Least squares of a constant

In summary, the conversation discusses the determination of the least square estimator of c in an observation where c is an unknown constant and e is the error with a given probability density function. The error cost function is minimized by taking the derivative with respect to c, and in the matrix case, the derivative is equal to zero. However, in the scalar case, it is unclear what to do with the distribution of the error. The conversation suggests that fixing e to 0 to minimize the error is not a viable solution and instead, the expected value for E(c) should be considered for minimization.
  • #1
cutesteph
63
0
Suppose we have an observation y = c+ e where c is an unknown constant and e is error with the pdf = exp(-e-1) for e >-1 . We want to determine the least square estimator of c given by the c* which minimizes the error cost function E(c) = .5(y-c)^2

Minimizing the error cost is done by taking the derivative wrt c so y=c. Shouldnt it take into account the distribution of the error?

I understand in the matrix case E(c) = T(e)e where T( ) is the transpose . where y=Hc+e

= T(y-Hc) (y-Hc) = T(y)y -T( x)Hy -T(y)Hx +T(x)T(H)Hx .
The derivative wrt x is -2T(y)H+2T(x)T(H)H= 0 => x=inverse(T(H)H)*T(H)y

I guess I am just confused on the scalar case on what to do.
 
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  • #2
cutesteph said:
Minimizing the error cost is done by taking the derivative wrt c so y=c. Shouldnt it take into account the distribution of the error?
This would fix e to 0. You cannot fix your random variable to minimize your error.
What is the expected value for E(c)? Afterwards you can try to minimize this.
 

1. What is the concept of "least squares of a constant" in statistics?

The concept of "least squares of a constant" is a statistical method used to find the best-fitting line or curve for a set of data points. It involves minimizing the sum of the squares of the differences between the observed data points and the predicted values from the model.

2. How is "least squares of a constant" calculated?

The calculation of "least squares of a constant" involves finding the value of the constant that minimizes the sum of the squares of the residuals (the differences between the observed data points and the predicted values). This can be done using various mathematical techniques such as gradient descent, normal equations, or matrix operations.

3. What is the purpose of using "least squares of a constant"?

The purpose of using "least squares of a constant" is to find the best approximation of a relationship between variables in a dataset. It is commonly used in regression analysis to determine the relationship between a dependent variable and one or more independent variables.

4. Can "least squares of a constant" be used for non-linear relationships?

Yes, "least squares of a constant" can be used for non-linear relationships by transforming the data or by using non-linear regression techniques. This allows for the best-fitting line or curve to be determined for non-linear relationships between variables.

5. What are the assumptions made when using "least squares of a constant"?

There are several assumptions made when using "least squares of a constant", including that the residuals are normally distributed, there is a linear relationship between the variables, and there are no influential outliers in the data. Violation of these assumptions can affect the accuracy and validity of the results.

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