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Let $X_{1},...,X_{n}$ be random variable from the probability density function:

$f(x|\beta)=\frac{\beta^{\alpha}}{\Gamma{\alpha}} x^{\alpha-1}e^{-\beta x}$ where $\alpha$ is known and $\beta>0$.

I know that $\sum_{i=1}^{n}X_{i}$ will be a complete Statistic since the gamma distribution is in exponential family. What I wanted to know is that whether $X_{1}$ is complete or not? Since, $X_{1}$ is from gamma family and gamma family is complete. Hence, $X_{1}$ is complete. Is my reasoning correct? If not, could you give me some examples which are complete Apart from the usual $\sum_{i=1}^{n}X_{i}$. Thanks in advance.

userNoOne
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  • Do you know the definition of "complete statistic"? – jbowman Jan 25 '18 at 03:53
  • Family of distribution ${f(x,\theta)}$ is complete for $\theta$ if $E(g(x))=0$ implies that $g(x)=0$ with probability one. – userNoOne Jan 25 '18 at 04:04
  • But the use of definition is irritating sometimes. The definition can be easily used in some distribution like Binomial, Poisson Etcetera but in some distribution the use of definition gets a little bit tricky. – userNoOne Jan 25 '18 at 04:05
  • What do you mean by "the gamma family is complete"? Completeness doesn't apply to distributions. – jbowman Jan 25 '18 at 04:24
  • Completeness apply for a family of distribution. We cannot say that gamma distribution is complete. We have to say that gamma family is complete for $\beta$ or we can say ${f(x|\beta), \beta>0}$ is complete. – userNoOne Jan 25 '18 at 04:32
  • Fundamentals of mathematical statistics by SC GUPTA. It's by an Indian author. – userNoOne Jan 25 '18 at 04:53
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    A statistic is not a family of distributions. The question is about a particular statistic, in this case $X_1$, not about whether the gamma family itself is complete. – jbowman Jan 25 '18 at 04:55
  • If a Statistic follow a certain distribution and if that distribution is complete, then the Statistic will be complete. For example, the family Poisson distribution with parameters $\lambda$ is complete and consider a Statistic $sum_{I=1}^{n}X_{i}$, then this Statistic again follows Poisson distribution with the parameter $\lambda$. Hence, $sum_{I=1}^{n}X_{i}$ will be complete. – userNoOne Jan 25 '18 at 04:58
  • Sorry, I was thinking specifically of the definition of a "complete statistic", and you responded with family of distributions, but the question is about the statistic, not the family. – jbowman Jan 25 '18 at 05:01
  • So, are you trying to say that there is no connection between the completeness of Statistic and corresponding family of distribution? Beacuse my teacher told us that, a Statistic is complete if the distribution corresponding to that Statistic is complete. – userNoOne Jan 25 '18 at 05:12

1 Answers1

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I am trying to help you on this. The Gamma distribution belongs to exponential family and has a minimal sufficient statistics. Theoretically, is a minimal sufficient statistic exists, then any complete statistic of the (Gamma or other probability distributions) should be the minimal sufficient statistic itself.

You could refer to the Basu's theorem again and again and figure out this. Hope it helps.

son520804
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  • Thanks for the reply, but minimal sufficient Statistic is not in our syllabus for the current semester. If you can explain, it without it, that would be very beneficial for me. – userNoOne Jan 25 '18 at 04:20
  • @RAHUlJHa: you were given a proper argument for the answer to the question. If you insist on an answer that fits your syllabus, you should seek it with your instructor or your fellow students rather than on this forum. – Xi'an Feb 01 '18 at 18:42