Plot confidence intervals/bounds of the estimated theoretical cdf

Hi,

Please understand that I am no mathematician or statistician, so I might mix up terms here. Any correction would be appreciated as I am here to learn!

I have experimental data that I have used to generate an ecdf and then fitted a theoretical cdf, by evaluating numerous distributions (e.g., Weibull, log-logistic etc.) using the AD-test to find the best theoretical distribution (e.g., Normal, Weibull, log-normal, logistic, log-logistic etc.).

I want to plot confidence bounds of the theoretical cdf (I want smooth confidence bounds, not the step-function shape of the ecdf), such that it should surround the fitted/theoretical curve (theoretical cdf)?

But using fitdist on the experimental data only returns the 95% confidence bounds for the distribution parameters of the theoretical distribution I use as input in fitdist.

I have been wrongly using the confidence bounds from fitdist, believing it would give me the confidence intervals of the theoretical (fitted) function/cdf (so that it can be compared to the generated/fitted theoretical cdf). But now learning that it is the confidence intervals of the distribution parameters only.

Is there a MATLAB-function that allows me to plot the true confidence interval of the theoretical cdf, such that I can plot the confidence bounds so that they surround the theoretical curve/cdf?

Thank you

Jia-Cheng

[Torsten]

I've never heard of confidence intervals for estimated cdfs. Can you give a link where you found this ?

[Star Strider]

One option could be to use the prctile function to get the (0.025 0.975) percentiles of the data.

It is distribution-independent, so the underlying data distribution is not relevant to its results.

[Jia Cheng Xu]

Sorry if I used the wrong terms and added confusion.

I have an ecdf based on experimental data, and have derived a theoretical cdf (I chose log-logistic cdf) and plot the log-logistic cdf based on its cdf equation with scale and shape parameter (I have to convert to alpha and beta, since MATLAB gives mu and sigma).

I am not sure if it is possible to find the 95th confidence interval of this theoretical cdf "curve" in MATLAB, and ideally I want the confidence bounds of the experimental data to evaluate my derived log-logistic cdf?

I had plotted the confidence intervals by using my log-logistic cdf equation but used the output from fitdist which I learned is the confidence interval of the distribution parameters, so I cannot just put them into my cdf and call them confidence bounds of the data as this is the uncertainty of the distribution parameters and not the data itself? Please correct me here, I am here to learn and any material on this would be helpful. For Weibull cdf, I think wblcdf can be used to get the confidence bounds of the data and not the distribution parameters? But doesn't exist for log-logistic cdf?

Is it possible to plot confidence bounds based on the experimental data, since fitdist only gives the confidence intervals of the distribution parameters. I want to find out how I can plot the confidence interval of the experimental data to see if my theoretical cdf is within the confidence bounds, I don't want the confidence bounds of the ecdf that is visualized with the step function characteristics.

Below is an example where blue ecdf and red theoretical cdf is plotted, then I wanted to add confidence intervals to visualize the theoretical cdf in relation to the experimental data through confidence bounds. But as you can see, the confidence bounds are intersecting since I used the confidence interval parameters from fitdist which is only for the distribution parameters.

Thank you!

[Jeff Miller]

Your plot shows two boundary CDFs. Was one computed with the lower bounds of the two parameters and the other computed with the upper bounds of the two parameters? If so, what happens if you compute boundary CDFs with the other two pairs of parameters--i.e., lower bound of parameter 1 and upper bound of parameter 2, or vice versa? I'm not familiar with the parameters of this distribution, but it is certainly possible that the leftmost/rightmost CDF curves will be found with the largest value of one parameter and the smallest value of the other parameter.

Like Torsten, I've never heard of confidence intervals for predicted CDFs. Your overall goal of somehow visualizing the range of predicted CDFs certainly makes sense, but it probably isn't technically correct to call this a CI for a predicted CDF.

[Paul]

I don't understand how there could be a confidence interval around the theoretical CDF. But one can get a confidence interval around the estimated, or empirical, CDF and the function ecdf can do exactly that. It sounds like you're aware of that functionality, but I don't understand what the objection is.