---
title: "Parameter Estimation"
output:
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    toc: true
vignette: >
  %\VignetteIndexEntry{Parameter Estimation}
  %\VignetteEngine{knitr::rmarkdown_notangle}
  %\VignetteEncoding{UTF-8}
  linkcolor: blue
link-citations: true
csl: american-statistical-association.csl
bibliography: bibliography.bib
---

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\def\eqdef{\mathrel{\raise0.4pt\hbox{$:$}\hskip-2pt=}}
\def\ev{{\rm E}}
\def\given{\mathrel|}

```{r setup, include = FALSE}
knitr::opts_chunk$set(echo = TRUE, message = FALSE, out.width = "75%",
                      fig.dim = c(6,5), fig.align = "center")
is_check <- ("CheckExEnv" %in% search()) ||
  any(c("_R_CHECK_TIMINGS_", "_R_CHECK_LICENSE_") %in% names(Sys.getenv()))
knitr::opts_chunk$set(eval = !is_check)

library(distfreereg)
```

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# Introduction

An essential part of the testing procedure implemented by `distfreereg()` is the
estimation of the model's parameters. When using the `lm` or `nls` methods, or
the corresponding `formula` methods, parameter estimation is handled using the
modeling functions themselves. Below is the setup used as needed thereafter.

```{r dfr_1}
set.seed(20240926)
n <- 3e2
true_mean <- function(X, theta) exp(theta[1]*X[,1]) - theta[2]*X[,2]^2
test_mean <- true_mean
theta <- c(3,-2)
Sigma <- rWishart(1, df = n, Sigma = diag(n))[,,1]
X <- matrix(rnorm(2*n), nrow = n)
Y <- distfreereg:::f2ftheta(true_mean, X)(theta) +
  distfreereg:::rmvnorm(n = n, reps = 1, SqrtSigma = distfreereg:::matsqrt(Sigma))

dfr_1 <- distfreereg(test_mean = test_mean, Y = Y, X = X,
                     covariance = list(Sigma = Sigma),
                     theta_init = rep(1, length(theta)))
```


# Modifying the Method

For the `function` method, parameters are estimated using the `optim()` function
by default. The default optimization method is `BFGS`, which works well for many
cases. The method can be changed using the `control` argument of
`distfreereg()`. For example, if we know beforehand that $\theta=(\theta_1,\theta_2)$
satisfies $2\leq\theta_1\leq4$ and $-3\leq\theta_2\leq-1$,

```{r dfr_2}
set.seed(20240926)
dfr_2 <- update(dfr_1, control = list(
  optimization_args = list(method = "L-BFGS-B", lower = c(2,-3), upper = c(4,-1))))
all.equal(dfr_1$theta_hat, dfr_2$theta_hat)
```

The results are not identical, but are nearly so.



# Estimating the Precision of Parameter Estimates

To estimate the precision with which the parameters have been estimated,
`vcov()` has a `distfreereg` method. When `test_mean` is a formula, an `lm`
object, or an `nls` object, the model is supplied to `vcov()` for appropriate
method dispatch. When `test_mean` is a function, then the method described in
@Vaart2007 is applied.

```{r vcov}
vcov(dfr_1)
```

An analogous method exists for `confint()`, which uses this covariance matrix
to calculate confidence intervals at a stated confidence level:

```{r confint}
confint(dfr_1, level = 0.9)
```





# Using a Different Optimization Function

By default, `distfreereg.function()` uses `optim()` to estimate the model's
parameters. Suppose, however, we wanted to use `nlminb()` instead. (This
function is selected to illustrate the use of another optimization function
only, and is not an implicit recommendation.) We can use `distfreereg()`'s
`control` argument to do this.

For a given optimization function, we need three things:

1. The name of the argument that specifies the function to be minimized.

1. The name of the argument that specifies the starting parameter values.

1. The name of the element in the returned value that contains the parameter
estimates.

In the case of `nlminb()`, a review of the help page indicates that these are
"`objective`", "`start`", and "`par`", respectively. We supply these three
names, along with the function `nlminb` itself, to the `control` argument as
follows.

```{r nlm, message = FALSE}
set.seed(20240926)
dfr_3 <- update(dfr_1, control = list(optimization_fun = nlminb,
                                      fun_to_optimize_arg = "objective",
                                      theta_init_arg = "start",
                                      theta_hat_name = "par"))
```

Once again, these are not identical, but are nearly so.

```{r}
all.equal(dfr_1$theta_hat, dfr_3$theta_hat)
```





# Detailed Output from the Optimization Process

Detailed output from `optim()`, helpful for diagnosing some parameter estimation
problems, is found in the `optimization_output` element of the output:
```{r}
dfr_3$optimization_output
```



# Checking Compatibility of Arguments

Before optimizing, `distfreereg.function()` verifies that the mean function can
be evaluated at `theta_init`.  This will detect some problems but does not
guarantee compatibility. This should be kept in mind when an opaque error
message arises. In the examples below, the starting vectors are either too short
or too long.

```{r optim_bad_theta_init}
try(update(dfr_1, theta_init = 1))# starting parameter vector too short!
try(update(dfr_1, theta_init = rep(1, 3)))# starting parameter vector too long!
```










# The Importance of Normality of Parameter Estimates

The theory behind the test implemented by `distfreereg()` requires that the
parameter estimates be approximately normally distributed around the true
parameter value. With regularity assumptions on the test mean function, this is
true in theory, but it must also be sufficiently close to true in practice. This
requires good optimization algorithms. In the example below, something goes
awry.

