Tests for the univ() function - Normal Distribution • jeksterslabRdata

library(testthat)
library(jeksterslabRdata)
context("Test univ Normal Distribution.")

Univariate Data Generation - Normal Distribution

Sample Size and Monte Carlo replications

Sample Size and Monte Carlo Replications
Variable	Description	Notation	Values
`n`	Sample size.	\(n\)	1000
`R`	Monte Carlo replications.	\(R\)	1000

Population Parameters

Population Parameters
Variable	Description	Notation	Values
`mu`	Population mean.	\(\mu\)	0
`sigma2`	Population variance.	\(\sigma^2\)	1
`sigma`	Population standard deviation.	\(\sigma\)	1

Sample Data

\[\begin{equation} X \sim \mathcal{N} \left( \mu = 0, \sigma^2 = 1 \right) %(\#eq:dist-norm-X-r) \end{equation}\]

The random variable \(X\) takes on the value \(x\) for each observation \(\omega \in \Omega\).

\(\omega\) refers to units or observations
\(\Omega\) refers to the collection of all units, that is, a set of possible outcomes or the sample space contained in the set \(S\).

A random variable acts as a function, inasmuch as, it maps each observation \(\omega \in \Omega\) to a value \(x\).

\[\begin{equation} x = X \left( \omega \right) %(\#eq:dist-random-variable-1) \end{equation}\]

\[\begin{equation} \mathbf{x} = \begin{bmatrix} x_1 = X \left( \omega_1 \right) \\ x_2 = X \left( \omega_2 \right) \\ x_3 = X \left( \omega_3 \right) \\ x_i = X \left( \omega_i \right) \\ \vdots \\ x_n = X \left( \omega_n \right) \end{bmatrix}, \\ i = \left\{ 1, 2, 3, \dots, n \right\} %(\#eq:dist-random-variable-2) \end{equation}\]

Monte Carlo Simulation

Monte Carlo Simulation Results
Variable	Description	Notation	Values	Mean of estimates	Variance of estimates	Standard deviation of estimates
`mu`	Population mean.	\(\mu\)	0	0.0007422	0.0009953	0.0315482
`sigma2`	Population variance.	\(\sigma^2\)	1	1.0014328	0.0020260	0.0450106
`sigma`	Population standard deviation.	\(\sigma\)	1	1.0004634	0.0005063	0.0225006

Sampling Distribution of Sample Means

\[\begin{equation} \hat{\mu} \sim \mathcal{N} \left( \mu, \frac{\sigma^2}{n} \right) %(\#eq:dist-sampling-dist-mean) \end{equation}\]

\[\begin{equation} \mathbb{E} \left[ \hat{\mu} \right] = \mu %(\#eq:dist-sampling-dist-mean-expected-value) \end{equation}\]

\[\begin{equation} \mathrm{Var} \left( \hat{\mu} \right) = \frac{\sigma^2}{n} %(\#eq:dist-sampling-dist-mean-var) \end{equation}\]

\[\begin{equation} \mathrm{se} \left( \hat{\mu} \right) = \frac{\sigma}{\sqrt{n}} %(\#eq:dist-sampling-dist-mean-se) \end{equation}\]

Sampling Distribution of Sample Variances

\[\begin{equation} \hat{\sigma}^2 \sim \chi^2 \left( k = n - 1 \right), \quad \text{when} \quad X \sim \left( \mu, \sigma^2 \right) %(\#eq:dist-sampling-dist-var) \end{equation}\]

\[\begin{equation} \mathbb{E} \left[ \hat{\sigma}^2 \right] = \sigma^2 %(\#eq:dist-sampling-dist-var-expected-value) \end{equation}\]

\[\begin{equation} \mathrm{Var} \left( \hat{\sigma}^2 \right) = \frac{ \left( n - 1 \right) \hat{\sigma}^2 }{ \sigma^2 }, \quad \text{where} \quad X \sim \mathcal{N} \left( \mu, \sigma^2 \right) %(\#eq:dist-sampling-dist-var-var) \end{equation}\]

When \(X \sim \mathcal{N} \left( \mu, \sigma^2 \right)\), \(\mathrm{Var} \left( \hat{\sigma}^2 \right) \sim \chi^2 \left( k = n - 1 \right)\). As \(k\) tends to infinity, the distribution of \(\mathrm{Var} \left( \hat{\sigma}^2 \right)\) converges to normality. When the \(X\) is not normally distributed, the variance of the sampling distribution of the sample variances takes on a slightly different functional form. Increase in sample size, however large it may be, does not help in approximating the normal distribution.

test_that("expected values (means) of muhat, sigma2hat, sigmahat", {
  expect_equivalent(
    round(
      means,
      digits = 0
    ),
    c(
      mu,
      sigma2,
      sigma
    )
  )
})

Tests for the univ() function - Normal Distribution

Ivan Jacob Agaloos Pesigan