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More on Lorenz Curve Interpolation

In the previous post the interpolation was demonstrated using a simulated data. Here we do an application to a real data set. The Data is for Urban China 2004-2005 obtained from PovcalNet website. In what follows we plot the Spline-interpolated curves vs their estimation using GB2 Continue reading

Lorenz Curves: Spline Interpolation

The standard practice of Lorenz curve interpolation relies on linear interpolations where each point is connected to the next one using a straight line. There are of course other ways of interpolation and a natural one is to use splines.  In this post, I introduce an existing package in “R” that can be used to interpolate Lorenz curves and to compute the corresponding CDF and PDF functions. Continue reading

Existence and Uniqueness of PPP Indexes

Most indexes used for computation of purchasing power parities (PPP) are defined in this way: Suppose there are  j=1,...,m  countries and  i=1,...,n  commodities. PPP for jth-country is defined as a weighted average of prices of commodities in that country deflated by  world average price of the respected commodities. Continue reading

Lorenz Curve Paper

I am happy to report that the Lorenz curve paper (joint with Bill Griffiths) which I consider one of my best papers is finalised and submitted. I will discuss the the paper and its results in future posts. For now, here is a link to the paper itself.

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