Consider a distribution for which the p.d.f. or the p.f. is f(x|), where the parameter is a k dimensional vector belonging to some parameter

Consider a distribution for which the p.d.f. or the p.f. is f(x|θ), where the parameter θ is a k dimensional vector belonging to some parameter space „¦. It is said that the family of distributions indexed by the values of θ in „¦ is a k-parameter exponential family, or a k parameter Koopman-Darmois family, if f(x|θ) can be written as follows for θ ˆˆ „¦ and all values of x:

Here, a and c1, . . . , ck are arbitrary functions of θ, and b and d1, . . . , dk are arbitrary functions of x. Suppose now that X1, . . . , Xn form a random sample from a distribution which belongs to a k-parameter exponential family of this type, and define the k statistics T1, . . . , Tk as follows:

Show that the statistics T1, . . . , Tk are jointly sufficient statistics for θ.

f(x|0) = a(8)b(x) expc;(e)d;(x) Li=1