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- The following topics will be covered in this lecture:
- The \( \chi^2 \) distribution
- The student-t distribution
- The F-distribution

While the

**normal distribution**is**frequently applied to describe the underlying distribution of a statistical experiment**,**asymptotic test statistics**are often based on a**transformation of a (non-) normal rv**.To get a better understanding of these tests, it will be helpful to study the \( \chi^2 \), t- and F-distributions, and their relations with the normal one.

We will begin with the

**\( \chi^2 \) distribution**, describing the**sum of the squares of independent standard normal rvs**.If \( Z_i \sim N(0, 1) \), for \( i = 1, \cdots, n \) are independent, then the rv \( X \) given by

\[ \begin{align} X = \sum_{i=1}^n Z_i \sim \chi^2_n \end{align} \] the

**\( \chi^2_n \) distribution in \( n \) total degrees of freedom**.This distribution is of particular interest since it describes the

**distribution of a sample variance as an unbiased estimator varying about the true parameter**.

- The pdf of the \( \chi^2 \) distribution is \[ \begin{align} f(z,n) = \frac{2^{-\frac{n}{2}} z^{\frac{n}{2} - 2}exp\left(-\frac{z}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}, \end{align} \]
- where \( \Gamma(k) \) is the classical “gamma function” given as, \[ \begin{align} \Gamma(z)=\int_0^\infty t^{z-1}\exp\left(-t\right)\mathrm{d}t. \end{align} \]
- The cdf of the χ2 distribution is \[ \begin{align} F(z,n)= \frac{\Gamma_z\left(\frac{z}{2}, \frac{z}{2}\right)}{\Gamma\left(\frac{n}{2}\right)} \end{align} \] where \[ \Gamma_z(\alpha) = \int_0^z t^{\alpha -1} \exp\left(-t\right)\mathrm{d}t \] is the incomplete gamma function.

The standard implemented functions for the \( \chi^2 \) distribution are as follow:

`dchisq(x, df)`

is the**pdf**;`pchisq(q, df)`

is the**cdf**;`qchisq(p, df)`

is the**quantile**;`rchisq(n, df)`

is the function for**generating a sample**.

Same as for other distributions, if

`log = TRUE`

in`dchisq`

function, then log density is computed, which is useful for maximum likelihood estimation.Similar to the functions for the t and F distributions, all the functions also have the parameter

`ncp`

which is the non-negative parameter of non-centrality,- this refers to when this rv is constructed from Gaussian rvs with non-zero expectations.

In the below we plot how the pdf of the \( \chi^2 \) changes for higher numbers of degrees of freedom.

- These are varied as
**n=5**,**n=10**,**n=15**and**n=25**.

- These are varied as

```
par(cex = 2.0, mar = c(5, 4, 4, 2) + 0.3)
z = seq(0, 50, length = 300)
df = c(5, 10, 15, 25)
colors = c("black", "red", "green", "blue")
plot(z, dchisq(z, df[1]), type = "l", xlab = "z", ylab = "pdf")
for (i in 2:4) { lines(z, dchisq(z, df[i]), col = colors[i])}
```

- In general, the \( \chi^2 \) pdf is bell-shaped and shifts to the right-hand side for greater numbers of degrees of freedom, becoming more symmetric.

- There are two special cases, namely n = 1 and n = 2.

```
par(cex = 2.0, mar = c(5, 4, 4, 2) + 0.3)
z = seq(0, 50, length = 300)
m = c(1, 2)
plot(z, dchisq(z, m[1]), type = "l", xlab = "z", ylab = "pdf", xlim = c(0, 10), xaxs = "i", yaxs = "i")
lines(z, dchisq(z, m[2]), col = "blue")
```

In the first case, the vertical axis is an asymptote and the distribution is not defined at 0.

In the second case, the curve steadily decreases from the value 0.5

- Respectively, using the
`pchisq`

function we can plot the cdf for each number of degrees of freedom**n=5**,**n=10**,**n=15**and**n=25**as

```
par(cex = 2.0, mar = c(5, 4, 4, 2) + 0.3)
z = seq(0, 50, length = 300)
df = c(5, 10, 15, 25)
colors = c("red", "green", "blue")
plot(z, pchisq(z, df[1]), type = "l", xlab = "z", ylab = "cdf")
for (i in 2:4) { lines(z, pchisq(z, df[i]), col = colors[i-1]) }
```

A distinctive feature of \( \chi^2 \) is that it is positive, due to the fact that it represents a sum of squared values.

The expectation and variance are both given by,

\[ \begin{align} \mathbb{E}\left[X\right] = n & & \mathrm{var}\left(X\right) = 2n \end{align} \]