Fisher-Neyman factoring criterion

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The Fisher-Neyman factoring criterion is a theorem that allows us to determine whether a T statistic fulfills the sufficiency property.

Intuitively, this theorem allows us to know if a statistic is a sufficient statistic. And, vice versa, without having information beforehand, trying to determine the existence of a sufficient statistic and its expression. See enough statistic

Fisher-Neyman factoring criterion formula

Formally, it is said that given a simple random sample (m.a.s.) of a random variable X with density function f (x; θ) with θ ∈ Ω. The statistic T = T (X1,…, Xn) is said to be sufficient for θ, if and only if, the density function of the sample can be written as:

f (x1,…, xn) = h (x1,…, xn) × g (T, θ)

To understand what each of the parts of this theorem mean, we are going to redefine it but with an example:

We randomly choose 100 students (simple random sample) and ask them what their annual spending on books is (random variable X). This variable will have a density function (see density function). We must then choose a sufficient statistic to calculate a parameter (θ) (The parameter θ will be the average of the annual expenditure on books).

The indicated formula is divided as follows:

  • f (x1,…, xn): It is the density function of the sample (density function of the sample on the random variable X).
  • h (x1,…, xn): It is a function that does not take negative values ​​only from the sample (the expenditure of the 100 students).
  • g (T, θ): It is a function that depends only on the chosen statistic (sample mean) and the parameter to be calculated (mean).

Carrying out the appropriate calculations the proof is obtained. This demonstration will not be seen here, as advanced knowledge of mathematics is required.

The Fisher-Neyman factoring criterion in practice

In this sense, taking into account the above, the most important thing is to understand that there are tools to check certain properties. Properties that are undoubtedly important when doing statistical studies.

Why is it the most important? Because we don't usually do proofs to see if a statistic is enough. We just know that it is enough. For example, mathematicians have already shown that the mean is a sufficient statistic. Therefore, we do not have to prove it.

In conclusion, the idea is to know the tool for informational purposes to understand some important concepts in statistical studies.

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