Rao-Blackwell Theorem

 Rao-Blackwell Theorem 

Calyampudi Radhakrishna Rao was born on 10 September 1920, in a 

small town in south India called Huvvinna Hadagalli, then in the integrated 

Madras Province of British India but now in the state of Karnataka. 

C. R. Rao has received numerous  awards for his precious work in statistics, including the Guy Medal 

in Silver of the Royal Statistical Society, London (1965), the Megnadh 

Saha Medal of the Indian National Science Academy (1969), the Jagdish 

Chandra Bose Gold Medal and cash award (1979), the Silver Plate bearing 

the monogram of the Andhra Pradesh Academy of Sciences (1984), the S. 

S. Wilks Medal of the American Statistical Association (1989), and the 

Mahalanobis Birth Centenary Gold Medal awarded by the Indian Science 

Congress (1996). The government of India honored C. R. Rao in 1968 by 

awarding him the title of Padma Bhushan, a high civilian award.

Let X and Y be random variables such that E(Y) = μ and var (Y) = σ² > 0 

Let E (Y |X = x) =  φ(x), then 

i) E [φ(x)] = p and (ii) var [φ(x)] ≤ var (Y) 

Thus, Rao-Blackwell theorem enables us to obtain MVUE through sufficient 

statistic. If a sufficient estimator exists for a parameter, then in our search for 

MVUE we may restrict ourselves to functions of the sufficient statistic. 

The theorem can be stated slightly different way as follows:

Let U = U(xl, x2, ... , xn) be an unbiased estimator of parameter  γ(θ) and let T 

= T(xl, x2, . . . , x,) be sufficient statistic for γ(θ). Consider the function φ(T) 

of the sufficient statistic defined as φ(T) = E (Y I T = t) which is independent 

of θ(since T is sufficient for γ(θ) . Then Eφ(T) =  γ(θ) and var φ(T) ≤ var (U).  

This result implies that starting with an unbiased estimator U, we can improve 

upon it by defining a function φ (T) of the sufficient statistic given as  φ(T)=

E (Y 1 T = t). This technique of obtaining improved estimator is called  

Blackwellisation. 

If in addition, the sufficient statistic T is also complete, then the estimator 

φ(T) discussed above will not only be an improved estimator over U but also 

the 'best (unique)' estimator.

The theorem is named after  Calyampudi Radhakrishna Rao and David Blackwell. The process of transforming an estimator using the Rao–Blackwell theorem is sometimes called Rao–Blackwellization. The transformed estimator is called the Rao–Blackwell estimator.

The Rao-Blackwell theory has many applications in statistics, including estimation of prediction error and producing estimates from sample survey data. For example, observations in adaptive sampling are found in sequence,each new observation depends on one or more characteristics from prior observations. Improved estimators can also be found by taking an average of estimators over every possible order.

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