Fundamental Statistics Theory Notes (8)

MLE continued

Definition:
The MLE is the value $\theta$ that maximized the likelihood function $L(\theta|x_1,x_2…x_n)$.
Denote estimate of $\theta$ by $\hat \theta$.

Definition:
The parameter space is the set of allowable values of the parameter $\theta$, which can be discrete or continuous.

Ex. Bin(n,p)

  • if n is unknown, n’s parameter space is discrete, n=1,2…
  • if p is unknown, p’s parameter space is continuous, $0\leq p \leq 1$

Strategies to find MLE

  1. discrete parameter: try different values
  2. continuous parameter, $\theta$: use calculus
    • usually easier to work with log of L, denoted by $l(\theta)$, “log-likelihood”, maximizing $l(\theta)$ is same as maximizing $L(\theta)$, since log is a monotonous function.
    • check $\frac{dl}{d\theta}=0$, solve $\theta$, check is $l’’(\theta)<0$, max by 2nd derivative order.
    • check boundry points

Ex. $x_1,x_2…x_n\overset{\text{i.i.d}}\sim Bern(p)$, parameter space: $0 \leq p \leq 1$
$L(p|x_1,x_2…x_n)=f(x_1,x_2…x_n|p)\overset{\text{i.i.d}}=\prod_{i=1}^n f(x_i|p)$
$= \prod_{i=1}^n \underbrace {p^{x_i}(1-p)^{1-x_i}}_{Bin(1,p), PMF} = p^{\sum x_i}(1-p)^{n-\sum x_i}$
$\Rightarrow L(0)=L(1)=0$, wherever $\sum x_i\neq0$, $\sum x_i \neq n$
Transform to log-likelihood function
$l(p) = logL(p)$
$l(p) = \sum x_i \cdot log(p) + (n-\sum x_i)log(1-p)$
$l’(p) = \frac{\sum x_i}{p}+\frac{n-\sum x_i}{1-p}(-1)\overset{\text{set}}=0$
$\Rightarrow \sum x_i(1-p)-(n-\sum x_i)p=0$
$\Rightarrow \sum x_i - np = 0 $
$\Rightarrow p = \frac{\sum x_i}{n}=\bar x$
check $l’’(\bar x)<0$, so MLE is $$\underbrace{\hat p}_{estimate\ of\ probability\ of\ success} = \underbrace{\bar x}_{observed fraction of success}$$.

German Tank Problems
During WW2, allies want to know how many tanks the Germans were prodcuing, based on serial numbers of caputured tanks, fortunately, some parts were numbered sequentially.

Data: n serial numbers randoms selected from the list ${1,2,\dots,N}$
$N = number\ of\ tanks$ (parameter of interest), find out the likelihood function of N.
Let $x_1,x_2…x_n$ be the serial number we observe
$L(N|x_1,x_2…x_n) = f(x_1,x_2…x_n|N)$
what does knowing $x_1$ tell us about N?
$N \geq x_1$, actually $N \geq max\lbrace x_i \rbrace$

They are $\left(N \atop n \right)$ differnet combinations of n serial number, each equally likely, so liklihood function is
$$\frac{1}{\left(N \atop n \right)}, N \geq max\lbrace x_i \rbrace$$
Note $\left(N \atop n \right)$ is increasing function of N when n is fixed.
To maximize $\frac{1}{\left(N \atop n \right)}$, we want smallest denominator.

Sufficient Statistics

$L(N|x_1,x_2…x_n) = \underbrace{\frac{1}{\left(N \atop n \right)}I(N\geq max\lbrace x_i \rbrace)}_{g(max\lbrace x_i \rbrace|\theta)}\cdot \underbrace{1}_{h(x_1,x_2…x_n}$
So $max\lbrace x_i \rbrace$ is sufficient for N.

Invariance Property of MLEs

If $\hat \theta$ is the MLE of $\theta$, then for any 1-1 function $g$, the MLE of $g(\theta)$ is $g(\hat \theta)$

Ex. $x_1,x_2…x_n \overset{\text{i.i.d}}\sim Expo(\beta)$, find MLE of $\underbrace{\beta}_{mean}$ and $\underbrace{\beta^2}_{variance}$

Parameter space: $\beta > 0$
$L(\beta|x_1,x_2…x_n)=f(x_1,x_2…x_n|\beta)\overset{\text{i.i.d}}=\sum_{i=1}^nf(x_i|\beta)$
$=\prod_{i=1}^n\frac{1}{\beta}e^{-\frac{x_i}{\beta}}=\frac{1}{\beta^n}e^{-\frac{\sum x_i}{\beta}}$
$\Rightarrow l’(\beta)=-\frac{n}{\beta}+\frac{\sum x_i}{\beta^2}\overset{\text{set}}=0$
$\Rightarrow -n\beta + \sum x_i = 0$
$\Rightarrow \beta = \frac{\sum x_i}{n} = \bar x$
remember to verify that $l’’(\bar x) < 0$, so we have a max MLE , $\hat \beta = \bar x$. By the invariance property, the MLE of $\beta ^2$ is $\bar x^2$.

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