Question

Suppose X?, X?, ..., X? is a random sample of size n from the uniform distribution U(0, ?), where ? is an unknown (positive) parameter. (a) The largest order statistic of the sample is denoted by X???. Show that, over the range 0 < x < ?, the c.d.f. of X??? is F_{X_{(n)}}(x; ?) = (x/?)?. (b) Obtain the p.d.f. of X??? and hence find the mean and variance of X???. (c) Find the value of c for which ??? = cX??? is an unbiased estimator of ?, and find its variance. (d) Find the variance of the unbiased estimator ??? = 2X?, where X? is the sample mean. [Hint: the variance of the uniform distribution is given on the sheet about common continuous distributions.] (e) Both estimators are unbiased — comment on their relative efficiency. What is the asymptotic relative efficiency of ??? to ????

          Suppose X?, X?, ..., X? is a random sample of size n from the uniform distribution U(0, ?), where ? is an unknown (positive) parameter.

(a) The largest order statistic of the sample is denoted by X???. Show that, over the range 0 < x < ?, the c.d.f. of X??? is
F_{X_{(n)}}(x; ?) = (x/?)?.

(b) Obtain the p.d.f. of X??? and hence find the mean and variance of X???.
(c) Find the value of c for which ??? = cX??? is an unbiased estimator of ?, and find its variance.
(d) Find the variance of the unbiased estimator ??? = 2X?, where X? is the sample mean.
[Hint: the variance of the uniform distribution is given on the sheet about common continuous distributions.]
(e) Both estimators are unbiased — comment on their relative efficiency. What is the asymptotic relative efficiency of ??? to ????

Suppose X?, X?, ..., X? is a random sample of size n from the uniform distribution U(0, ?), where ? is an unknown (positive) parameter.

(a) The largest order statistic of the sample is denoted by X???. Show that, over the range 0 < x < ?, the c.d.f. of X??? is
FX(n)(x; ?) = (x/?)?.

(b) Obtain the p.d.f. of X??? and hence find the mean and variance of X???.
(c) Find the value of c for which ??? = cX??? is an unbiased estimator of ?, and find its variance.
(d) Find the variance of the unbiased estimator ??? = 2X?, where X? is the sample mean.
[Hint: the variance of the uniform distribution is given on the sheet about common continuous distributions.]
(e) Both estimators are unbiased — comment on their relative efficiency. What is the asymptotic relative efficiency of ??? to ????

Added by Dustin H.

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