate a mean, a variance and a standard deviation.
- Describe how a sample moment differs from a population moment.
- Explain what the bias of an estimator is and what it indicates about the estimator.
- Interpret the claim that the estimator of the mean is BLUE.
- Define the consistency of an estimator and describe why the idea is useful.
- Apply the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) to a sample mean, saying what each result implies.
- Use data to estimate the skewness and kurtosis of a random variable and interpret the results.
- Use sample data to estimate quantiles, the median among them.
- Apply the CLT after estimating the means of two variables from sample data.
- From sample data, calculate estimates of how two random variables covary and how they correlate.
- Relate coskewness and cokurtosis back to the skewness and kurtosis of a single variable.