Point Estimation and Estimators: Estimator vs Estimate

Understand Point Estimation and Estimators

Key rules to remember

Estimator for a parameter

θ̂ = g(X₁, X₂, ..., Xₙ)

A function of the random sample. It is a random variable. Uppercase X is used.

Estimate

θ̂ = g(x₁, x₂, ..., xₙ)

The same function applied to observed values. It is a single number. Lowercase x is used.

Sample mean

X̄ = (1 ÷ n) Σ Xᵢ

Common estimator of the population mean μ.

Sample variance

S² = (1 ÷ (n − 1)) Σ (Xᵢ − X̄)²

Common estimator of σ². Uses divisor n − 1.

Mean and variance of X̄

E(X̄) = μ and Var(X̄) = σ² ÷ n

For independent observations with common mean μ and variance σ².

Standard error

SE(θ̂) = √Var(θ̂)

The standard deviation of the estimator's sampling distribution.

Bias

bias(θ̂) = E(θ̂) − θ

Defined from the sampling distribution. Studied further under unbiasedness.

How to solve Point Estimation and Estimators questions

1. 1Identify the parameter you want to estimate and write it down with its symbol, for example λ or μ.
2. 2State the model for the data, for example X₁, ..., Xₙ independent and identically distributed Poisson(λ). Say if independence is assumed.
3. 3Write the estimator as a function of the random variables X₁, ..., Xₙ. Use uppercase letters.
4. 4If data is given, substitute the observed values to get the estimate. Use lowercase letters and give units.
5. 5To find the sampling distribution, use known results: sums of independent normals are normal, sums of independent Poissons are Poisson, and so on. Otherwise find the mean and variance using expectation rules.
6. 6Compute E(θ̂) and Var(θ̂) and the standard error if asked. Compare E(θ̂) with θ to comment on bias.
7. 7State the conclusion in words, linking the estimator to the parameter, and give the final numerical estimate with the correct notation.

Quickest way: Estimator or estimate in 20 seconds

Common mistakes in Point Estimation and Estimators

• Using the words estimator and estimate as if they mean the same thing.

  In everyday speech both mean a guess, and textbooks sometimes use the same hat symbol for both.

  Fix: Estimator is the random rule with uppercase X. Estimate is the number from data with lowercase x. Write which one you mean.

• Treating the parameter as random when finding the sampling distribution.

  Students mix this with Bayesian ideas, where the parameter has a distribution.

  Fix: In classical estimation the parameter is fixed. The randomness comes only from the sample.

• Calling a quantity a statistic when it includes an unknown parameter, such as (X̄ − μ).

  The expression looks like a function of the sample.

  Fix: A statistic must be computable from the data alone. (X̄ − μ) is not a statistic, but it can be part of a pivotal quantity.

• Confusing the sampling distribution with the distribution of the data.

  Both involve the same population, so they seem the same.

  Fix: The data distribution describes one observation. The sampling distribution describes the estimator over repeated samples of size n. For example, X̄ has variance σ²/n, not σ².

• Giving the variance of X̄ as σ² instead of σ²/n, or using the wrong divisor in S².

  Students rush and forget that averaging reduces variability.

  Fix: Write Var(X̄) = Var(ΣXᵢ) ÷ n² = nσ² ÷ n² = σ²/n. Use n − 1 in S² unless told otherwise.

• Stating that a sampling distribution is normal without a reason.

  The central limit theorem is over-applied.

  Fix: X̄ is exactly normal only if the data are normal. Otherwise it is approximately normal for large n by the CLT. Say which applies.

Worked examples