Fractile Graphical Analysis Some Applications

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Fractile Graphical Analysis Some Applications

• Bodhisattva Sen.
• Statistics Department.

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Topics

• Motivation.
• Fractile Graphical Analysis concepts and
  properties.
• Applications - Banking sector examples.
• Some issues and discussion.

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Motivation

• Introduced by Mahalanobis(1960).
• X total expenditure per capita per household,
• Y fraction of expenditure on food articles.
• Data (X,Y) over two time points.
• Is the community economically well-off now?
• How are the poor people doing?

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Problem?

• How do you standardize the covariate (X)?
• How do make meaningful inference from the graphs
  when the covariate (X) is changing?

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To compare two regression functions when the
covariate (X) is getting transformed.

• Could have looked into g(x) E(YXx).
• Instead we work with m(t) E(YF(X)t), fractile
  graph (where F is the distribution function of
  X).

fractile graphs
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Alternatives?

• Transform everything to 1960 rupees (base year).
• Usual Standardization (with mean and standard
  deviation).
• Use some specific transformation like log(X)
  too naïve!

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WHY Fractile Graphical Analysis?

• Idea Regress Y on F(X) (nonparametric notion of
  standardization).
• Suppose f is any strictly increasing function.
  Then F(x) G(f(x)) where F and G are the
  distribution functions of X and f(X)
  respectively.
• Thus, E(YF(X)t) E(YG(f(X))t), i.e.,
  the fractile graphs remain invariant under
  strictly increasing transformations.

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Implications

• Very strong notion of standardization.
• The distribution of the covariate might be
  very different in the two populations, but still
  this standardization works!
• But F(X) is always Uniform(0,1).
• Thus, FGA does the required standardization of
  the covariate.

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Example I Interest rate distribution for loan
accounts

• YInterest rate vs Xloan amount (for different
  states, at different time points).
• Some surmises
• Bigger borrowers have received better terms!
• State wise variation in interest rate (important
  policy implications)!

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Example II Credit-Deposit ratio.

• XTotal Credit and
• Y Credit-Deposit Ratio.
• Has the credit-deposit ratio changed over the
  years over different states?

For the years - 1996 and 2003.
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Example III Sales and profit data for Private
Companies in India

• X total sales and Y of profit (of sales).
• How does the profitability change with changing
  size of the companies over years?

For the years - 1996 and 2003.
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Important Issues

• How to compute the fractile graphs?
• How to compare two fractile graphs?
• Can we extend it to the multiple covariate setup,
  i.e., when X(X1,X2,,Xd)? (its motivation and
  technical problems).
• (1) There is no standard notion of multivariate
  quantile.
• (2) Curse of dimensionality (sparseness of data
  points)!

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Discussion

• What can be its other applications?
• (Maybe in demography and epidemiology??)
• What are the other ways to solve the problem?
• Some interesting data sets!
• What next?
• Thank you!