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How do you find the minimum MSE?
One way of finding a point estimate ˆx=g(y) is to find a function g(Y) that minimizes the mean squared error (MSE). Here, we show that g(y)=E[X|Y=y] has the lowest MSE among all possible estimators. That is why it is called the minimum mean squared error (MMSE) estimate. h(a)=E[(X−a)2]=EX2−2aEX+a2.
What is the expected value of a linear combinations of random variables?
The notion of the average value of a random variable, which we have defined as the expected value of a random variable, behaves in the same fashion. If Y = aX +b, then E(Y) = aE(X) + b. More generally, this property extends to linear combinations of random variables.
Is MMSE linear?
Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the Wiener–Kolmogorov filter and Kalman filter.
How are linear combinations of random variables expressed?
Mathematically linear combinations can be expressed as shown in the expression below: Y = c 1 X 1 + c 2 X 2 + ⋯ + c p X p = ∑ j = 1 p c j X j = c ′ X. Here what we have is a set of coefficients c 1 through c p that is multiplied bycorresponding variables X 1 through X p.
How to calculate the MSE of a random variable?
ˆXL = g(Y) = aY + b, where a and b are some real numbers to be determined. More specifically, our goal is to choose a and b such that the MSE of the above estimator MSE = E[(X − ˆXL)2] is minimized.
Which is the optimal MSE for linear MMSE?
ˆXL = g(Y) = aY + b, where a and b are some real numbers to be determined. More specifically, our goal is to choose a and b such that the MSE of the above estimator MSE = E[(X − ˆXL)2] is minimized. We call the resulting estimator the linear MMSE estimator. The following theorem gives us the optimal values for a and b .
How to calculate the mean and variance of a linear combination?
Then, the mean and variance of the linear combination Y = ∑ i = 1 n a i X i, where a 1, a 2, …, a n are real constants are: respectively. Let’s start with the proof for the mean first: Now for the proof for the variance.