# Questions tagged [mixture]

A mixture distribution is one that is written as a convex combination of other distributions.

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### Maximum likelihood function for mixed type distribution

In general we maximize a function $$L(\theta; x_1, \ldots, x_n) = \prod_{i=1}^n f(x_i \mid \theta)$$ where $f$ is probability density function if the underlying distribution is continuous, and a ...
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### How to derive the MLE of a Gaussian mixture distribution

In my self-study, I consider a Gaussian mixture distribution: $$p(x)= p(k=1) N(x|\mu_1,\sigma^2_1) + p(k=0) N(x|\mu_0,\sigma^2_0)$$ where $p(k=1)+p(k=0)=\pi_1+\pi_0=1$. I am now asked to do three ...
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### Difference between a mixture of distributions and a convolution. Interpretation in a applied setting

From what I could gather Mixture: if $X_i\sim^{iid} f_i$, then W is a mixture with $f_W =\sum \frac{f_i}{n}$. This definition could also be for the CDF instead of the density. Convolution: To make it ...
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### Finding a Scalable Approach to a modified coin-flip problem using MLE model

and thank you in advance for your help! I am interested in applying MLE to estimate parameters in a "modified" coin flip model, but have been having difficulties scaling the solution. The problem ...
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### Defining overlapping periods

I have a dataset containing the abundance of migrating bird species. In the figure below you can see that there are two "bell" shapes that are overlapping somewhere around September. One of the bell ...
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### Expectation and variance of sample mean with random sample size

I have a question regarding sampling where the sample size itself is a random variable. Say I have two sub-populations $A$ and $B$ from which I can sample a real valued random variable with ...
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### Building a Regression Tree with one Gaussian Mixture Model at each node

I am trying to build a regression tree that outputs both a mean and a covariance matrix for each leaf of the tree. Ideally I would be able to have a Gaussian Mixture Model at each leaf. A first ...
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### Advantages and disadvantages of EM algorithm vs trust region methods for nonlinear optimization

I have a set of observations X that I believe were generated by a mixture of several probability distributions (specifically, two von mises and one uniform). I'd like to find the maximum likelihood ...
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### variance of a mixture in terms of the mean and variance of each component

I am following the MATLAB example to fit a mixture of two normal distribution that you can find here At some point it is defined the inital guess for the standard deviation as: ...
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### Looking for proof of conditional dependence, when the conditioning variables are linearly related

Suppose we have three random variables, $X$, $Y_1$, and $e$ (for error). Variable $e$ is independent of $X$ and $Y_1$, but $X$ and $Y_1$ are dependent. Further suppose we construct a new mixture ...
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### Probability distribution on a subset of a simplex

I want to define a probability distribution on a subset of a simplex. for example, on a 3-simplex, we know that $x_1+x_2+x_3+x_4=1$ and $X \sim Dirichlet$. Would it be possible to constraint $X$ more (...
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### Combining monthly distributions of data with different right-censoring levels for prediction

I have a number of pricing datasets with integer values (prices in round dollars) for the same product which are right-censored at different levels (one level per dataset). I would like to combine ...
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### How to combine two beta-binomial distributions

Say I have the following situation. I have two weighted coins: Coin 1: In the past I've seen this coin flipped 10 times, 8 of which it came up heads. So I can model the probability of $n$ heads out ...
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### Models for nonnegative (incl. zero) positively skewed multivariate time series (trade volumes)

I want to build a Monte Carlo simulation that is based in part on share amounts that are traded in the market for a set of stocks. I need to be able to take into account the co-dependence of trade ...
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### Tail dependence of mixture copulas

I am currently using (multivariate) mixture copulas to model a financial data set. The mixture has two components as follows: $$C_{mixture}=wC_1+(1-w)C_2$$ where $C_1$ and $C_2$ are copulas. I have ...
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### solution to mixture CDF / inverse CDF of finite mixture

I am currently numerically solving the follwing equation, which is a convex combination/finite mixture of two marginal CDFs $F(x)$ and $G(x)$ (actually they are joint CDFs but all arguments but x are ...
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### Estimate weighted variances in mixture models

Given a generic mixture model $X$ of $k$ components, with distribution $$f(x)=ā_i\pi_if_i(x),$$ It is easy to show that the $k-th$ moment is just the weighted mean of the $k-th$ moments of the ...
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### weights in a mixture Gaussian model

If the variance of a random variable is proportional to its mean, then what is the best way of making a mixture distribution that will faithfully reconstruct a data set coming from a mixture model. ...