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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.!

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  • $\begingroup$ 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
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    $\begingroup$ @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
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    $\begingroup$ @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
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    $\begingroup$ @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
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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.

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  • $\begingroup$ 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

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