The K distribution

Definition

Suppose X ∼ χm2 is a Chi-squared random variate on m degrees of freedom. Then the distribution of $$Y = \sqrt{\frac{X}{m}}$$ is the Kay distribution on m degrees of freedom, written as Y ∼ Km. Its density is $$f(y) = \left\{\begin{array}{lcl} \frac{m^{\frac{m}{2}} y ^{m-1} e^{-\frac{1}{2} m y^2}} {2^{\frac{m}{2}-1} \Gamma(\frac{m}{2})} &~~~& \text{for} ~~ 0 \le y < \infty \\ &&\\ 0 && \text{otherwise}. \end{array} \right. $$

The Km density has some very attractive features over the χm2 density:

  • Km has a much more symmetric density than had the χm2, for any m;
  • like a χm2 density a Km density, as m → ∞ Km becomes more symmetric and nearly normally distributed but does both faster than a χm2;
  • as m↑, the Km density concentrates around the value y = 1, rather than heading off to ∞ like the χm2;
As m increases, Km has better properties

As m increases, Km has better properties

These values were calculated using the dkay(...) density function. For example, dkay(1.0, df=10) = 1.7546737.

Normal theory relations

Perhaps the most obvious relation between a normal random variate and a Km is that if Z ∼ N(0, 1), then |Z| ∼ K1, the half-normal.

More important in applications is that distribution of the estimator of the sample standard deviation is proportional to a Km. To be precise, if Y1, …, Yn are independent and identically distributed as N(μ, σ2) random variates, with realizations y1, …, yn and the usual estimates μ̂ = ∑yi/n and $\widehat{\sigma} = \sqrt{\sum (y_i - \widehat{\mu})^2/(n-1)}$, then the corresponding estimators μ̃ and σ̃ are distributed as $$ \widetilde{\mu} \sim N(\mu, \frac{\sigma^2}{n}) ~~~~~\text{and} ~~~~~ \frac{\widetilde{\sigma}}{\sigma} \sim K_{n-1}. $$ The latter shows that Km is used for inference (e.g. tests and confidence intervals) about σ.

This is handy because the Km quantiles vary much less than do those of χm2. For example, condider the following table of the cumulative distribution.

df p=0.05 p=0.5 p=0.95
1 0.0627068 0.6744898 1.959964
2 0.2264802 0.8325546 1.730818
3 0.3424648 0.8880642 1.613973
4 0.4215220 0.9160641 1.540108
5 0.4786390 0.9328944 1.487985
6 0.5220764 0.9441152 1.448654
7 0.5564364 0.9521263 1.417601
8 0.5844481 0.9581311 1.392269
9 0.6078297 0.9627987 1.371090
10 0.6277180 0.9665308 1.353035
15 0.6957463 0.9777136 1.290886
20 0.7365735 0.9832962 1.253205
25 0.7644974 0.9866425 1.227232
30 0.7851255 0.9888719 1.207932
35 0.8011601 0.9904636 1.192858
40 0.8140839 0.9916570 1.180662

Unlike the χm2 distribution, the quantiles in this table stabilize, allowing 1 ± 0.20 being not a bad rule of thumb for a 90% probability of the ratio σ̃/σ.

These values were calculated using the qkay(...) quantile function. For example, qkay(0.05, df=5) = 0.478639. These would be used to construct interval estimates for σ.

To get observed significance levels, the cumulative distribution function pkay(...) would be used. For example, SL = 1- pkay(1.4, df=10) = 1 - 0.9667287 = 0.0332713.

The Student t distribution

For the standard normal theory, the Student tm distribution can be defined as follows. If Z ∼ N(0, 1) and Y ∼ Km is distributed independently of Z, then the ratio $$T=\frac{Z}{Y} = \frac{N(0,1)}{K_m} = t_m$$ which is fairly easy to remember.

For the estimators from the above model $$\frac{\widetilde{\mu} - \mu} {\widetilde{\sigma} / \sqrt{n}} = \frac{ \frac{\widetilde{\mu}-\mu} {\sigma/\sqrt{n}} } {\frac{\widetilde{\sigma}} {\sigma} } = \frac{N(0,1)}{K_{n-1}} = t_{n-1} $$ is used to construct interval estimates and tests for the value of the parameter μ.

The functions

As with every other distribution in R four functions are provided for the Km distribution. These are

  • dkay(x, df=m, ...) which evalutes the density of Km at x,
  • pkay(x, df=m, ...) which evalutes the distribution of Km at x,
  • qkay(p, df=m, ...) which evalutes the quantile of Km at the proportion p,
  • rkay(n, df=m, ...) which generates n pseudo-random realizations from Km.

The parameters in the ellipsis include a non-centrality parameter. All functions rely on the corresponding χm2 functions in base R.

We briefly illustrate each below.

The density dkay(x, df, ...)

x <- seq(0,2,0.01)
plot(x, dkay(x, df=10), type="l", col="steelblue", 
     main="Density", xlab="x", ylab="f(x)")
abline(v=1.0, lty=2, col="grey")

The cumulative distribution function pkay(x, df, ...)

x <- seq(0,2,0.01)
plot(x, pkay(x, df=10), type="l", col="steelblue", 
     main="Distribution", xlab="x", ylab="F(x)")
abline(v=1.0, lty=2, col="grey")

The quantile function qkay(p, df, ...)

x <- seq(0,2,0.01)
p <- pkay(x, df=10)
plot(p, qkay(p, df=10), type="l", col="steelblue", 
     main="Quantile function", xlab="p", ylab="Q(p)")
abline(h=1.0, lty=2, col="grey")

Pseudo-random realizations rkay(n, df, ...)

x <- rkay(1000, df=10)
hist(x, col="steelblue", 
     main="Pseudo-random numbers", xlab="x")
abline(v=1.0, lty=2, col="grey")