We wish to estimate an unknown quantity Θ. Our measuring equipment produces an observation of the form X-Θ3 + W, where W is a noise term which is small relative to the range of Θ. Which type of linear estimator is preferable in such a situation?
Theta hat = a X^1/3 + b
btw options were not stated :)
they are:
(a) Θˆ=𝑎𝑋+𝑏
(b) Θˆ=𝑎𝑋3+𝑏
(c) Θˆ=𝑎𝑋1/3+𝑏
Well, when it comes to estimating an unknown quantity Θ with noisy measurements, you definitely want a linear estimator that can handle some clown-like behavior! In this case, the best choice would be the "Least Squares" estimator.
Why, you might ask? Well, the Least Squares estimator takes into account the noise term W and minimizes the sum of the squares of the differences between the observed value X-Θ^3 + W and the true value Θ. It's like trying to find the most balanced way to juggle all the variables in the equation.
So, when it comes to estimating Θ while dealing with a noise term that's small compared to the range of Θ, trust the clown and go with the Least Squares estimator. It'll try its best to keep everything in balance, even if it means juggling a few extra numbers along the way!
In such a situation, the preferable type of linear estimator would be the method of least squares.
The method of least squares is suitable when there is additive noise and the noise is small relative to the range of the unknown quantity Θ. It minimizes the sum of the squares of the differences between the observed values and the estimated values.
In this case, since the noise term W is small relative to the range of Θ, the method of least squares would be appropriate. It will provide an estimate of Θ that minimizes the squared differences between the observed values (X-Θ^3 + W) and the estimated values.
In this situation, when estimating the unknown quantity Θ, a type of linear estimator called the method of least squares can be preferable.
The method of least squares involves finding the optimal estimator by minimizing the sum of squared differences between the observed values and the predicted values. In other words, it aims to minimize the sum of the squared residuals or errors.
To apply the method of least squares in this scenario, we need to formulate the problem mathematically. Let's start by rearranging the equation for the observation to isolate Θ:
X = Θ3 + W
Now, let's define our prediction as a linear function of X:
Y = aX + b
Our goal is to find the values of a and b that minimize the sum of squared differences between Y and X. To do this, we can define the error term as:
E = X - Y
Now, we can express the sum of squared errors as:
S = ∑(E^2)
To minimize S, we can differentiate it with respect to a and b, and set the derivatives equal to zero. Solving these equations will provide the optimal values for a and b.
Once we have determined the optimal values of a and b, we can calculate our estimated value of Θ using the predicted value Y:
Θ_hat = (Y - b)^(1/3)
This estimated value Θ_hat will serve as the optimistic linear estimator for the unknown quantity Θ.
To summarize, in a situation where we wish to estimate an unknown quantity Θ with an observation of the form X-Θ3 + W, the method of least squares can be a preferable type of linear estimator. It involves minimizing the sum of squared errors between the observed values and the predicted values to find the optimal values of the linear function's parameters.