Question

Suppose we observe independent pairs (Xi,Yi) where each (Xi,Yi) has a uniform distribution in the circle of unknown radius θ and centered at (0,0) in the plane.

Show that for the method of moments estimator and the maximum likelihood estimator, it is the case that the distribution of ˆθ/θ does not depend on θ. Explain why this means we can write

MSEθ(ˆ θ) = θ2*MSE1(ˆ θ) where here the subscript θ means "under the assumption that the true value is θ," so MSE1 denotes the mean square error under the assumption that θ = 1. From this, explain why it suffices that we compare the two estimators when θ = 1

Answer 1

The method of moments estimator and the maximum likelihood estimator are two commonly used techniques to estimate the unknown parameter, θ, in a given distribution. In this case, we are interested in estimating the radius, θ, of a circle based on independent pairs (Xi, Yi), which have a uniform distribution within the circle centered at (0,0).

To understand why the distribution of ˆθ/θ does not depend on θ for both estimators, let's look at the method of moments estimator first. This estimator matches the moments of the observed data to the moments of the theoretical distribution. In our case, the first moment of the uniform distribution within the circle is related to the radius, θ. Therefore, the method of moments estimator for θ will depend on the first moment of the observed data.

Similarly, the maximum likelihood estimator maximizes the likelihood function, which is a measure of how likely the observed data is given a specific value of θ. In our case, the likelihood function will depend on the radius, θ, as it determines the probability density within the circle. Therefore, the maximum likelihood estimator for θ will also depend on θ.

Now, since the distribution of ˆθ/θ does not depend on θ for both estimators, it means that the ratio of the estimated value to the true value, ˆθ/θ, has a distribution that is independent of θ. This implies that the mean square error (MSE) of the estimator, denoted as MSEθ(ˆθ), is proportional to the square of the MSE under the assumption that θ = 1, denoted as MSE1(ˆθ). Mathematically, this can be written as

MSEθ(ˆθ) = θ²* MSE1(ˆθ).

From this relationship, we can see that by comparing the two estimators when θ = 1, we can determine their relative performance in terms of mean square error. Since the distribution of ˆθ/θ is independent of θ, comparing the estimators for θ = 1 gives us insight into their performance across different values of θ. This simplifies the analysis and allows us to assess the estimators' accuracy and precision without considering different values of θ.

In summary, the distribution of does not depend on θ for the method of moments estimator and the maximum likelihood estimator. This allows us to write MSEθ(ˆθ) = θ² * MSE1(ˆθ), and comparing the estimators when θ = 1 is sufficient to assess their performance.