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Let X1, X2, . . ., Xn be independent, but not identically distributed, samples. All these...

Let X1, X2, . . ., Xn be independent, but not identically distributed, samples. All these Xi ’s are assumed to be normally distributed with

Xi ∼ N(θci , σ^2 ), i = 1, 2, . . ., n,

where θ is an unknown parameter, σ^2 is known, and ci ’s are some known constants (not all ci ’s are zero). We wish to estimate θ.

(a) Write down the likelihood function, i.e., the joint density function of (X1, . . ., Xn), and identify a sufficient statistic.

(b) Use your sufficient statistic to construct MVUE for θ.

Math   Statistics-And-Probability   
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