3 Kernels and Their Symmetry,U Statistic

Mr Taranga Mukherjee

1 A short recap

So far we have learnt the following

Functionals are the analogues of parameter of interest. Statistical functionals are the analogues of statistic.

Functionals are linear and non linear.

Functionals and statistical functionals form the basis for nonparametric inference.

2 Inferential problems in nonparametrics

As in the parametric inference, we have three di erent types of inferential problems:

i.Estimation

ii.Hypothesis Testing

iii. Con dence Interval estimation

In this module we start with nonparametric estimation.

3 Estimability

We have already introduced functional and a statistical functional. Suppose X1; X2; :::; Xn are iid observations from F and F is a functional de ned on F, the class of all absolutely continuous DFs.

Then F is said to be estimable if there exists a (X1; X2; :::; Xk),k n, such that, EF f (X1; X2; :::; Xk)g = F , for all F 2 F.

If the above holds for some   and k,  F is called a regular functional is called a kernel for estimation of F . Minimum value of k, ensuring the above is called the degree of the kernel.

4 Sum / Di erence of the kernels

If i is a kernel of degree ki for i(F ); i = 1; 2, then we are interested to know degree of 1 2. Observe that 1 2 is a kernel for 1(F ) 2(F ) Few observations can be common between 1 and 2.

5 Product of kernels

If i is a kernel of degree ki for i(F ); i = 1; 2, then we are interested to know the degree of 1 2. Suppose there are some common observations in 1 and 2. Then 1 and 2 are, in general, dependent. Thus product of them is not a kernel for 1(F ) 2(F ). Suppose 1(F ) 2(F ) is the functional of interest. Then 1 2 is a kernel for 1(F ) 2(F ), if they are based on di erent sets of observations. Assume that the sets of observations corresponding to k1 and k2 are di erent Then 1 2 involves k1 + k2 distinct observations. Degree of 1 2 is k1 + k2.

2 Median, quantiles or any function of them does not depend on the order of the data and hence are symmetric kernels.

3 I(x1 + x2 > 1) is a symmetric kernel.

Asymmetric kernels

A kernel (x1; x2; :::; xk) is asymmetric if it is not permutation invariant, i.e. (xi1 ; xi2 ; :::; xik ) and (x1; x2; :::; xk) are not the same for some permutation (i1; i2; :::; ik) of f1; 2; ::::; kg.

1    (x1; x2) = x21     x1x2 is an asymmetric kernel as  (x1; x2) 6=   (x2; x1).

2  x1 x2 and x1  are further examples of asymmetric kernels.x2

3 As another example, (x1; x2; x3) = x21 +x22 +x23 x1x3 is asymmetric as (x1; x2; x3) = (x3; x2; x1) but (x1; x2; x3) 6= (x3; x1; x2).

A useful result

Result:  For every regular functional, there exists a symmetric kernel.

Proof: Suppose X1; ::; Xn are iid observations from F and F is a regular functional. If (x1; x2; :::; xk) is an asymmetric kernel then E (x1; x2; :::; xk) = F . As the observations are iid,E (Xi1 ; Xi2 ; :::; Xik ) =  F for any permutation (i1; i2; :::; ik) of f1; 2; ::::; kg. We construct the symmetric function s(x1; x2; :::; xk) = k1! Pi1;i2;:::;ik (xi1 ; xi2 ; :::; xik ). Then E s(X1; X2; :::; Xk) =  F .

Without any loss of generality, kernels can be taken as symmetric.

8 U statistic

We consider the set up F=class of all absolutely continuous DFs and F = P (X1 a), a is known. We observe that F = EI(X1 a). This suggests to take (X1) = I(X1 a), as a kernel is a symmetric kernel of degree k = 1. Then the corresponding U statistic is:

with degree 2 and for the second component, a kernel is X1X2 with degree 2. For F , the kernel is the sum X2, which has degree 1. 1

We at once observe the following:

The kernel is a sum of two kernels, each of degree 2. But the resultant kernel is of degree

1. Thus the degree of the sum of two kernels can be strictly less than 2. The inequality can be strict in Result. Again, one of the kernels is symmetric and the other asymmetric. But the resultant kernel is symmetric. In particular the sum of two symmetric kernels are always symmetric.

Example 6

We consider the set up F=class of all bivariate absolutely continuous DFs. (Xi; Yi); i = 1; 2; ::; n are iid observations from F 2 F and F = EF (X1Y1); V ar(X + Y ).

De ne a new variable Zi = XiYi; i = 1; 2; ::; n. Then Zi are iid observations from some univariate absolutely continuous DF G. Thus F reduces to G = EG(Z1). Then the corre-