---
title: "Estimation with `maxlogL` objects"
author: "Jaime Mosquera"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Estimation with `maxlogL` objects}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

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

These are basic examples which shows you how to solve a common maximum likelihood estimation problem with `EstimationTools`:

## Estimation in regression models

<!-- The data is from [NIST](https://www.itl.nist.gov/div898/handbook/apr/section2/apr221.htm#Example). They generated  20 random Weibull failure times with a parameter `shape=1.5` and `scale=500`. The test time is 500 hours, 10 of these failure times are right censored. The observed times are, to the nearest hour: 54, 187, 216, 240, 244, 335, 361, 373, 375, and 386. -->

We generate data from an hypothetical failure test of approximately 641 hours with 40 experimental units, 20 from group 1 and 20 from group 2. Lets assume a censorship rate of 10%, and regard that the data is right censored. Six of the 20 data points are shown just bellow:

```{r, echo=TRUE, warning=FALSE, message=FALSE}
if (!require('readr')) install.packages('readr')
library(readr)

urlRemote <- "https://raw.githubusercontent.com/"
pathGithub <- 'Jaimemosg/EstimationTools/master/extra/'
filename <- 'sim_wei.csv'
myURL <- paste0(urlRemote, pathGithub, filename)
data_sim <- read_csv(myURL)

data_sim$group <- as.factor(data_sim$group)
head(data_sim)
```

The model is as follows:

$$
f(t|\alpha, k) = \frac{\alpha}{k} \left(\frac{t}{k}\right)^{\alpha-1} \exp\left[-\left(\frac{t}{k}\right)^{\alpha}\right]
$$

\centering
$$
\begin{aligned}
T &\stackrel{\text{iid.}}{\sim} WEI(\alpha,\: k), \\
\log(\alpha) &= 1.2 + 0.1 \times group \quad  (\verb|shape|),\\
k &= 500 \quad (\verb|scale|).
\end{aligned}
$$

The implementation and its solution is printed below:

```{r example1, message=FALSE, warning=FALSE}
library(EstimationTools)

# Formulas with linear predictors
formulas <- list(scale.fo = ~ 1, shape.fo = ~ group)

# The model
fit_wei <- maxlogLreg(formulas, data = data_sim,
                      y_dist = Surv(Time, status) ~ dweibull,
                      link = list(over = c("shape", "scale"),
                                  fun = rep("log_link", 2)))
summary(fit_wei)
```

## Estimation in distributions

$$
\begin{aligned} 
X &\sim N(\mu, \:\sigma^2), \\
\mu &= 160 \quad (\verb|mean|), \\
\sigma &= 6 \quad (\verb|sd|).
\end{aligned}
$$

The solution for a data set generated with size $n=10000$ is showed below

```{r example2, message=FALSE, warning=FALSE}
x <- rnorm( n = 10000, mean = 160, sd = 6 )
fit <- maxlogL( x = x, dist = "dnorm", link = list(over = "sd", fun = "log_link") )
summary(fit)
```

