Let&nbsp;X1,⋯,Xn&nbsp;be a random sample from&nbsp;f(x∣θ)=θxθ−1&nbsp;for&nbsp;0&lt;x&lt;1. Find a confidence interval for&nbsp;θ&nbsp;using as pivotal quantity a function of the maximum likelihood estimator for&nbsp;θ.

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Let&amp;nbsp;X1,⋯,Xn&amp;nbsp;be a random sample from&amp;nbsp;f(x∣θ)=θxθ−1&amp;nbsp;for&amp;nbsp;0&amp;lt;x&amp;lt;1. Find a confidence interval for&amp;nbsp;θ&amp;nbsp;using as pivotal quantity a function of the maximum likelihood estimator for&amp;nbsp;θ.

Drake Huang · Accepted Answer

Step 1Assuming you want a confidence interval for θ, you may use the CLT or a known chi distribution for variances and some integrals to get an interval for the parameter. First, we have thatθ^=−n∑ln⁡ xi.For the expected value we have the integralE[X]=∫01θx(θ−1)x dθ=θ1+θAnd for the second momentμ2=∫01θxθ−1x2 dθ=θ2+θSo the variance isVar(X)=θ2+θ−(θ1+θ)2==θ(θ+1)2(θ+2).Now then, we know that the following quotient has a chi-squared distribution, so in this caseC=(n−1)S2σ2∼χ{n−1}2  and  C=(n−1)S2θ(θ+1)2(θ+2)where S2=1n−1∑(Xi−X―)2. Finally we can work with the random variable C distributed as a chi squared. We want a confidence interval of (1−α). ThenP(χ{n−1}2(α2)&amp;lt;C&amp;lt;χ{n−1}2(1−α2))=1−αdenoting that χ{n−1}2(β) is the β percentile of the χ2 which may be found in reference tables.From here we work with the interval only substituting the percentiles with 1Dt and 1Dπ, lower and superior.1Dl&amp;lt;C&amp;lt;1Ds⇒1Dl&amp;lt;(n−1)S2θ(θ+1)2(θ+2)&amp;lt;1DsDl&amp;gt;θ(θ+1)2(θ+2)(n−1)S2&amp;gt;DsDl((n−1)S2)&amp;gt;θ(θ+1)2(θ+2)&amp;gt;Ds((n−1)S2)(Dl((n−1)S2))&amp;gt;θ(θ+1)2(θ+2)&amp;gt;(Ds((n−1)S2))(θ^+1)2(θ^+2)(Dl((n−1)S2))&amp;gt;θ&amp;gt;(θ^+1)2(θ^+2)(Ds((n−1)S2))where, taking the before definition and derivation, θ^=−n∑ln⁡ xi and S2=1n−1∑(Xi−X―)2. When we sent the θ to the other side, we simply used its estimate to work around the problem. (However a mathematician should confirm this is valid!) Either way, if you find a better known distribution that may include θ, then try it and tell us if it is simpler!

Let \(\displaystyle{X}_{{{1}}},\cdots,{X}_{{{n}}}\) be a random sample from

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