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Math Help - method of moments estimator for NBinom

  1. #1
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    method of moments estimator for NBinom

    Derive the method of moments estimator for pi for a sample of size n from a NBinom(r, pi)-distribution. (Treat r as known, as it would be in a typical situation where you would be collecting data by repeating the Bernoulli trials.)

    I have no idea where to start. Help!
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  2. #2
    Moo
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    Hello,

    It's just basic use of the method of moments oO

    Let X ~ NBinom(r,p). Then \displaystyle E[X]=r\cdot\frac{p}{1-p} (you can compute that yourself, or find that on the wikipedia).

    Then we approximate by taking the empirical mean instead of the expectation and consider the estimator of p instead of p. Which gives \displaystyle \bar X_n=\frac{\sum_{i=1}^n X_i}{n}=r\cdot\frac{\hat p}{1-\hat p}

    Hence \displaystyle \hat p=\frac{\bar X_n}{r+\bar X_n}

    and that's all...
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