I have run into a problem in statistics which might interest people here, and also I’d quite like to know if there is a good solution.
In charm mixing we try to measure mixing parameters imaginatively named x and y. (They are normalised mass and width differences of mass eigenstates, but this is not important to the problem.) In the most experimentally-accessible decay channel, however, we are not sensitive to x and y directly, but to rotated quantities
y′=ycosdeltaxsindelta
where the strong phase delta is unknown. In fact, the situation is a bit worse than this; we get our result from a fit to a sum of terms, where one term is proportional to y’ and the other to (x’^2+y’^2). Consequently the experimental result is in fact a value for y’ and a value for x’^2. Because of uncertainty, the x’^2 value may be negative, which is unphysical—x’ cannot be imaginary, but if it’s close to zero you can make a mistake and measure x’^2 as negative.
Now, for comparison with other experiments we’d like to make a confidence-limit contour in (x,y) space. It seems that this contour ought to be an annulus, a doughnut shape, since we don’t know anything about delta and we can therefore only tell how far we are from the origin. However, I cannot figure out how to make a contour which consistently (for all values of delta) has 68% coverage. Does anyone have an insight?
I’m curious about this too, not because I’m working on any problems like this, but just because it sounds interesting. I have no insights, but the popular Feldman and Cousins paper about building confidence belts that don’t stray into unphysical ranges might be helpful. Ditto the papers citing that one.
Thank you, that paper contained the solution. The trick is to consider r^2=x’^2+y’^2 as the variable of interest, and note that it may be measured negative; then construct the confidence bands using the ordering principle given in their section III, with a numerical rather than analytical calculation of the likelihood ratios since the probability depends on x’^2 and y’ in a complicated way rather than straightforwardly on the distance from zero. But that’s all implementation details, the concept is exactly what Feldman and Cousins outline.
I have run into a problem in statistics which might interest people here, and also I’d quite like to know if there is a good solution.
In charm mixing we try to measure mixing parameters imaginatively named x and y. (They are normalised mass and width differences of mass eigenstates, but this is not important to the problem.) In the most experimentally-accessible decay channel, however, we are not sensitive to x and y directly, but to rotated quantities
y′=ycosdelta xsindelta
where the strong phase delta is unknown. In fact, the situation is a bit worse than this; we get our result from a fit to a sum of terms, where one term is proportional to y’ and the other to (x’^2+y’^2). Consequently the experimental result is in fact a value for y’ and a value for x’^2. Because of uncertainty, the x’^2 value may be negative, which is unphysical—x’ cannot be imaginary, but if it’s close to zero you can make a mistake and measure x’^2 as negative.
Now, for comparison with other experiments we’d like to make a confidence-limit contour in (x,y) space. It seems that this contour ought to be an annulus, a doughnut shape, since we don’t know anything about delta and we can therefore only tell how far we are from the origin. However, I cannot figure out how to make a contour which consistently (for all values of delta) has 68% coverage. Does anyone have an insight?
I’m curious about this too, not because I’m working on any problems like this, but just because it sounds interesting. I have no insights, but the popular Feldman and Cousins paper about building confidence belts that don’t stray into unphysical ranges might be helpful. Ditto the papers citing that one.
Thank you, that paper contained the solution. The trick is to consider r^2=x’^2+y’^2 as the variable of interest, and note that it may be measured negative; then construct the confidence bands using the ordering principle given in their section III, with a numerical rather than analytical calculation of the likelihood ratios since the probability depends on x’^2 and y’ in a complicated way rather than straightforwardly on the distance from zero. But that’s all implementation details, the concept is exactly what Feldman and Cousins outline.
No problem! I was wondering if I was wasting your time with a shot in the dark—glad to hear it helped.