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Is a minimal sufficient statistic unique?
If both T and S are minimal sufficient statistics, then by definition there is a one-to-one function ψ such that T = ψ(S); hence, the minimal sufficient statistic is unique in the sense that two statistics that are functions of each other can be treated as one statistic.
Are unbiased estimators are always a function of sufficient statistics?
Definition. for any other unbiased estimator. and conditioning any unbiased estimator on it. Further, by the Lehmann–Scheffé theorem, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVUE estimator.
How to figure out the statistic F ( x1 x2 xn )?
A statistic T = r(X1,X2,···,Xn) is sufficient if and only if the joint density can be factored as follows: f(x1,x2,···,xn|θ) = u(x1,x2,···,xn)v(r(x1,x2,···,xn),θ) (2) where u and v are non-negative functions.
When is the estimator of σ2 unbiased?
The estimator ˆθ is unbiased if its bias is zero, i.e., if E[ˆθ] = θ. Example 3.2 We saw in (2.3) and (2.4) that the sample variance S2 was not an unbiased estimator of σ2, whereas the sample quasivariance S ′ 2 was unbiased.
When to use the z distribution in Stat?
The \\(z\\) distribution is used on rare occasions when the population is normal and the population standard deviation is known. Note that for this course the one sample mean \\(z\\) test is optional; it used only in specific cases where the population is known to be normally distributed and when the population standard deviation (\\(\\sigma\\)) is known.
How to use t distribution to approximate sampling distribution?
In order to use the t distribution to approximate the sampling distribution either the sample size must be large (\\(\\ge\\ 30\\)) or the population must be known to be normally distributed. The possible combinations of null and alternative hypotheses are: