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I found a problem from some notes i found online, here is a screenshot: enter image description here

I am trying to understand this question, it seems this function they define as the LIP() function is basically the quartile/quantile function. And it seems the question says that the mean does not affect the LIP when we go from a uniform distribution to Log-Normal distribution. It looks like one has to do some kind of analytical mathematical algebraic or maybe calculus based manipulation.

So the means does not affect the 'shape' of the distribution it seems. Not sure how or what I should do to the Log-Normal Distribution function. SO need some help here.

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    $\begingroup$ No calculus is needed, nor is this a special fact about lognormal distributions. Just observe that the effect of $\mu$ is to multiply the unit of measurement of income by the factor $\exp(-\mu)$ and that the LIP does not depend on how income is measured. $\endgroup$
    – whuber ♦
    Commented Sep 29, 2018 at 18:30
  • $\begingroup$ OK, thanks for this information whuber, that this is not a special fact about log-normal distribution. For the Y-income distribution, i know i can break up the exponential into 2 parts, one with mean and another exponential with sigma*Z, but I am still don't understand your statement when you say: " Just observe that the effect of μ is to multiply the unit of measurement of income by the factor exp(−μ) " $\endgroup$
    – Palu
    Commented Sep 29, 2018 at 18:43

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The LIP is not quite equivalent to the quantile function, thanks to that phrase "... fraction of incomes that are below $\alpha$ times the $\beta$ quantile of incomes." The choice of a Uniform distribution as an example was unfortunate one, as it resulted in an LIP that was indeed equal to the same quantile as the value that resulted from the "below $\alpha$ times" part of the calculation.

However, if we, for example, choose an Exponential distribution for income with a mean of $1$, and calculate the LIP($1/2$,$1/2$) cutoff, we find that the cutoff is at the $29.3^{rd}$ percentile of the distribution - the $50^{th}$ percentile is at $0.693$, half of that is $0.347$, and $0.347$ happens to be at the $29.3^{rd}$ percentile of the standard exponential.

As @whuber has observed, all that having a nonzero $\mu$ does is rescale the lognormal variate, multiplying it by $\exp\{\mu\}$; consequently, the resulting LIP remains the same. This is what you want; the LIP should be unchanged regardless of whether income is measured in dollars or in thousands of dollars, for example.

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  • $\begingroup$ Thanks for the clarification about the LIP() function jbowman, i was wondering about it, was not fully confident about it. Just not sure how to mathematically demonstrate how the mean does not affect the the LIP when using the log-normal distribution. $\endgroup$
    – Palu
    Commented Sep 29, 2018 at 18:48
  • $\begingroup$ But in your own example jbowman, when you used the Exponential distribution, so the 50th percentile has a value on the x-axis of 0.693 and when we take half of that x-value we get 0.347, for this we get 29.3rd percentile which is not like the quartile of being at 25% proportion it is higher at 29.3% , so you demonstrated that the exponential does affect the LIP. I hope i am interpreting this correctly. But in the log-normal case this should not happen?, hence the log-normal is special in this way? $\endgroup$
    – Palu
    Commented Sep 29, 2018 at 18:52
  • $\begingroup$ The LIP will of course depend on the distribution, but not on scale parameters associated with the distribution. For an example of this, choose an Exponential distribution with a mean of 100 and repeat the exercise. The cutoff will still be at the $29.3^{rd}$ percentile; the $50^{th}$ percentile is at 69.3, half of that is 34.7, and 34.7 is at the $29.3^{rd}$ percentile of an Exponential(100) distribution. $\endgroup$
    – jbowman
    Commented Sep 29, 2018 at 19:51
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    $\begingroup$ That's because the shape didn't change, just the scale. It's akin to converting from dollars to cents; the percentage of people in poverty doesn't change just because you multiplied all the incomes by \$100. But if you change the shape, as we did by going from Lognormal to Exponential, then the LIP will change too. $\endgroup$
    – jbowman
    Commented Sep 29, 2018 at 19:53
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    $\begingroup$ Yes, that's correct! $\endgroup$
    – jbowman
    Commented Sep 29, 2018 at 21:07

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