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Thread: Suggest an Unbiased estimator for θ

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    Exclamation Suggest an Unbiased estimator for θ




    Prove that the maximum of the random sample of size 'n' from uniform distribution over [0,θ] is a biased but a consistent estimator of θ. Suggest a n unbiased estimator of θ. Is it consistent?

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    Now suppose we have the estimate \hat{\theta} = X_{(n)}

    We can use the simple theory in order statistic to evaluate its C.D.F.:
    F_{X_{(n)}} (x) = \Pr\{X_{(n)} \leq x\}
= \Pr\left\{\bigcap_{i=1}^n (X_i \leq x) \right\}
= \prod_{i=1}^n \Pr\{ X_i \leq x \} 
= \left( \frac {x} {\theta} \right)^n \forall x \in [0, \theta]

    Then differentiating it with respect to x to obtain its p.d.f.:
    f_{X_{(n)}} (x) = \frac {dF_{X_{(n)}} (x)} {dx} 
= \frac {n} {\theta^n} x^{n-1} \forall x \in [0, \theta]

    Now we can calculate the its expected value to see whether it is a biased estimator:
    E[X_{(n)}] = \int_0^{\theta} x \frac {n} {\theta^n} x^{n-1} dx
= \frac {n} {(n+1) \theta^n} \left. x^{n+1} \right|_{x=0}^{x=\theta}
= \frac {n\theta} {(n+1)}
    which underestimate the parameter \theta

    However, it is consistent since
    \lim_{n \to \infty} E[X_{(n)}] = \theta
    (convergence in mean implies convergence in probability as well)

    We can correct the bias by multiplying back the factor
    \frac {n+1} {n} so that it is unbiased but still consistent.
    i.e. Use \frac {n+1} {n} X_{(n)}

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    can u suggest any book for reference......for Good Estimators

    Thank u

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    You can refer to the book "Statistical Inference" by
    George Casella and Roger L. Berger

    There are many nice properties that you want an estimator to acheive.
    So you need to know in which way your estimator is optimal.
    Note the new estimator in this question is the UMVU estimator,
    one type of the most frequently used estimator.

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