User:Hilikus44

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One-Dimensional Examples of Sufficient Statistics[edit]

Normal Distribution[edit]

If are independent and normally distributed with expected value θ (a parameter) and known finite variance , then is a sufficient statistic for θ.

To see this, consider the joint probability density function of . Because the observations are independent, the pdf can be written as a product of individual densities, ie -

Then, since , which can be shown simply by expanding this term,

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

Since does not depend on the parameter and depends only on through the function

the Fisher–Neyman factorization theorem implies is a sufficient statistic for .

Exponential Distribution[edit]

If are independent and exponentially distributed with expected value θ (an unknown real-valued positive parameter), then is a sufficient statistic for θ.

To see this, consider the joint probability density function of . Because the observations are independent, the pdf can be written as a product of individual densities, ie -

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

Since does not depend on the parameter and depends only on through the function

the Fisher–Neyman factorization theorem implies is a sufficient statistic for .


Two-Dimensional Examples of Sufficient Statistics[edit]

Uniform Distribution (with two parameters)[edit]

If are independent and uniformly distributed on the interval (where and are unknown parameters), then is a two-dimensional sufficient statistic for .

To see this, consider the joint probability density function of . Because the observations are independent, the pdf can be written as a product of individual densities, ie -

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

Since does not depend on the parameter and depends only on through the function ,

the Fisher–Neyman factorization theorem implies is a sufficient statistic for .

Gamma Distribution[edit]

If are independent and distributed as a , where and are unknown parameters of a Gamma distribution, then is a two-dimensional sufficient statistic for .

To see this, consider the joint probability density function of . Because the observations are independent, the pdf can be written as a product of individual densities, ie -

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

Since does not depend on the parameter and depends only on through the function ,

the Fisher–Neyman factorization theorem implies is a sufficient statistic for .