Confidence intervall of quantil

Hi, I am trying to find the 95% confidence interval for my 99% quantile. I am using the 3 parameter Weibull distribution to get the quantile. Now I am trying to get the confidence interval. I have read that it is possible to get it with an anonymous function and with bootci while giving the data as input and getting the quantil as output. Unfortunately, I do not know how exactly this works.

clear all
close all
clc;
    


C = wblrnd(3,1:100)


custompdf = @(x,a,b,c) (x>c).*(b/a).*(((x-c)/a).^(b-1)).*exp(-((x-c)/a).^b);
customcdf = @(x,a,b,c) (x>c).*1-exp(-((x-c)/a).^b);
    
opt = statset('MaxIter',1e5,'MaxFunEvals',1e5,'FunValCheck','off');
params = mle(C,'pdf',custompdf,'start',[mean(C) mean(C) min(C)/max(C)],'Options',opt,'LowerBound',[0 0 -Inf],'UpperBound',[Inf Inf min(C)]);

a_parameter = params(1,1);
b_parameter = params(1,2);
c_parameter = params(1,3);

x = [c_parameter+eps(c_parameter):0.1:max(C)*1.5];

    quantile_Weibull3 = fsolve(@(x) customcdf(x, a_parameter, b_parameter, c_parameter) - 0.99, mean(C));

[Jeff Miller]

Why do you start by generating random values for C? Normally a confidence interval is generated from some specific set of observed data values, so generating them randomly is confusing. Possibly I just don't understand the question...

Answer 1

[Prathamesh]

Hi,

I understand that you want to know how the code works and how to find 95% confidence interval for your 99% quantile.

Pointers to explain the code and confidence interval calculation:

1. Random data samples are generated from a Weibull distribution and PDF and CDF are defined for a three-parameter Weibull distribution.
2. Options for MLE estimation are set specifying the maximum number of iterations and function evaluations and disables the check for valid function values.
3. Parameters of the three-parameter Weibull distribution are estimated using MLE and are extracted from “params” array.
4. Then the “fsolve” function is used to find the value of x for which the CDF equals 0.99. This corresponds to the quantile for a cumulative probability of 0.99 and the quantile is acquired.
5. Now you can use “bootci” function to calculate 95% bootstrap confidence interval.

nboot = 1000;

bootfcn = @(x) quantile(x, 0.99);

ci = bootci(nboot, bootfcn, data);

To get more information about working of “bootci” function and how the confidence interval is calculated for different functions refer to the documentation attached below.

https://www.mathworks.com/help/stats/bootci.html#mw_61dcd345-cab1-4e4c-a516-b314fd007c86