For any vectors u, v and w, show

2022-04-06

For any vectors u, v and w, show that the vectors u+v, u+w and v+w form a linearly dependent set.

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asked 2022-04-04

Find a 95 confidence interval for θ based on inverting the test statistic statistic θ^.
For our data we have YiN(θxi,1) for i=1,,n.
Therefore it can be proven that the MLE for θ is given by
θ^=xiYixi2
To find the confidence interval I should invert the test statistic θ^.
The most powerful unbiased size α=0,05 test for testing
H0:μ=μ0 vs. H1:μμ0
where X1,,Xn  iid  n(μ,σ2) has acceptance region
A(μ0)=x:|x¯-μ0|1,96σ/n.
Substituting my problem (I think) we get that the most powerful unbiased size α=0,05 test for testing
H0:θ=θ^ vs. H1:θθ^
has acceptance region {A(θ^)={y:|yθ^|1,96n}
or equivalently, A(θ^)=y:ny¯-1,96nxiθ^ny¯+1,96nxi
Substituting θ^=xiYixi2 we obtain
A(θ^)=y:ny¯-1,96nxiΣxiYiΣxi2ny¯+1,96nxi
This means that my 10,05=0,95(95%) confindence interval is defined to be
C(y)={θ^:yA(θ^)}
But I can't seem to find anything concrete and I feel that I've made mistakes somewhere. What to do?

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