How to differentiate the distribution function of lognormal distribution with respect to its parameters? What solution will we get? I know if differentiate wrt variable, we will get density function.!

enter image description here

  • $\begingroup$ Please present your notation. For the standard notation you will find the equations at mathworld.wolfram.com. $\endgroup$ – Semoi Aug 12 '17 at 10:42
  • 1
    $\begingroup$ @Semoi yes I have added now..Sorry i dont know how to write equations in this page.. so I have uploaded image of equations. $\endgroup$ – Huchesh Aug 12 '17 at 12:28
  • $\begingroup$ @Huchesh what is the context for your question? $\endgroup$ – user137329 Aug 12 '17 at 12:52
  • 1
    $\begingroup$ @phdmba7of12 actually i am using truncated lognormal distribution (TLD). In truncated distribution (density of the distribution divided by the distribution function of the distribution at the specific point) , denominator is distribution function which is in the form of cumulative distribution function in case of lognormal distribution. And i want to estimate the parameters of the LTD but due to the not explicit form of likelihood equations (LE's), i have to use numerical methods like newton raphson method. In NR method, i have to take the derivatives of LE's. conti.. $\endgroup$ – Huchesh Aug 15 '17 at 3:54
  • 1
    $\begingroup$ @phdmba7of12 So i am unable to differentiate that distribution function with respect to parameters. I hope, u got my point. $\endgroup$ – Huchesh Aug 15 '17 at 3:55

Well, as you stated the cumulative distribution function is $$F(x) = \int_{-\infty}^x f(z) dz = \Phi\big[\frac{\ln(x)-\mu}{\sqrt{2}\sigma}\big]$$

Now I would differentiate the error fct using the chain rule $$\frac{\partial}{\partial y}\Phi[z] = \frac{\partial \Phi[z]}{\partial z} \cdot \frac{\partial z}{\partial y} = f(z) \cdot \frac{\partial z}{\partial y}$$ So all you need to do is to calculate the partial derivatives \begin{align} \frac{\partial }{\partial \mu} \big[\frac{\ln(x)-\mu}{\sqrt{2}\sigma}\big] &= ... \\ \frac{\partial }{\partial \sigma} \big[\frac{\ln(x)-\mu}{\sqrt{2}\sigma}\big] &= ... \end{align} which is straight forward.

  • $\begingroup$ can u do in detail please?? $\endgroup$ – Huchesh Aug 15 '17 at 3:42
  • $\begingroup$ what second order differentiation of the above equations?? $\endgroup$ – Huchesh Aug 15 '17 at 14:19
  • $\begingroup$ Well, $\frac{\partial}{\partial \mu} \mu = 1$ and $\frac{\partial}{\partial \sigma} \sigma^{-1} = -\sigma^{-2}$. $\endgroup$ – Semoi Aug 15 '17 at 17:31
  • $\begingroup$ I am asking what is second order differentiation of distribution function with respect to parameters? I have mentioned one of above comments why am doing this.? Using the Newton Raphson method, i am estimating the parameters so i have to form a hassian matrix which second order derivative matrix. $\endgroup$ – Huchesh Aug 15 '17 at 18:03
  • $\begingroup$ logl=-Nlog σ\left(2πright)-∑i filog xi-0.5∑ifn fi eft((log xi-μ/σ2right)-Nlog Φ eft((log T-μ/σright) $\endgroup$ – Huchesh Aug 16 '17 at 4:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.