so the pdf is $$f(x)=\frac{1}{2}\frac{1}{\sqrt{2\pi}}\exp^{-\frac{x^2}{2}}+\frac{1}{2}\frac{1}{\sqrt{2\pi}\sigma}\exp^{-\frac{(x-\mu)^2}{2\sigma^2}} $$ $\mu$ and $\sigma$ are unknown,prove that they don't have MLE
it is a gaussian mixture model I guess? And I know there are no analytic solve for the derivative of log likehood function. But the text seems to mean that there are no numerical solve either. And I really don't know any methodology for prove MLE don't exist
One way to prove that MLE doesn't exist is to show that the density or mass function doesn't have a global maxmium.
Here is the joint density function : $$ \boldsymbol{f}_{\boldsymbol{X}_1,..,\boldsymbol{X}_{\boldsymbol{n}}}\left( \boldsymbol{x}_1,...,\boldsymbol{x}_{\boldsymbol{n}}|\boldsymbol{\mu },\boldsymbol{\sigma }^2 \right) =\prod_{\boldsymbol{i}=1}^{\boldsymbol{n}}{\left( \frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}}\exp \left\{ -\frac{\boldsymbol{x}_{\boldsymbol{i}}^{2}}{2} \right\} +\frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}\boldsymbol{\sigma }}\exp \left\{ -\frac{\left( \boldsymbol{x}_{\boldsymbol{i}}^{2}-\boldsymbol{\mu } \right) ^2}{2\boldsymbol{\sigma }^2} \right\} \right)} $$
After observing the value of X, we have the Log likelihood:
$$ \boldsymbol{L}\left( \boldsymbol{\mu },\boldsymbol{\sigma }^2|\boldsymbol{x}_1,...,\boldsymbol{x}_{\boldsymbol{n}} \right) =\sum_{\boldsymbol{k}=1}^{\boldsymbol{n}}{\ln \left( \frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}}\exp \left\{ -\frac{\boldsymbol{x}_{\boldsymbol{i}}^{2}}{2} \right\} +\frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}\boldsymbol{\sigma }}\exp \left\{ -\frac{\left( \boldsymbol{x}_{\boldsymbol{i}}^{2}-\boldsymbol{\mu } \right) ^2}{2\boldsymbol{\sigma }^2} \right\} \right)} $$ if μ happens to be the value of some x, for example, μ = x1, then we have: $$ \begin{align} \boldsymbol{L}=&\,\,\underset{\boldsymbol{part}\,\,\boldsymbol{A}}{\underbrace{\ln \left( \frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}}\exp \left\{ -\frac{\boldsymbol{x}_{\boldsymbol{i}}^{2}}{2} \right\} +\frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}\boldsymbol{\sigma }} \right) }} \\ &\,\, +\underset{\boldsymbol{part}\,\,\boldsymbol{B}}{\underbrace{\sum_{\boldsymbol{k}=2}^{\boldsymbol{n}}{\ln \left( \frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}}\exp \left\{ -\frac{\boldsymbol{x}_{\boldsymbol{i}}^{2}}{2} \right\} +\frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}\boldsymbol{\sigma }}\exp \left\{ -\frac{\left( \boldsymbol{x}_{\boldsymbol{i}}^{2}-\boldsymbol{x}_1 \right) ^2}{2\boldsymbol{\sigma }^2} \right\} \right)}}} \end{align}\\ $$
if we set σ → 0+: $$ \begin{align} \boldsymbol{part}\,\,\boldsymbol{A}&\rightarrow +\infty \\ \boldsymbol{part}\,\,\boldsymbol{B}&\rightarrow \sum_{\boldsymbol{k}=2}^{\boldsymbol{n}}{\ln \left( \frac{1}{2}\frac{1}{\sqrt{2\boldsymbol{\pi }}}\exp \left\{ -\frac{\boldsymbol{x}_{\boldsymbol{i}}^{2}}{2} \right\} \right)} \end{align}\\ $$
Thus you can say the Log likelihood doesn't have a global maxmium and the density function doesn't have a global maxmium and MLE equivalently.