Show that t(Xn) = {Xn:1,... ,Xn:n} is a symmetric function of Xn = (X1,... ,Xn) and, conversely, that any symmetric function of Xn is a function of t(Xn). Also show that t(Xn) is complete for the...

Show that t(Xn) = {Xn:1,... ,Xn:n} is a symmetric function of Xn = (X1,... ,Xn) and, conversely, that any symmetric function of Xn is a function of t(Xn). Also show that t(Xn) is complete for the class of distribution functions on Rn when F is (a) any discrete distribution or (b) any absolutely continuous distribution function. Is t(Xn) a sufficient statistics? Is it minimal sufficient in general? [Lehmann 2005]