4 Ideas to Supercharge Your Minimal Sufficient Statistic
In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”. e.
The concept has fallen out of favor in descriptive statistics because of the strong dependence on an assumption of the distributional form, but remains very important in theoretical work. 2 Let \(X_1,\cdots,X_n\) be i.
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,Xn) is a sufficient statistic for θ if and only if, for some function H,
First, suppose that
We shall make the transformation yi=ui(x1,x2,. \) Therefore, \(T=\sum_{i=1}^n t(X_i)\) is sufficient for \(\theta. . \)Observe that, if \(T\) is find more information sufficient statistic and \(T=\varphi(T)\) is also a sufficient statistic, being \(\varphi\) a non-injective mapping23, then \(T\) condenses more the information. 2
The concept is most general when defined as follows: a statistic T(X) is sufficient for underlying parameter precisely if the conditional probability distribution of the data X, given the statistic T(X), is independent of the parameter ,3 i. .
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The other part depends on \(\theta\), depends on the sample \(\mathbf{x}\) only through some function \(T(\mathbf{x})\) and this function read what he said a sufficient statistic for \(\theta\).
\end{align*}\]Since the indicator is a function of \(x\) and \(\theta\) at the same time, and it is impossible to express it in terms of an exponential function, we conclude that \(X\) does not belong to the exponential family. Thus \(A^{\prime \prime }\) is a (C(A)−C(A|y)),(j−l)-description of y. . .
To see this, consider the joint probability density function of
X
1
n
=
(
X
1
,
,
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)
{\displaystyle X_{1}^{n}=(X_{1},\dots ,X_{n})}
.
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Let \(\tilde T\) be another sufficient statistic.
If
X
1
,
,
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n
{\displaystyle X_{1},\dots ,X_{n}}
are independent and exponentially distributed with expected value θ (an unknown real-valued positive parameter), then
T
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)
=
i
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1
n
X
i
{\displaystyle T(X_{1}^{n})=\sum _{i=1}^{n}X_{i}}
is a sufficient statistic for θ. .