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Complete sufficient statistic of multivariate t-distribution
Step-by-step statistics solution: Complete sufficient statistic of multivariate t-distribution
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1. What is being asked?
We have a random vector
[ X=(X_{1},\dots ,X_{n})\sim t_{n,\nu}\bigl(0,\sigma^{2}I_{n}\bigr), \qquad \nu>4, ]
i.e. a multivariate Student‑(t) distribution with
- location vector (0),
- scale matrix (\sigma^{2}I_{n}) (the only unknown parameter), and
- (\nu) degrees of freedom (known).
From the factorisation theorem the statistic
[ T(X)=\sum_{i=1}^{n}X_{i}^{2} ]
is sufficient for (\sigma^{2}).
The question is: Is (T) also complete for the one‑parameter family ({\sigma^{2}>0})?
In other words, we must decide whether the following implication holds:
[ \boxed{\; E_{\sigma^{2}}\bigl[g(T)\bigr]=0\;\text{for every }\sigma^{2}>0\; \Longrightarrow\;g(T)=0\;\text{a.s.} \;} ]
If it holds, (T) is a complete sufficient statistic; otherwise it is not.
2. Full solution
2.1 A useful representation of the multivariate (t)
Let
- (Z\sim N_{n}(0,I_{n})) (standard normal vector),
- (W\sim \chi^{2}_{\nu}) (chi‑square with (\nu) d.f.),
independent of each other.
Then
[ X\;\stackrel{d}{=}\; \frac{\sigma}{\sqrt{W/\nu}}\; Z . ]
Consequently
[ T=\sum_{i=1}^{n}X_{i}^{2} =\sigma^{2}\,\frac{\lVert Z\rVert ^2}{W/\nu}. ]
Now (\l
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