[提问]两个不同正态总体的样本混在一起，如何推测两分布的特征

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第一次发帖，有纰漏的话见谅。

现在假设有两个正态总体，他们一个均值和标准差分别是mu1,sigma1,mu2,sigma2
从两个总体中抽取若干个样本，然后混在一起。那么，怎么从现在这个混合样本的特征，去推断两个总体的特征？
也就是求一下mu1,sigma1,mu2,sigma2?

大神们还请不吝赐教。

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关键词：在一起 Sigma 第一次发帖 GMA MA1 如何 样本

To be added，w can be treated as a random variable, which generally follows Binomial (or Bernoulli) distribution w/ parameter pi. But we cannot observe w. In practice, w is replaced with its conditional mean, i.e., E(w|y;mu,sigma,pi) = Prob(w=1|y;mu,sigma,pi) = pi*f(y;mu1,sigma1)/[pi*f(y;mu1,sigma1)+(1-pi)*f(y;mu2,sigma2)]. In the EM algorithm, the parameters relevant to y (mu and sigma) are estimated first; then pi is estimated. This procedure is repeated until convergence condition is satisfied.

This case involves incomplete data. One strategy is to introduce an indicator vector w, where w_i = 1 if obs i belongs to population 1 and w_i = 0 otherwise; prob(w=1)=pi.
Then write the joint density distribution of (y_i, w_i), for example, f(y,w;mu,sigma,pi)=pi*f(y;mu1,sigma1)+(1-pi)*f(y;mu2,sigma2), as well as the corresponding log-likelihood function. Estimate the LL function using Expectation-Maximization (EM) Algorithm.
You can find the details of EM estimation from statistical/econometric textbooks.

mannlee 发表于 2014-5-27 07:06
This case involves incomplete data. One strategy is to introduce an indicator vector w, where w_i =  ...

谢谢，很有启发~