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How do you show minimal sufficiency?
To show minimal sufficiency, suppose X’ and Y’ were another sample from the same distribution. If the ratio of pdfs, L(θ;x,y)L(θ;x’,y’) is independent of θ iff T1(X,Y)=T1(X’,Y’) and T2(X,Y)=T2(X’,Y’), then (T1,T2) is minimal sufficient.
How do you show that something is a sufficient statistic?
For observations X1,…,Xn, the statistic T, is called a sufficient statistic if equation (1) is a function of the values, t, of the statistic and does not depend on the value of the parameter θ. Thus, by the law of total probability Pθ{X1 = x1,…,Xn = xn} = P{X1 = x1,…,Xn = xn|T(X) = T(x)}Pθ{T(X) = T(x)}.
Is minimal sufficient statistic complete?
A minimal sufficient statistic is not necessarily complete. In this example, X1,…,Xn is a random sample from uniform(θ,θ +1), θ ∈ 勿, and we showed that T = (X(1),X(n)) is the minimal sufficient statistic for θ.
What is sufficiency of an estimator?
Sufficient estimators exist when one can reduce the dimensionality of the observed data without loss of information. Thus sufficiency refers to how well an estimator utilizes the information in the sample relative to the postulated statistical model.
Why is sufficiency important in statistics?
1 Introduction. Statistical sufficiency is a concept in the theory of statistical inference that is meant to capture an intuitive notion of summarizing a large and possibly complex set of data by relatively few summary numbers that carry the relevant information in the larger data set.
What do you mean by sufficiency in statistics?
In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”.
What is the sufficiency principle?
Sufficiency. SUFFICIENCY PRINCIPLE: If T(X) is a sufficient statistic for θ, then any inference about θ should depend on the sample X only through the value T(X). That is, if x and y are two points such that T(x) = T(y), then any inference about θ should be the same whether X = x or X = y is observed.
Is a sufficient statistic unbiased estimator?
Any estimator of the form U = h(T) of a complete and sufficient statistic T is the unique unbiased estimator based on T of its expectation. Hence, if T is complete and sufficient, U = h(T) is the MVUE of its expectation.