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Questions related to sufficient statistics and confidence interval

  1. May 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Suppose X1, X2, ..., Xn constitute a random sample from a population following N(μ,θ) where μ is known.

    (a)Find a sufficient statistic for θ.
    (b)Use the maximum likelihood estimator of θ to construct a confidence interval for θ with confidence level 1-α.

    2. Relevant equations
    3. The attempt at a solution
    For part (a), I have tried to use the factorization criterion to find the sufficient statistic for θ but I have difficulty in separating θ from the exponential function as θ is the denominator. Can someone teach me how to get a function that depends on Xi only!

    For part (b), the maximum likelihood estimator of θ is (1/n)*Σ(Xi-μ)^2. I am not sure about whether the information that Σ(Xi-μ)^2/θ follows chi square distribution with n degrees of freedom can help us to find the confidence interval! Can someone teach me how to get the confidence interval as well?
     
  2. jcsd
  3. May 8, 2014 #2

    Ray Vickson

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    Use a chi-squared table to find points ##a## and ##b## such that
    [tex] P(\chi^2(n) \leq a) = \frac{\alpha}{2}, \; P(\chi^2(n) \geq b) = \frac{\alpha}{2}[/tex]
    Thus
    [tex] P\left( a \leq \frac{\sum(X_i - \mu)^2}{\theta} \leq b \right) = 1 - \alpha.[/tex]
    Now turn that into a ##1-\alpha## probability statement about ##\theta##.
     
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