Some Interesting Limits

While working on the Lorenz curve paper, we came up with several interesting limits involving CDF, Quantile and Lorenz functions which I haven’t seen in references.

Let  i=1,...,N and points z_i s are chosen in a way that F(z_i)=i/N where N is the number of observations. Then we have the following limit as N \rightarrow \infty.

  •  N\sum_{i=0}^{N-1} \big\{\widehat{F} (z_{i+1})-F(z_{i+1})-\widehat{F} (z_{i})+F(z_i))\big\}^2\rightarrow 1

It can be shown that this is equivalent to this limit

  •  [\widehat {\bold c}-{F}(\bold z)]'\bold\Omega_1^{-1} [\widehat {\bold c}-{F}(\bold z)]\rightarrow 1

where \widehat {\bold c}=\big\{\widehat{F} (z_i),i=1,...,N\big\} and \Omega_1 is the covariance matrix of \widehat {\bold c}.

 

Now let c_i=i/N. Then we can have the following limit involving quantiles

  •  N\sum_{i=0}^{N-1} \big\{\dfrac{\widehat{F}^{-1} (c_{i+1})-{F}^{-1}(c_i+1)}{G_{i+1}}-\dfrac{\widehat{F}^{-1} (c_{i})-F^-1(c_i))}{G_i}\big\}^2 \rightarrow 1

where G_i is derivative of the quantile function with respect to c_i. This formula can also be written in this  matrix form

  •  [\widehat {\bold z}-{F}^{-1}(\bold c)]'\bold\Omega_2^{-1}[\widehat {\bold z}-{F}^{-1}(\bold c)] \rightarrow 1

where \widehat {z_i}=\widehat{F^{-1}} (c_i)  and  \Omega_2 is the covariance matrix of empirical quantiles.

However, such quadratic forms do not always converge to one. For example, for a Lorenz curve it seems that we have

  •  [\widetilde{\bold y} -L(\bold c)]'\bold\Omega_L^{-1}[\widetilde{\bold y} -L(\bold c)] \rightarrow 1.73

This is because \Omega_L becomes singular when N goes towards infinity. I don’t know how to prove this limit however!

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