Which statement correctly describes the relationship between MSE and the true parameter?

Which statement correctly describes the relationship between MSE and the true parameter?

• A. The MSE is a function of the true parameter.
• B. The MSE is independent of the true parameter.
• C. The MSE equals the variance of the estimator.
• D. The MSE equals the bias of the estimator.

Answer: (A)

Mean squared error is defined as E[(θ̂ − θ)²], where θ is the true parameter. This quantity inherently depends on θ because both the estimator’s bias (E[θ̂] − θ) and its variance can change as θ changes. The standard relationship MSE(θ) = Var(θ̂) + [Bias(θ̂, θ)]² makes this dependence explicit: unless both the bias and the variance are somehow constant with respect to θ, the MSE will vary with the true parameter. In an unbiased case (Bias = 0), MSE reduces to Var(θ̂), but even then the dependence on θ isn’t guaranteed to disappear in general models; if the variance itself can vary with θ, the MSE remains a function of θ. The statement that the MSE equals the variance or equals the bias is incomplete; MSE combines both components.

Mean squared error is defined as E[(θ̂ − θ)²], where θ is the true parameter. This quantity inherently depends on θ because both the estimator’s bias (E[θ̂] − θ) and its variance can change as θ changes. The standard relationship MSE(θ) = Var(θ̂) + [Bias(θ̂, θ)]² makes this dependence explicit: unless both the bias and the variance are somehow constant with respect to θ, the MSE will vary with the true parameter.

In an unbiased case (Bias = 0), MSE reduces to Var(θ̂), but even then the dependence on θ isn’t guaranteed to disappear in general models; if the variance itself can vary with θ, the MSE remains a function of θ. The statement that the MSE equals the variance or equals the bias is incomplete; MSE combines both components.