#
STAT111 2019-03-07
Thu, Mar 7, 2019

Statistic is a function of the data

Estimator is to estimate the estimand

The estimate is where you compute the estimator

# Mixture model

You are mixing more than one distribution.

You want to mix a normal with a Cauchy distribution.

The number of steps somebody takes is a degenerate distribution mixed with a Poisson.

You take a mixture of two binomials, it is no longer binomial

# Mixture model example

There are k groups $1,2,\cdots,k$

Then we have $k\geq2$, where within group k, there is a density $f_{\theta_k)(x)$

Each individual has probability $p_k$ of being in group k. Then we will have a multinomial distribution. We face a challenge which is we don’t know which group each individual is in.

The difference between these two cases:

We will assume $n$ individuals, all of whom are independent. Let’s assume independence.

We wish we knew what group person 1 was in:

$$\sum_{k = 1}^K p*kf*{\theta k}(x_i)$$

Our likelihood function is $$L(\theta, p) = \pi$$

# Clustering vs Classification

Clustering is different from classification

If we have a mixture of 2 normals, then if k = 2:

$N(\mu_k, \sigma_k^2)$ and $k = 1,2,$

# Questions

- Why are the observations true for $X
*{1}, \ldots, X*{n} \sim N(0, \theta)$:

$$
\operatorname{Var}\left(X*{1}^{2}\right)=E\left(X*{1}^{4}\right)-\left(E\left(X_{1}^{2}\right)\right)^{2}=2 \theta^{2} :
$$