Archive for November, 2015

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!

Back Again

I am back again with the promise of regular updates starting with a good news. After a very lengthy process, this paper joint with Prasada Rao is officially accepted for publication in the Journal of Econometrics. Continue reading

Red Back in our backyard

We were doing this experiment with my daughter but if you look closely, you can see something is resisting not to come out of the tap with the water.

RedBack2

It turned out to be a redback spider which I hadn’t seen before !

RedBack