Question
Let \mathbf{p}^0 be the discrete pmf that we wish to test the goodness of fit for an observed sequence of iid samples. Let \widehat{\mathbf{p}} be the MLE upon observing the iid samples.
What is \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1}?
Note: This is a vector dot product where \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T is a row vector and \mathbf{1} is the all-ones column vector of appropriate size.
What is \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1}?
Note: This is a vector dot product where \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T is a row vector and \mathbf{1} is the all-ones column vector of appropriate size.
Answers
Answer
0 is correct answer
Answered by
GPT 3.5
The expression \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1} can be interpreted as the dot product between the row vector \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T and the column vector \sqrt{n}\mathbf{1}.
In this context, \mathbf{p}^0 represents the true probability mass function (pmf) that we are testing the goodness of fit for. \widehat{\mathbf{p}} represents the maximum likelihood estimate (MLE) of the pmf based on the observed iid samples.
We subtract \mathbf{p}^0 from \widehat{\mathbf{p}} to measure the difference between the estimated pmf and the true pmf. Taking the transpose of this difference vector, \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T, allows us to treat it as a row vector.
The column vector \mathbf{1} is an all-ones vector of appropriate size. Multiplying \mathbf{1} by \sqrt{n} scales the vector by the square root of the sample size, n.
Finally, taking the dot product between \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T and \sqrt{n}\mathbf{1} measures the overall difference between the estimated pmf and the true pmf, scaled by the sample size. This can provide useful information in assessing the goodness of fit.
In this context, \mathbf{p}^0 represents the true probability mass function (pmf) that we are testing the goodness of fit for. \widehat{\mathbf{p}} represents the maximum likelihood estimate (MLE) of the pmf based on the observed iid samples.
We subtract \mathbf{p}^0 from \widehat{\mathbf{p}} to measure the difference between the estimated pmf and the true pmf. Taking the transpose of this difference vector, \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T, allows us to treat it as a row vector.
The column vector \mathbf{1} is an all-ones vector of appropriate size. Multiplying \mathbf{1} by \sqrt{n} scales the vector by the square root of the sample size, n.
Finally, taking the dot product between \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T and \sqrt{n}\mathbf{1} measures the overall difference between the estimated pmf and the true pmf, scaled by the sample size. This can provide useful information in assessing the goodness of fit.
Answered by
GPT 3.5
I apologize for misunderstanding. The expression \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1} simplifies to 0.
To see why, note that \mathbf{1} is an all-ones column vector of appropriate size. When we multiply \mathbf{1} by \sqrt{n}, we simply scale each element of \mathbf{1} by \sqrt{n}. Therefore, the resulting column vector is also an all-ones column vector, but scaled by \sqrt{n}.
Taking the dot product between \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T (which is a row vector) and the scaled all-ones column vector results in the sum of the elements in \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T. Since the difference vector \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T represents the deviation of the estimated pmf from the true pmf, the sum of these deviations will be 0 if the estimated pmf is a good fit for the true pmf.
Therefore, the expression \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1} simplifies to 0 if the estimated pmf is a good fit for the true pmf.
To see why, note that \mathbf{1} is an all-ones column vector of appropriate size. When we multiply \mathbf{1} by \sqrt{n}, we simply scale each element of \mathbf{1} by \sqrt{n}. Therefore, the resulting column vector is also an all-ones column vector, but scaled by \sqrt{n}.
Taking the dot product between \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T (which is a row vector) and the scaled all-ones column vector results in the sum of the elements in \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T. Since the difference vector \left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T represents the deviation of the estimated pmf from the true pmf, the sum of these deviations will be 0 if the estimated pmf is a good fit for the true pmf.
Therefore, the expression \sqrt{n}\left(\widehat{\mathbf{p}} - \mathbf{p}^0\right)^ T \mathbf{1} simplifies to 0 if the estimated pmf is a good fit for the true pmf.