---
title: "Sampling and ML estimation"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Sampling_and_ML_estimation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Using TruncExpFam

We recommend installing the stable, peer-reviewed version of TruncExpFam,
available on CRAN:

```{r install, eval=FALSE}
install.packages("TruncExpFam")
```

After successful installation, the package can be loaded with

```{r setup}
library(TruncExpFam)
```

# Sampling from a truncated distribution

TruncExpFam comes equipped with functions to generate random samples from no
less than 12 different probability distributions from the truncated exponential
family. You can read more about them by running `?rtrunc` on your R console.

As an example, we will sample 100 values from a chi-square
distribution with 14 degrees of freedom:

```{r sample}
x <- rtrunc(100, family = "chisq", df = 14)
```

## Different ways to do the same thing

By default, however, `rtrunc()` doesn't generate a truncated distribution. As
a matter of fact, the code above will generate the exact same sample as if
were drawn from `stats::rchisq()`, watch:

```{r stats}
set.seed(3067)
x2 <- rtrunc(20, "chisq", df = 14)

set.seed(3067)
x3 <- rchisq(20, 14)

identical(x2, x3)
```

Oh, wait... Those objects are supposed to be identical! What happened? Let's
investigate:

```{r investigate}
x2
x3
str(x2)
str(x3)
class(x2)
class(x3)
```

OK, so you can tell that the generated numbers are the same, but `x2` and `x3`
are not literally the same objects because the former has a different class.
These `trunc_*` classes are actually very special, because they contain some
extra information about the distribution that a simple vector does not. One
can access such information using `print(x2, details = TRUE)`:

```{r print}
print(x2, details = TRUE)
```

Just to be sure that the sample itself matches:

```{r conclusion}
identical(as.vector(x2), x3)
```

Speaking of alternative ways to generate the same sample, for the sake of
convenience and of users familiar with the sampling functions
from the stats package, the wrapper function `rtruncchisq()` is also available.
The results, as you can see below, are identical:

```{r wrapper}
set.seed(2912)
x4 <- rtrunc(1e4, "chisq", df = 14)

set.seed(2912)
x5 <- rtruncchisq(1e4, df = 14)

identical(x4, x5)
```

## Actually sampling from a truncated distribution

So far, all samples generated are actually not truncated. This is because,
by default, the truncation limits `a` and `b` are set to the limits of the
distribution support, which are 0 and Inf for the chi-squared distribution.

Let us use a simpler distribution for this second example by sampling from
Poisson(10):

```{r pois}
y1 <- rtruncpois(1e4, 10)
summary(y1)
var(y1)
```

As expected, the values are all larger than 0 and the mean and variance are 10.
If we wanted to generate instead from a Poisson(10) truncated at, say,
8 and 20, we would run:

```{r truncate}
y2 <- rtruncpois(1e4, 10, a = 9, b = 20)
summary(y2)
var(y2)
```

Notice how, even with a large sample, the observed mean and variance are still
quite far from 10.

# Recovering the original parameters

One reliable method of estimating the original lambda used to generate `y2` is
by running the `mlEstimationTruncDist()` function:

```{r ml}
lambda <- mlEstimationTruncDist(y2, print.iter = TRUE)
lambda
```

More information about that function and how you can tweak it is available on
`?mlEstimationTruncDist`.