---
title: "MLE Gamma"
author: "Prof. Eric A. Suess"
format: 
  html:
    self-contained: true
---

## MME fitted gamma

$X_1, X_2, ...,X_n$ iid $Gamma(\alpha,\lambda)$, find the MMEs.

$f(x|\alpha,\lambda)= \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}$

```{r}
n <- 100

alpha <- .5; a.min <- 0.1; a.max <- 2;
lambda <- 2; l.max <- 10


# generate random data

x <- rgamma(n, shape = alpha, rate = lambda)

x.orig <- x	# save a copy of the original data
```

# make a histogram

use the density estimator since this is a continuous random variable

```{r}
hist(x.orig, probability=T)
lines(density(x.orig))
```

# MMEs

```{r}
sigma2.hat <- (1/n)*sum( (x - mean(x))^2 )

alpha.hat.mme <- mean(x)^2/sigma2.hat
alpha.hat.mme
lambda.hat.mme <- mean(x)/sigma2.hat
lambda.hat.mme
```


The AIC and BIC are defined as follows, with $n$ the sample size and $k$ the number of parameters in the model.

$$AIC = -2\log(L(\hat{\theta})) + 2k$$

$$BIC = -2\log(L(\hat{\theta})) + k\log(n)$$


```{r}
library(fitdistrplus)

fit.gam <- fitdist(x.orig, "gamma", method = "mme")
summary(fit.gam)
```


plot the fitted model to the simulated data

```{r}
denscomp(list(fit.gam), legendtext = c("Gamma"))
```


```{r}
cdfcomp(list(fit.gam), legendtext = c("Gamma"))
```

```{r}
qqcomp(list(fit.gam), legendtext = c("Gamma"))
```

```{r}
ppcomp(list(fit.gam), legendtext = c("Gamma"))
```

### MLE fitted gamma

```{r}
fit.gam <- fitdist(x.orig, "gamma", method = "mle")
summary(fit.gam)

```

plot the fitted model to the simulated data

```{r}
denscomp(list(fit.gam), legendtext = c("Gamma"))
```

```{r}
cdfcomp(list(fit.gam), legendtext = c("Gamma"))
```

```{r}
qqcomp(list(fit.gam), legendtext = c("Gamma"))
```

```{r}
ppcomp(list(fit.gam), legendtext = c("Gamma"))
```