Fisherian reduction for the t-Test

In the second chapter of (Cox 2006) the authors talks about a Fisherian reduction which I think of as a framework of doing inference given a sufficient statistic $S$. An interesting point here is that one can use the conditional distribution of the data, $X_1,\dots ,X_n$ say, on $S$ to evaluate the fit of the model.

In this post I want to explore this concept in the setting of a standard $t$-Test, i.e. we have $X_i\stackrel{\text{i.i.d}}{\sim }\mathcal{N}(\mu ,{\sigma }^2)$. The parameter of interest is of course $\mu$ and $S=({\overline{X}}_n,{\hat{\sigma }}_n^2)$ is a sufficient statistic, with ${\hat{\sigma }}_n^2$ the empirical variance.

To apply the Fisherian reduction we thus need to find the conditional distribution of $X=(X_1,\dots ,X_n)$ on ${\overline{X}}_n$, i.e.$X|{\overline{X}}_n$. For this, let $A=(A_1,B)\in {\mathbf{R}}^{n\times n}$ be an orthogonal matrix whose first column is $$A_1=(\frac{1}{\sqrt{n}},\dots ,\frac{1}{\sqrt{n}}).$$

Then $Y=A^{T}X\sim \mathcal{N}(\mu A^T\mu \mathbf{1},{\sigma }^2{I}_n)$ and $Y=(\sqrt{n}{\overline{X}}_n,Z)$ where $Z\sim \mathcal{N}\left({\sigma }^2{I}_{n-1}\right)$, $Z$ and ${\overline{X}}_n$ being independent.

Transforming back we obtain the conditional distribution we sought: $$X|{\overline{X}}_n\sim AY|{\overline{X}}_n\mathbf{1}\sim {\overline{X}}_n+BZ$$

The catch here is that $B$ is a $n\times (n-1)$ dimensional matrix, so $X|{\overline{X}}_n$ has a normal distribution of dimension $n-1$, e.g. the variance-covariance matrix is rank-deficient.

Let’s verify this through simulation.

mu <- 10
sigma <- 1
n <- 2
m <- 1000
x <- matrix(rnorm(n * m, mu, sigma), nrow= n)

y <- t(t(x) - colMeans(x))

transformed <- c(matrix(c(1/sqrt(2), -1/sqrt(2)), nrow = 1) %*% y)

plot(y[1,], y[2,])

hist(transformed)
print(mean(transformed))
print(sd(transformed))

[1] 0.007553103
[1] 1.05018

References

Cox, D. R. 2006. Principles of Statistical Inference. Cambridge ; New York: Cambridge University Press.