In CDF we use integrals for continuous data sets. How can we calculate it for a discrete data set ?
Given a random variable X, its cdf is the function F(x) = Prob(X
Given a random variable X, its cdf is the function F(x) = Prob(X
You may find the following link useful. It described also a method for calculating the confidence interval of the calculated CDF.
http://www.epa.gov/nhrlsup1/arm/documents/m02.pdf
Go once through the Glivenko-Cantelli Theorem on empirical distribution function.
Vahid Nouri, I have an hetherodox answer: If you have N values of variable and estimate U (mean) as the average of those N values, then you are assuming that each one has frequence 1/N and average value Xi. Then you may build the Lorenz curve for your N points. The derivate of a continuous Lorenz curve ordered from high to low values gives the CDF (x) = Prob(X >= x) where X is variable "in medias" and x is cummulative frequence, 0
@Emiio Chaves: The Lorentz curve is a transform of the cumulative distribution function which (a) is only defined for distributions with finite mean, and (b) is invariant under scale transformations. This means that not every distribution has a Lorentz curve, and that all distributions got by scale transformation from one another have the same Lorentz curve.
Added after subsequent comment by Emilio - sorry for my misleading misprints!!! I meant Lorenz curve, not Lorentz curve. https://en.wikipedia.org/wiki/Lorenz_curve
Hi Richard. There is a 1905 Nobel prize of physics Hendrik Antoon Lorentz, but I refer to Max Otto Lorenz, a less important man that is mentioned in econometric distributions and lived in US. I do not have the mathematic level to speak about Lorentz work, but I have researched about Otto Lorenz, who only wrote one confusing paper about statistics and became famous for that paper where he proposed a simple curve named Lorenz Curve that plots the cummulative distribution of income versus cummulative population for ascending order of incomes, based on Prusian data. I use a different version for descending order that may be expressed as an exponential function of a function:L(Xac) = Xac ^ F(X ac). ... where L(X) is the fraction of the total distributed variable and Xac is the accumulated fraction of receptors people. Give a look to the short document I sent to RG and check the simple context and method I deal with. Thanks, emilio
Sorry, that was just a stupid misprint by me! I too was referring to the Lorenz curve of the economist Lorenz. Not to the (Dutch) physicist Lorentz. I will correct my earlier posting.
Thanks for your correction Richard, but do not worry: I made the same mistake 20 years ago in a paper, nobody noticed it. That experience is the reason I recognized the problem at once. Let´s laugh about it, emilio
Richard, you say that "This means that not every distribution has a Lorenz curve". I think different: if there is distribution there is something to share among some receptors population that are asigned to it. In my opinion any distribution data of a measured variable with counted population must have a Lorenz curve. And this is a nice adimensional graph of them that is central in analysis because it gives the value of the integral of CDF for all fractions from zero to one, or total population fraction. Of course it is independent of the media real value.
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