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Definition: A statistic T is called complete if E[g(T)]=0 for all and some function g implies that P(g(T)=0)=1 for all.
It is easy to see that if f(t) is a one to one function and T is a sufficient statistic, then f(T) is a sufficient statistic. In particular, we can multiply a sufficient statistic by a nonzero constant and get another sufficient statistic.
Sufficiency pertains to data reduction, not merely estimation. A sufficient statistic need not estimate anything. For example, if X1,In are did samples drawn from an exponential distribution with unknown mean, then X is sufficient for, but so is (X1++Xn1,In).
x is a sufficient statistic if it retains all the information about the population mean that was contained in the original data points. According to statistician Ronald Fisher, no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter.
A sufficient statistic T : X T is minimal sufficient if for any sufficient statistic U : X U there is a measurable function g : UT such that T = g(U) ex|(· |)-a.s. for all .
In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.
The minimal sufficient statistic is not unique. But, the minimal sufficient partition is unique. {1+(yo )2}. The ratio is a constant function of if T(Y n)=(Y(1), ×··, Y(n)).
If both T and S are minimal sufficient statistics, then by definition there is a one-to-one function such that T = (S). Hence, the minimal sufficient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic.
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