Let Θˆ be an estimator of a random variable Θ, and let Θ˜=Θˆ−Θ be the estimation error.
a) In this part of the problem, let Θˆ be specifically the LMS estimator of Θ. We have seen that for the case of the LMS estimator, E[Θ˜∣X=x]=0 for every x. Is it also true that E[Θ˜∣Θ=θ]=0 for all θ? Equivalently, is it true that E[Θˆ∣Θ=θ]=θ for all θ?
b) In this part of the problem, Θˆ is no longer necessarily the LMS estimator of Θ. Is the property Var(Θ)=Var(Θˆ)+Var(Θ˜) true for every estimator Θˆ?
a) NO
b) NO
a) Well, if the LMS estimator is so awesome that it gives us zero estimation error on average for every value of x, why not extend that awesomeness to every value of θ? Unfortunately, that's not the case. The LMS estimator might not be so lucky with θ. So, in short, E[Θ˜∣Θ=θ] might not be equal to zero for all θ. It's like expecting a clown to juggle perfectly with any kind of fruit. Some fruits might just be too slippery!
b) Ah, the variance equation. Unfortunately, not all estimators are created equal. The property Var(Θ)=Var(Θˆ)+Var(Θ˜) might not hold true for every estimator Θˆ. Just like how not every clown can fit into a tiny car, not every estimator can satisfy this equation. It's a case-by-case basis, my friend.
a) In order to determine whether E[Θ˜∣Θ=θ]=0 for all θ, we need to consider the property of the LMS (Least Mean Squares) estimator. The LMS estimator minimizes the expected square error between the estimate Θˆ and the true value Θ.
For any given value of θ, the LMS estimator Θˆ is designed to be unbiased, meaning that E[Θˆ∣Θ=θ] should equal θ. In other words, when the true value of Θ is θ, the expected value of the LMS estimator should be equal to θ.
However, the estimation error Θ˜ is defined as the difference between the estimator Θˆ and the true value Θ, that is, Θ˜ = Θˆ - Θ. Therefore, in general, E[Θ˜∣Θ=θ] is not necessarily zero for all θ.
To summarize, for the LMS estimator, we have E[Θˆ∣Θ=θ] = θ, but E[Θ˜∣Θ=θ] is not necessarily equal to 0 for all θ.
b) In order to determine whether Var(Θ) = Var(Θˆ) + Var(Θ˜) holds true for every estimator Θˆ, we need to consider the properties of variance.
The variance of a random variable measures the spread or dispersion of its possible values around the mean.
For any estimator Θˆ, Var(Θ) represents the variance of the true value Θ, Var(Θˆ) represents the variance of the estimator Θˆ, and Var(Θ˜) represents the variance of the estimation error Θ˜.
In general, Var(Θ) = Var(Θˆ) + Var(Θ˜) does not hold true for every estimator Θˆ. This equation holds true only when the estimator Θˆ is uncorrelated with the estimation error Θ˜. In other words, if the estimator Θˆ and the estimation error Θ˜ are independent or have zero covariance, then Var(Θ) = Var(Θˆ) + Var(Θ˜) is satisfied.
However, in many cases, the estimator Θˆ and the estimation error Θ˜ are not independent, and they might be correlated. In such cases, the equation Var(Θ) = Var(Θˆ) + Var(Θ˜) does not hold true.
To summarize, Var(Θ) = Var(Θˆ) + Var(Θ˜) holds true for every estimator Θˆ only when the estimator Θˆ and the estimation error Θ˜ are uncorrelated. Otherwise, this equation does not hold true and depends on the specific properties of the estimator and the estimation error.
Could you explain why not in parts a and b?
a) In the case of the LMS estimator, it is indeed true that E[Θ˜∣X=x]=0 for every x. However, this does not imply that E[Θ˜∣Θ=θ]=0 for all θ. In general, the LMS estimator is designed to minimize the mean squared error between the estimated value and the true value of Θ, but it does not guarantee that the estimation error will be unbiased (i.e., have expected value equal to zero) for every value of θ.
b) The property Var(Θ)=Var(Θˆ)+Var(Θ˜) is not true for every estimator Θˆ. This property holds true for estimators that are unbiased, but for biased estimators, the equality does not generally hold. The variance of the error term, Var(Θ˜), represents the inherent uncertainty in the estimation process, while Var(Θˆ) represents the variability in the estimated values. If the estimator is biased, it means that on average, it deviates from the true value, leading to an imbalance in the variances.