STAT111 2019-03-07
Zhu, Justin

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:

$$\sum_{k = 1}^K pkf{\theta k}(x_i)$$

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

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

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