The following data are observations of the random variable...

The following data are observations of the random variable $X=(1-W) Y_{1}+$ $W Y_{2}$, where $W$ has a Bernoulli distribution with probability of success $0.70 ; Y_{1}$ has a $N\left(100,20^{2}\right)$ distribution; $Y_{2}$ has a $N\left(120,25^{2}\right)$ distribution; $W$ and $Y_{1}$ are independent; and $W$ and $Y_{2}$ are independent. Data are in the file mix668.rda. $\begin{array}{ccccccc}119.0 & 96.0 & 146.2 & 138.6 & 143.4 & 98.2 & 124.5 \\ 114.1 & 136.2 & 136.4 & 184.8 & 79.8 & 151.9 & 114.2 \\ 145.7 & 95.9 & 97.3 & 136.4 & 109.2 & 103.2 & \end{array}$ Program the EM algorithm for this mixing problem as discussed at the end of the section. Use a dotplot to obtain initial estimates of the parameters. Compute the estimates. How close are they to the true parameters? Note: assuming the $\mathrm{R}$ vector $\mathrm{x}$ contains the sample on $X$, a quick dotplot in $\mathrm{R}$ is computed by plot $\left(\operatorname{rep}(1,20)^{2} \mathrm{x}\right)$.

The following data are observations of the random variable $X=(1-W) Y_{1}+$ $W Y_{2}$, where $W$ has a Bernoulli distribution with probability of success $0.70 ; Y_{1}$ has a $N\left(100,20^{2}\right)$ distribution; $Y_{2}$ has a $N\left(120,25^{2}\right)$ distribution; $W$ and $Y_{1}$ are independent; and $W$ and $Y_{2}$ are independent. Data are in the file mix668.rda. $\begin{array}{ccccccc}119.0 & 96.0 & 146.2 & 138.6 & 143.4 & 98.2 & 124.5 \\ 114.1 & 136.2 & 136.4 & 184.8 & 79.8 & 151.9 & 114.2 \\ 145.7 & 95.9 & 97.3 & 136.4 & 109.2 & 103.2 & \end{array}$
Program the EM algorithm for this mixing problem as discussed at the end of the section. Use a dotplot to obtain initial estimates of the parameters. Compute the estimates. How close are they to the true parameters? Note: assuming the $\mathrm{R}$ vector $\mathrm{x}$ contains the sample on $X$, a quick dotplot in $\mathrm{R}$ is computed by plot $\left(\operatorname{rep}(1,20)^{2} \mathrm{x}\right)$.

Key Concepts

Mixture Models

Mixture models are statistical models that assume data are generated from a combination of several underlying probability distributions. Instead of fitting one single distribution to the entire data set, a mixture model represents the data as a weighted sum of multiple component distributions, each of which may capture different underlying data subpopulations or processes.

Expectation-Maximization (EM) Algorithm

The EM algorithm is an iterative method for finding maximum likelihood estimates when the data are incomplete or have latent (hidden) variables. It alternates between an expectation (E) step, which computes the expected value of the log-likelihood with respect to the current estimates of the latent variables, and a maximization (M) step, which updates the parameter estimates by maximizing this expected log-likelihood.

Latent Variables

Latent variables are unobserved random variables that influence the observed data. In mixture models, the component membership is typically treated as a latent variable because it indicates which underlying distribution each observation is drawn from, even though this information is not directly observed.

Gaussian Distribution

The Gaussian or normal distribution is a continuous probability distribution characterized by its mean and variance. It is frequently used in statistical modeling due to its properties and the central limit theorem, and it often serves as a component distribution in mixture models for modeling continuous data.

Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution for a binary random variable with only two possible outcomes, often referred to as 'success' and 'failure'. It plays a crucial role in mixture models when one of the variables indicates membership in a component, with the probability of success defining the mixing proportion.

Data Visualization for Initial Parameter Estimation

Data visualization techniques, such as dotplots, are used to obtain a preliminary understanding of the data distribution. They help in forming initial estimates of model parameters, which are important for initializing iterative algorithms like the EM algorithm to ensure better convergence and avoid local optima.

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