Suppose that Y 1 , Y 2 , …, Y n is a random sample from a probability density function in the (one-parameter) exponential family so that f ( y | θ ) = { α ( θ ) b ( y ) e − [ c ( θ ) d ( y ) ] , α ≥ y ≤ b , 0 , elsewhere, where a and b do not depend on θ . Show that ∑ i = 1 n d ( Y i ) is sufficient for θ .
Suppose that Y 1 , Y 2 , …, Y n is a random sample from a probability density function in the (one-parameter) exponential family so that f ( y | θ ) = { α ( θ ) b ( y ) e − [ c ( θ ) d ( y ) ] , α ≥ y ≤ b , 0 , elsewhere, where a and b do not depend on θ . Show that ∑ i = 1 n d ( Y i ) is sufficient for θ .
Solution Summary: The author explains the probability density function of, and how it can be written as follows.
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