Hi everyone,
My data exhibits bimodal distribution. Both of them are beta distribution at different intervals and does not overlap. My doubt is how to obtain the first moment (mean) ,second moment (variance), third moment and so on of the total distribution?
Thank you in advance
Sam
You say your data exhibit a bimodal distribution. I assume now that your problem is to determine sample estimates (not population estimators) of the moments. The fact that the distribution has two modes should not be a hindrance to the question of sample estimation of the moments. The sample moments may be estimated as usual. For example mean = (X1+X2+...+Xn)/n where n is the sample size and variance = SIGMA(Xi)/(n-1) irrespective of the nature of the underlying population.
The natural question I have when I see a bimodal distribution, especially with the separation of modes you describe, is "Are there sub-populations you can identify that explains the groupings?" If there is finding the moments for the sub populations separately will be more meaningful.
Bimodality points towards a mixed population. As Ronald has mentioned your population may be of mixed beta type
Thank you for your replies . My total distribution is clearly a distribution which consists of two sub population following beta distribution. So my doubt is, let's say I find the moments for each sub population, is their anyway to combine them to find the moments of the total distribution?
so express your pdf as the mixture of two beta and fine the mixing prportion
Thank you subrata. Can you please point me to some references on how to obtain this missing proportion?
you need to estimate this mixing coefficient (say \alpha) considering it as another parameter in your likelihood using your sample data!
Thank you very much for the idea, I will look into it for more detail
The general formula of moments of mixture say a F + (1-a) G of two distributions F and G follows from the definition of moments and general properties of integrals. Accordingly, the moment of order k of the mixture equals to
m_k = \int_R x^k d( a F(x) + (1-a) G(x) ) =
= a \int_R x^k dF(x) + (1-a) \int_R x^k dG(x)
= a m_k(F) + (1-a) m_k(G)
(the Stieltjes integral's notation applied). Obviously. existence of the moments involved should exist. Note, however, that the formula is invalid for central moments,unless the means (= moments of order 1) are equal.
Probably, no particular reference can be easily found. One can try LadyWiki under mixture of probability measures , see e.g.
https://en.wikipedia.org/wiki/Mixture_distribution#Moments
With respect to estiamtes of the proportion a , the advice by Subrata, for samples with separted subsamples (which is the case in question due to separability of the supprors of the beta distributions) , the MLE equals s a* = n_1/n . This is due to the fact that n_1 is of binomal probability distribution with parameter a , if the sample size equals n. Simple reference:
https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Discrete_distribution.2C_finite_parameter_space
Regards, Joachim
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