## The Setup

In this example, we use a different error generating function from the default,
specifically one that uses a block-diagonal covariance matrix and generates
errors corresponding to each block using a multivariate $t$ distribution.
Further, we use the default method for `optim()`, namely Nelder--Mead.

First, define the function and true parameter vector.


```{r block_diagonal_setup_func_pars}
true_func <- function(X, theta) theta[1] + theta[2]*X[,1] + X[,1]^2
theta <- c(2,5)
```

Next, define the error distribution function. This first specifies the dimension
`matdim` of the blocks. The function `block_t()` generates the errors by
repeated calls to `mvtnorm::rmvt()` with the appropriate scale matrix to obtain
the correct covariance structure. (Recall that the errors for each repetition
done by `compare()` are stored in the columns of the error matrix.)
```{r block_diagonal_setup_err_dist}
matdim <- 10
block_t <- function(n, reps, blocks, df){
  matsqrt_list <- distfreereg:::covsqrt(blocks, tol = -(.Machine[["double.eps"]])^0.25)
  output <- matrix(NA, nrow = n, ncol = reps)
  for(i in seq_len(reps))
    output[,i] <- as.vector(
      sapply(matsqrt_list,
             function(x) distfreereg:::rmvt(1, df = df, SqrtSigma = x*sqrt((df-2)/df))))
  return(output)
}
```

Finally, define a function that extracts the parameter estimates from a
`distfreereg` object, and a function that generates covariates, a covariance
matrix, and then runs `compare()`.

```{r block_diagonal_define_compare}
get_theta_hat <- function(dfr) dfr$theta_hat

create_cdfr <- function(n){
  X <- as.matrix(rexp(n, rate = 1))
  Sig <- lapply(1:(n/matdim), function(x) rWishart(1, df = matdim, diag(matdim))[,,1])

  return(compare(theta = theta,
                 true_mean = true_func, test_mean = true_func,
                 true_X = X, X = X,
                 true_covariance = list(Sigma = Sig), covariance = list(Sigma = Sig),
                 theta_init = c(1,1),
                 err_dist_fun = "block_t",
                 err_dist_args = list(blocks = Sig, df = 4),
                 reps = 1e3,
                 prog = Inf,
                 manual = get_theta_hat,
                 control = list(optimization_args = list(method = "Nelder-Mead")))
         )
}
```

## The Simulation

Now, we set the seed, run the simulation, and extract the saved parameter
estimates.

```{r run_with_14}
set.seed(32)
cdfr_seed14 <- create_cdfr(500)
theta_seed14_1 <- sapply(cdfr_seed14$manual, function(x) x[[1]])
theta_seed14_2 <- sapply(cdfr_seed14$manual, function(x) x[[2]])
```

Since we are using the same function for `true_mean` and `test_mean`, the
cumulative distribution functions of the observed and simulated statistics
should be nearly the same. The following plot is therefore alarming.

```{r cdf_plot}
plot(cdfr_seed14)
```

We can see that there is a problem with the parameter estimates, namely that
their densities are not approximately normally distributed about the true
parameter values.

```{r density_plots}
plot(density(theta_seed14_1))
plot(density(theta_seed14_2))
```

## The Most Likely Source of the Problem

The problem seems to be the use of Nelder--Mead here, since the default method
selected by `distfreereg()`, BFGS, does not result in such distributions in
similar cases. This is, of course, no guarantee that BFGS will always be better,
but it does appear to be a more sensible default here.

## Two Possible Objections

The keen-eyed reader will note two possible objections to the analysis above:
perhaps the sample size is insufficient, and perhaps the input of `set.seed()`
was carefully selected to obtain these results.

The first of these objections can be addressed by showing the results with a
different seed, where the plots suggest no substantial problems.

```{r sample_size_not_the_problem}
set.seed(1)
cdfr_seed1 <- create_cdfr(500)
theta_seed1_1 <- sapply(cdfr_seed1$manual, function(x) x[[1]])
theta_seed1_2 <- sapply(cdfr_seed1$manual, function(x) x[[2]])
plot(cdfr_seed1)
plot(density(theta_seed1_1))
plot(density(theta_seed1_2))
```

A formal test that the samples have come from the same distribution is also
reassuring.

```{r kstest}
ks.test(cdfr_seed1)
```

The second possible objection, namely that the seed was carefully selected to
produce these results, is only half true. It was selected, but it was not
difficult to find. Considering integers from 1 to 40 as inputs to `set.seed()`,
at least 16, 26, 27, and 32 produce poor results. The phenomenon is not
difficult to reproduce.









# References
