MSE estimation with random variables

In summary, the problem at hand involves finding the coefficients for a given equation and determining the variance of the variable U_K. The approach so far has involved setting the derivative of the expressions involving the coefficients equal to 0 and using the formula (b-a)^2/12 for finding the variance, but this calculation seems to be incorrect. Any assistance with solving this problem would be appreciated.
  • #1
60
2
Homework Statement
Please see attached problem
Relevant Equations
MSE = E( (Xhat[k] - X[k])^2 )
Hello all, I would appreciate any guidance to the following problem. I have started on parts (a) and (b), but need some help solving for the coefficients. Would I simply take the expressions involving the coefficients, take the derivative and set it equal to 0 and solve? I believe I also need the variance (for part (c) for instance) ##Var(U_K)## but using the formula (b-a)^2/12 (which would be (3-(-3)^2/12 = 3) from the uniform distribution seems wrong. Any help appreciated

Problem:
1669065615464.png

Approach so far:
1669068082207.png
 
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  • #2
(a) $$U_K = \alpha + \beta x_1 + \gamma x_2 + \delta x_3$$(b) $$\frac{1}{n}\sum_{i=1}^{n}(U_K - \hat{U_K})^2 = \frac{1}{n}\sum_{i=1}^{n}(\alpha + \beta x_{1i} + \gamma x_{2i} + \delta x_{3i} - \hat{U_K})^2$$(c)Var(U_K) = ?
 

1. What is MSE estimation?

MSE estimation is a statistical method used to estimate the mean squared error (MSE) of a random variable. It involves calculating the squared difference between the observed values and the predicted values of a random variable, and then taking the average of these squared differences.

2. How is MSE estimation used in scientific research?

MSE estimation is commonly used in scientific research to evaluate the performance of statistical models. It allows researchers to assess how well a model fits the data and make comparisons between different models.

3. Can MSE estimation be applied to any type of random variable?

Yes, MSE estimation can be applied to any type of random variable, including continuous, discrete, and multivariate variables. It is a versatile and widely used method in statistical analysis.

4. What are the advantages of using MSE estimation?

One of the main advantages of MSE estimation is that it takes into account both the bias and variance of a model, providing a more comprehensive measure of its performance. It is also a relatively simple and easy to interpret metric.

5. Are there any limitations to MSE estimation?

While MSE estimation is a useful tool, it does have some limitations. For example, it assumes that the errors in the data are normally distributed, which may not always be the case. Additionally, it can be sensitive to outliers in the data, so it is important to check for and address any influential data points before using MSE estimation.

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