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Does minimal sufficient statistics implies complete? If true prove it, otherwise give an example

Does minimal sufficient statistics implies complete? If true prove it, otherwise give an example

Math   Statistics-And-Probability   
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Answer #1

Explanation

Bahadur's theorem says that every bounded complete sufficient statistics is also minimal sufficient.Since any minimal sufficient statistics by definition is afunction of any sufficient statistics, thus if a complete sufficient statistics exists, that minimal sufficient statistic must be a function of it and is therefore complete sufficient too..

Proof:-

A sufficient statistic T: X tends to T is minimal sufficient if for any..if we show that U(x) =U(y) implies y belongs to D(x) , then it follows that U(x)= U (y)

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