Factorisation Theorem for Sufficiency

[Coolster7]

I'm having trouble with applying this theorem to likelihood functions in order to obtain a sufficiency statistic for the relevant variables.

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The factorisation theorem being:

Under certain regularity conditions;

T(X) is sufficient for θ ⇔ f(x|θ) = h(x)g(t(x),θ)

for some functions h and g.

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The main problem I'm having is when to allow h(x) = 1.

For example in the exponential distribution you get a likelihood function: f(x|θ) = θⁿ(1-θ)^{\sum(x_i) - n}

you set h(x) =1 here and g(x) = (t,θ) = θⁿ(1-θ)^{t - n} where t = \sum(x_i)

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However, in the Poisson distribution you get a likelihood function: 1/(x₁!x₂!) x e^-2θθ^x₁+x₂.

here you set h(x) = 1/(x₁!x₂!) and not h(x) = 1.

Is this because h(x) has to be a constant or involving just x?

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So say for example you had a likelihood function:

f(x|θ) = xⁿσ^-2ne^-0.5nx2σ-2

using the Factorisation method would you let h(x) = 1 with g(x, σ) = f(x|θ)

and say x is sufficient for σ

OR would you let h(x) = xⁿ and g(x, σ) = σ^-2ne^-0.5nx2σ-2

and say x is sufficient for σ

Note: Obviously there is the same outcome for the sufficiency statistic, but in a different problem this may not be the case.

Can anyone help me please?

 

[h6ss]

However, in the Poisson distribution you get a likelihood function: 1/(x₁!x₂!) x e^-2θθ^x₁+x₂.

here you set h(x) = 1/(x₁!x₂!) and not h(x) = 1.

Is this because h(x) has to be a constant or involving just x?

The idea is to have h(x) not depending on the parameter. Whether it's a constant or not.

 

[Coolster7]

The idea is to have h(x) not depending on the parameter. Whether it's a constant or not.

Ah I see, thanks for this I understand now.. it's simple really.

 

[Coolster7]

The idea is to have h(x) not depending on the parameter. Whether it's a constant or not.

Actually just one more question. What about n? Would n need to be sufficient for the parameter or is it treated as constant/number?

 

[Coolster7]

Actually just one more question. What about n? Would n need to be sufficient for the parameter or is it treated as constant/number?

Anyone?