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Thread: Hypothesis Testing

  1. #1
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    Hypothesis Testing

    Let $\displaystyle X_{1}, ... , X_{n}$ be a random sample from an exponential distribution with mean $\displaystyle \theta$. Show that the likelihood ratio test of $\displaystyle H_{0} : \theta = \theta_{0}$ against $\displaystyle H_{1} : \theta = \theta_{1}$ has a critical region of the form $\displaystyle \sum^{n}_{i=1} x_{i} \leq c_{1}$ or $\displaystyle \sum^{n}_{i=1} x_{i} \geq c_{2}$. How would you modify this so that chi-square tables could be easily used?
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  2. #2
    MHF Contributor matheagle's Avatar
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    The sum of indep EXP(theta's) is a $\displaystyle \Gamma(n,\theta)$

    Call this sum of X_i's X and now transform to the chi square

    A $\displaystyle \chi^2_a=\Gamma(a/2,2)$

    So do some calculus one with the substitution $\displaystyle x/\theta=y/2$
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