Logistic Distribution

[vasvet]

Hi All,

If .the logistic distribution is defined as it corresponding quantile function:


How can I show that q is strictly increasing and compute logistic distribution function and it's density function?

How to compute VaR and ES of r.v. with logistic distribution?

Thank you for your help

 

[Detective]

Did you mean to write F inverse = ln((alpha)/(1-alpha))? I.e. the logit probability function?

To prove g(x) is strictly increasing, show g’(x) > 0 for all x.

To get CDF you can find the inverse of the function you have, take derivative you get PDF.

VAR: By definition VAR is based on the quantile. https://en.m.wikipedia.org/wiki/Value_at_risk

See mathematical definition

ES: You’ll need to take an integral.

https://en.m.wikipedia.org/wiki/Expected_shortfall

See formal definition.

 

[sapozan]

Thank you @Detective . The function appears to be correct as it is presentes as a quantile function. Would ah inverse of an inverse just be an actual CDF ?
In this case F=ln((alpha)/(1-alpha))

Am I correct here ?

 

[Detective]

Thank you @Detective . The function appears to be correct as it is presentes as a quantile function. Would ah inverse of an inverse just be an actual CDF ?
In this case F=ln((alpha)/(1-alpha))

Am I correct here ?

These are well known functions. Inverse of logit function is sigmoid function.

https://en.m.wikipedia.org/wiki/Sigmoid_function

 

[sapozan]

Awesome thank you @Detective,

Is it possible to derive the actual distribution function from the above quantile function and not the other way round?

I am not too sure if the above formula is exactly correct so i will have to double check this.

 

[Detective]

Awesome thank you @Detective,

Is it possible to derive the actual distribution function from the above quantile function and not the other way round?

I am not too sure if the above formula is exactly correct so i will have to double check this.

Not sure if I follow, you have a function,

y= f(x) = ln(x/(1-x)), with 0<x<1 and you want to find inverse?

The way I was taught was to replace “y” with “x” and then solve for y. Getting rid of LN will involve taking EXP.

 

[sapozan]

Thank you @Detective, will give it a go