Frequentist Confidence Interval for a RNG
January 25th, 2019 at 12:32:20 AM
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I am wondering if an expert in frequentist statistical inference can tell me how to calculate the confidence interval for the following problem:
I program a computer to randomly select a number b which is uniform on the interval [0,1]. The number b will represent the bias of a coin toss coming up heads. In other words, b = 1/2 represents a fair coin, and b = 1 would mean the coin always comes up heads, and so forth..
Now the computer generates simulated coin tosses using the b value, and reports a string of results, for example HHTHTHTHHHTTTTT... for say 100 tosses.
Please note that this is not the standard textbook example of estimating the proportion in a sample, as we are given extra information that the bias is selected from a uniform distribution. An expert in frequentist inference has told me that the standard method for calculating confidence intervals is not appropriate, but the expert was unable to tell me the correct way to calculate confidence intervals for this problem.
I am hoping that an expert frequentist can shed some light on this problem.
Thanks,
Tom
I program a computer to randomly select a number b which is uniform on the interval [0,1]. The number b will represent the bias of a coin toss coming up heads. In other words, b = 1/2 represents a fair coin, and b = 1 would mean the coin always comes up heads, and so forth..
Now the computer generates simulated coin tosses using the b value, and reports a string of results, for example HHTHTHTHHHTTTTT... for say 100 tosses.
Please note that this is not the standard textbook example of estimating the proportion in a sample, as we are given extra information that the bias is selected from a uniform distribution. An expert in frequentist inference has told me that the standard method for calculating confidence intervals is not appropriate, but the expert was unable to tell me the correct way to calculate confidence intervals for this problem.
I am hoping that an expert frequentist can shed some light on this problem.
Thanks,
Tom
January 25th, 2019 at 6:46:57 AM
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Welcome to the forum Tom. I hope a frequentist comes along to answer your question. I can't speak to what they would say. I also don't understand why you couldn't use standard Bayesian formulas to estimate a confidence interval.
Let's say the distribution was 60 heads and 40 tails.
The variance of each flip = E(x^2) - (E(x))^2 = 0.6 - 0.36 = 0.24
standard devation = 0.4899
A 95% confidence interval should be
0.6 +/- 1.96 * 0.24/sqrt(99) = 0.6 +/- 0.965 = 0.6 +/- 0.0965 = 0.5035 to 0.6965.
I know I'm not factoring into this the statement that the mean was taken from the uniform distribution, but I personally don't know how to mix that into this or even if it's kosher to do so. I'm already acknowledging the unknown true mean in subtracting one from the sample size. I am just doing my best with the actual data. I'm sure that would make the hair on the back of a frequentist's neck stand up, but I find them to be quick to criticize and slow to offer actual answers.
Let's say the distribution was 60 heads and 40 tails.
The variance of each flip = E(x^2) - (E(x))^2 = 0.6 - 0.36 = 0.24
standard devation = 0.4899
A 95% confidence interval should be
0.6 +/- 1.96 * 0.24/sqrt(99) = 0.6 +/- 0.965 = 0.6 +/- 0.0965 = 0.5035 to 0.6965.
I know I'm not factoring into this the statement that the mean was taken from the uniform distribution, but I personally don't know how to mix that into this or even if it's kosher to do so. I'm already acknowledging the unknown true mean in subtracting one from the sample size. I am just doing my best with the actual data. I'm sure that would make the hair on the back of a frequentist's neck stand up, but I find them to be quick to criticize and slow to offer actual answers.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
