Final answer:
Using the Fisher-Neyman factorization theorem, we can demonstrate that the sample mean μ is a sufficient estimator for the parameter θ in an exponential population. The joint probability density function of a random sample from an exponential population factorizes into a part that is a function of μ and θ, and a part that is independent of θ, which confirms the sufficiency of the sample mean.
Step-by-step explanation:
To show that μ is a sufficient estimator of the parameter θ in an exponential population, we can use the concept of sufficiency defined by the Fisher-Neyman factorization theorem. This theorem states that an estimator is sufficient if the joint probability density function (PDF) can be factorized into two parts: one part that is dependent on the sample data and the parameter, and another part that is dependent only on the sample data.
For an exponential population with rate parameter θ (where θ is the inverse of the mean), the probability density function is given by:
f(x|θ) = θ * e^(-θ2 * x) for x ≥ 0
If X₁, X₂, ..., Xn is a random sample from this population, then the joint PDF of the sample is given by:
L(θ) = θ^n * e^(-θ * ∑Xi)
where ∑Xi is the sum of the sample observations. This can be further factorized into:
L(θ) = θ^n * e^(-θ * n * μ) * e^(-θ * (∑Xi - n * μ))
Where μ is the sample mean. As the last part of the expression is not dependent on θ, we can see that the sample mean μ is a sufficient statistic for the parameter θ in the exponential distribution, according to the factorization theorem.