Confidence interval for parameter \(\displaystyle{X}_{{1}},{X}_{{2}},\cdots,{X}_{{n}}\)- i.i.d observations \(\displaystyle{X}_{{1}}=\xi+\eta\)

Marzadri9lyy

Marzadri9lyy

Answered question

2022-03-25

Confidence interval for parameter
X1,X2,,Xn- i.i.d observations
X1=ξ+η where ξ~N(θ2,θ2+1),η={0,1/24θ,1/2,ξ and η and η are independent.
Find α-confidence interval.
The first that I need to do is to find some estimate for θ. The only one that I find is 5XS2+110 but it is difficult to find distribution.
Is it possible to do something else?

Answer & Explanation

Carter Lin

Carter Lin

Beginner2022-03-26Added 13 answers

A natural choice arises from the method of moments estimator:
X¯=E[ξ+η]=θ2+2θ,
thus if θ>0,
θ^=1+X1.
Then, what is the variance of θ^? You could use the delta method:
Var[g(X)](g'(E[X]))2Var[X].
This will give you a Wald-type asymptotic confidence interval.

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