I want to verify the fitting of the demand data with the stuttering poisson distribution by k-s test, in the excel. but without the cumulative distribution function of stuttering poisson distribution, can any one guide me it how to get the cumulative distribution function (CDF) of stuttering poisson distribution?
Hi you might start with this link: https://www.google.com/search?client=firefox-b-1-d&q=moments+of+stuttering+poisson+distribution+
I seriously doubt that EXCEL will be up to this task. May the force be with you. David Booth
This has been used long before its formal proof for approximating the probability function of some mixed Poisson distributions. For example some scholars in the 70's proposed the use of the cumulative distribution function of a Gamma distribution as an approximation to the cumulative distribution function of the negative binomial distribution. The resemblance is much greater for larger values of the mean. If the mean is small, there is a high probability of observing a 0 value, i.e. P (0) is large. This is not true for many continuous densities, and thus the approximation is poor. Some also discussed the use of the lognormal distribution instead of the Poisson-Lognormal distribution. Therefore under fairly general assumptions, the order of convergence of a mixed Poisson distribution to its mixing distribution. For example, since the Beta distribution tends to the Gamma distribution for certain parameter values, the negative binomial distribution can be regarded as a limiting case of the Poisson-Beta distribution.
Something like this MATLAB code
% Stuttering Poisson Distribution,Draft,Tong Wu.
% Parameter
Rho = 0.98;
Lambda_t = 4;
% Initial
R_0 = exp(-Lambda_t);
% Assume R1 to R3
N = 100;
% Creat an array to store R_n
R_n=zeros(1,N);
% Creat an array to store Factor 1: (1-rho)*Lambda_t/n
Factor_1 = zeros(1,N);
Factor_1(1)=(1-Rho)*Lambda_t*(1/1);
R_n(1) = Factor_1(1) * 1 * Rho^(1-1)* R_0; %j=1 to n, but n = 1
for n=2:N
Factor_1(n)=(1-Rho)*Lambda_t*(1/n);
R_n(n) = R_n(n-1);
for j=2:n
if n-j ~=0
R_n(n) = R_n(n)+ j*Rho^(j-1)*R_n(n-j);
else
R_n(n) = R_n(n)+ j*Rho^(j-1)*R_0;
end
R_n(n) = R_n(n)*Factor_1(n);
end
end
% R_n(n) is the probability that n units are demanded in the time
% interval of length t
% plot PDF
figure(1);
x=1:N;
y=R_n(1:N);
plot(x,y,'b--o');
title('pdf stuttering poisson')
xlabel('n')
ylabel('R_n')
% Plot CDF
C_n = zeros(1,N);
C_n(1) = R_n(1);
for n=2:N
C_n(n)=C_n(n-1)+R_n(n);
end
figure(2)
z=C_n(1:N);
plot(x,z,'b--o');
title('cdf stuttering poisson')
xlabel('n')
ylabel('C_n')
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