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 Ω. It is said that the family of distributions indexed by the values of θ in Ω is a k-parameter exponential family, or a k parameter Koopman-Darmois family, if f(x|θ) can be written as follows for θ ∈ Ω 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.

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