Find a 95 confidence interval for&nbsp;θ&nbsp;based on inverting the test statistic statistic&nbsp;θ^.For our data we have&nbsp;Yi∼N(θxi,1)&nbsp;for&nbsp;i=1,…,n.Therefore it can be proven that the MLE for&nbsp;θ&nbsp;is given byθ^=∑xiYi∑xi2To find the confidence interval I should invert the test statistic&nbsp;θ^.The most powerful unbiased size&nbsp;α=0,05&nbsp;test for testingH0:μ=μ0&nbsp;vs.&nbsp;H1:μ≠μ0where&nbsp;X1,…,Xn∼&nbsp;&nbsp;iid&nbsp;&nbsp;n(μ,σ2)&nbsp;has acceptance regionA(μ0)=x:|x¯-μ0|≤1,96σ/n.Substituting my problem (I think) we get that the most powerful unbiased size&nbsp;α=0,05&nbsp;test for testingH0:θ=θ^&nbsp;vs.&nbsp;H1:θ≠θ^has acceptance region&nbsp;{A(θ^)={y:|y―−θ^|≤1,96n}or equivalently,&nbsp;A(θ^)=y:ny¯-1,96nxi≤θ^≤ny¯+1,96nxiSubstituting&nbsp;θ^=∑xiYi∑xi2&nbsp;we obtainA(θ^)=y:ny¯-1,96nxi≤ΣxiYiΣxi2≤ny¯+1,96nxiThis means that my&nbsp;1−0,05=0,95(95%)&nbsp;confindence interval is defined to beC(y)={θ^:y∈A(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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Find a 95 confidence interval for&amp;nbsp;θ&amp;nbsp;based on inverting the test statistic statistic&amp;nbsp;θ^.For our data we have&amp;nbsp;Yi∼N(θxi,1)&amp;nbsp;for&amp;nbsp;i=1,…,n.Therefore it can be proven that the MLE for&amp;nbsp;θ&amp;nbsp;is given byθ^=∑xiYi∑xi2To find the confidence interval I should invert the test statistic&amp;nbsp;θ^.The most powerful unbiased size&amp;nbsp;α=0,05&amp;nbsp;test for testingH0:μ=μ0&amp;nbsp;vs.&amp;nbsp;H1:μ≠μ0where&amp;nbsp;X1,…,Xn∼&amp;nbsp;&amp;nbsp;iid&amp;nbsp;&amp;nbsp;n(μ,σ2)&amp;nbsp;has acceptance regionA(μ0)=x:|x¯-μ0|≤1,96σ/n.Substituting my problem (I think) we get that the most powerful unbiased size&amp;nbsp;α=0,05&amp;nbsp;test for testingH0:θ=θ^&amp;nbsp;vs.&amp;nbsp;H1:θ≠θ^has acceptance region&amp;nbsp;{A(θ^)={y:|y―−θ^|≤1,96n}or equivalently,&amp;nbsp;A(θ^)=y:ny¯-1,96nxi≤θ^≤ny¯+1,96nxiSubstituting&amp;nbsp;θ^=∑xiYi∑xi2&amp;nbsp;we obtainA(θ^)=y:ny¯-1,96nxi≤ΣxiYiΣxi2≤ny¯+1,96nxiThis means that my&amp;nbsp;1−0,05=0,95(95%)&amp;nbsp;confindence interval is defined to beC(y)={θ^:y∈A(θ^)}But I can&#039;t seem to find anything concrete and I feel that I&#039;ve made mistakes somewhere. What to do?

Malia Booth · Accepted Answer

You have&amp;nbsp;θ^=∑ixiYi∑ixi2.Although non-linear in&amp;nbsp;(x1,…,xn), this is linear in&amp;nbsp;(Y1,…,Yn), and that&#039;s why it&#039;s a &quot;linear&quot; model (it&#039;s not because one is fitting a straight line; if that were true then least-squares fitting of polynomials would be considered non-linear regression, but it&#039;s linear regression). So you have a linear combination of independent normally distributed random variables, where the coefficients are constant (i.e. not random). Being a linear combination of independent normally distributed random variables with constant coefficients makes its distribution easy to find:θ^~N(∑i(xiE⁡(Yi))∑ixi2,∑i(xi2var⁡(Yi))(∑ixi2)2)=N(θ,1∑ixi2)&amp;nbsp;since&amp;nbsp;E⁡(Yi)=xiθ&amp;nbsp;and&amp;nbsp;var⁡(Yi)=1.So&amp;nbsp;Pr(θ−A1∑ixi2&amp;lt;θ^&amp;lt;θ+A1∑ixi2)=0.95where A is such that&amp;nbsp;Pr(−A&amp;lt;Z&amp;lt;A)=0.95. ConsequentlyPr(θ^−A1∑ixi2&amp;lt;θ&amp;lt;θ^+A1∑ixi2)=0.95and that is your confidence interval.

Find a 95 confidence interval for \(\displaystyle\theta\)

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