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I 0' ’i // 7.5 Exponential families of Disfhbabons Consider a family f($;6! : 6 E Q) of probability density or mass functions, where Q is the interval set 9 : {6 : 7 < 6 < 6}, where 'y and 6 are known constants
(they may be ioo), and where ﬁx; 9) Z { eXPlP(9lK($l + 5(a) + q(6)] a: e 5 O elsewhere, (751) where S is the support of X. In this section we will be concerned with a particular
class of the family called the regular exponential class. Deﬁnition 7.5.1 (Regular Exponential Class). A pdf of the form (7.5.1) is
said to be a member of @ regular exponential " ; of probability density or mass functions if 1. S, the support of X, does not depend upon 6, 2. p(6) is a nontrivial continuous function of 6 E Q, 5’. Finally, (a) ifX is a continuous random variable then each of K’(ac) i 0 and S($)
is a continuous function of a: E 8, (b) if X is a discrete random variable then K is a nontrivial function of
a: E S. For example, each member of the family {f(:c;6) : 0 < 6 < 00}, where f(:c;6) is N (0, 6), represents a regular case of the exponential class of the continuous type
because 1 —a:
f($;6) 2 Tree 2/29 1 .
exp (—Eémz — logv27r6>, —oo < a: < oo. II On the other hand, consider the uniform density function given by ﬂag) 2 { exp{——log 6} a: 6 (0,6) 0 elsewhere. This can be written in the form (7.5.1) but the support is the interval (0,6) Which
depends on 6. Hence, the uniform family is not a regular exponential family. Consider a family { f (:13; 0) : 49 E Q} of probability density or mass functions,
Where Q is the interval set 9 = {6 : 7 < 6 < 5}, Where *y and 6 are known constants
(they may be too), and Where . _ eXp[p(0)K(w) + 3(93) + 9(9)] 03 E 5
ﬂw’g) — { 0 elsewhere, (7'5'1)
Where 8 is the support of X.
Let X 1, X 2, . . . ,Xn denote a random sample from a distribution that represents
a regular case of the exponential class. The joint pdf or pmf of X1, X2, . . . ,Xn is n exp [p(6) Z + Z + nq(6) 1 Theorem 7.5.2. Let f(a:;6), *y < 9 < 6, be a pdf or pmf of a random variable X
whose distribution is a regular case of the exponential class. Then if X1,X2, . . . ,Xn (where n is a ﬁxed positive integer) is a random sample from the distribution of X ,
n _
the statistic Y1 : ZK is a suﬁicient statistic for 6’ and the family {fy1(y1; 0) : 1
7 < 0 < 6} of probability density functions of Y1 is complete. That is, Y1 is a
complete sufﬁcient statistic for 6. ...
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 Spring '08
 AlexeiStepanov
 Statistics, Normal Distribution, Probability, Probability theory, probability density function, exponential class

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