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) For the LMS estimator Θˆ, it is true that E[Θ˜∣X=x]=0 for every x. However, it is not necessarily true that E[Θ˜∣Θ=θ]=0 for all θ. This means that the estimation error Θ˜ may not be centered around zero when conditioned on the true value of Θ. In other words, the LMS estimator may have a bias, even though its conditional expectation is zero for every observed value of X.
b) The property Var(Θ)=Var(Θˆ)+Var(Θ˜) is not necessarily true for every estimator Θˆ. This property represents the decomposition of the total variance of Θ into the variance of the estimator Θˆ and the variance of the estimation error Θ˜. It holds true in some cases, such as when the estimator is unbiased and the estimation error is uncorrelated with the estimator. However, for estimators that have bias or correlated estimation error, this property may not hold true. Hence, it depends on the specific characteristics of the estimator Θˆ and the estimation error Θ˜.
a) To answer this question, we need to understand the properties of the LMS estimator and the concept of conditional expectation.
The LMS (Least Mean Squares) estimator of Θ, denoted by Θˆ, is a type of estimator that minimizes the mean square error between the estimated value and the true value of Θ. In other words, it tries to find an estimate Θˆ that minimizes E[(Θˆ - Θ)^2].
The estimation error, denoted by Θ˜, is defined as the difference between the estimator and the true value: Θ˜ = Θˆ - Θ.
Now, the key property of the LMS estimator is that for every value of X (the observed data), the conditional expectation of the estimation error, given X=x, is zero: E[Θ˜ | X=x] = 0.
The question asks if the same property holds when we condition on Θ instead of X: E[Θ˜ | Θ=θ] = 0 for all θ.
In general, for any estimator Θˆ, it is not guaranteed that E[Θ˜ | Θ=θ] would be zero for all θ. It depends on the specific estimator and the underlying distribution of the random variable. Therefore, it is not necessarily true that E[Θ˜ | Θ=θ] = 0 for all θ for every estimator, including the LMS estimator.
b) To answer this question, we need to understand the properties of variance and the relationship between the true value (Θ), the estimator (Θˆ), and the estimation error (Θ˜).
The variance of a random variable measures the spread or dispersion of its values. It is a measure of how much the values of the random variable deviate from its expected value.
The property Var(Θ) = Var(Θˆ) + Var(Θ˜) is not true for every estimator Θˆ. This property holds true only if the estimator Θˆ is an unbiased estimator of Θ (i.e., E[Θˆ] = Θ) and the estimation error Θ˜ is uncorrelated with the estimator Θˆ.
If the estimator Θˆ is unbiased, it means that on average, the estimated value is equal to the true value: E[Θˆ] = Θ. In this case, the variance of the estimator Var(Θˆ) quantifies the variability in the estimated values around the true value.
The variance of the estimation error Var(Θ˜) quantifies the variability in the difference between the estimator and the true value. It measures how much the estimated values differ from the true value on average.
So, if the estimator is unbiased and the estimation error is uncorrelated with the estimator, then Var(Θ) = Var(Θˆ) + Var(Θ˜) holds.
However, if the estimator is biased (E[Θˆ] ≠ Θ) or the estimation error is correlated with the estimator, then Var(Θ) = Var(Θˆ) + Var(Θ˜) may not hold.
In conclusion, the property Var(Θ) = Var(Θˆ) + Var(Θ˜) is not true for every estimator, and its validity depends on the specific properties of the estimator and the estimation error.
a) No, it is not necessarily true that E[Θ˜∣Θ=θ]=0 for all θ. While the LMS estimator has the property that E[Θ˜∣X=x]=0 for every x, this does not imply that E[Θ˜∣Θ=θ]=0 for all θ. It is possible for the estimation error to have a non-zero mean even when conditioned on the true parameter value.
b) Yes, the property Var(Θ)=Var(Θˆ)+Var(Θ˜) is true for every estimator Θˆ. This can be shown by using the definition of variance and properties of conditional expectations:
Var(Θ) = E[(Θ - E[Θ])^2]
= E[(Θ - E[Θˆ + Θ˜])^2]
= E[(Θ - E[Θˆ] - E[Θ˜])^2]
= E[(Θ - Θ)^2 + (Θ - E[Θˆ])^2 + (Θ - E[Θ˜])^2 + 2(Θ - E[Θˆ])(Θ - E[Θ˜])]
= Var(Θˆ) + Var(Θ˜) + 2E[(Θ - E[Θˆ])(Θ - E[Θ˜])]
= Var(Θˆ) + Var(Θ˜)
So, the variance of the estimator Θ is equal to the sum of the variances of its estimation error and the true parameter.