Eleisha's Segment 19: The Chi-Square Statistic

To Calculate

1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity $({\mathbf {x} -\mathbf {\mu } })^{T}{\mathbf {\Sigma } }^{-1}({\mathbf {x} -\mathbf {\mu } }),$ where $\mathbf {x}$is a random draw from the multivariate normal, is $\chi ^{2}$ distributed.

Must prove that $({\mathbf {x} -\mathbf {\mu } })^{T}{\mathbf {\Sigma } }^{-1}({\mathbf {x} -\mathbf {\mu } })$ is $\chi ^{2}$ distributed.

Since $\mathbf {x}$ is a random draw from the mulitvariate normal, x can be written as $\mathbf {x=Ly+\mu }$, where y is filled with $y_{i}{\text{'s}}$ drawn from a normal with a \mu zero and \sigma one. (Shown in lecture 17)

If you manipulate $\mathbf {x=Ly+\mu }$ you get $\mathbf {Ly=x-\mu }$

Therefore

$({\mathbf {x} -\mathbf {\mu } })^{T}{\mathbf {\Sigma } }^{-1}({\mathbf {x} -\mathbf {\mu } })=\mathbf {(Ly)^{T}\Sigma ^{-1}(Ly)=(y^{T}L^{T})\Sigma ^{-1}(Ly)=(y^{T}L^{T})(LL^{T})^{-1}(Ly)=y^{T}y} =\sum y_{i}{\text{'s}}$

Since the $y_{i}{\text{'s}}$ are drawn for the normal distributions with mean \mu zero and \sigma one, \sum y_i \text{'s} is the sum of squared t-values.

This means that t $({\mathbf {x} -\mathbf {\mu } })^{T}{\mathbf {\Sigma } }^{-1}({\mathbf {x} -\mathbf {\mu } })$ is $\chi ^{2}$ distributed.

To Think About

1. Why are we so interested in t-values? Why do we square them?

2. Suppose you measure a bunch of quantities $x_{i}$, each of which is measured with a measurement accuracy $\sigma _{i}$ and has a theoretically expected value $\mu _{i}$. Describe in detail how you might use a chi-square test statistic as a p-value test to see if your theory is viable? Should your test be 1 or 2 tailed?

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