Prove that ML estimate of a function of a parameter, $g(\theta)$ is the function of its ML estimate $g(\hat{\theta})$ As titles say, given $\hat{\theta}$ is the Maximum Likelihood estimate of a parameter $\theta$, how to prove that Maximum Likelihood estimate of $g(\theta)$ is $g(\hat{\theta})$. Additionally, is this property applied to Bayesian estimate? I think given that $\hat{\theta}$ is the Maximum Likelihood estimate of function $f(x_1, x_2, ..., x_n)$, then $\frac{\partial f(x_1, x_2, ..., x_n)}{\partial \theta}|_{\theta=\hat{\theta}}=0$. Then we can continue proving that $\frac{\partial g(f(x_1, x_2, ..., x_n))}{\partial g(\theta)}|_{g(\theta)=g(\hat{\theta})}=0$
Question about Casella and Berger's proof of MLE invariance In Casella and Berger, p. 320, they have a proof of the invariance of the MLE. Let $g: \theta \mapsto \eta$ be a function. They define the induced likelihood as  $$ L^*(\eta \mid X) = \sup_{\{\theta: g(\theta) = \eta\}} L(\theta \mid X). $$  This ensures that $g$ is one-to-one with respect to the likelihood functions because if $g$ is not one-to-one, there may be multiple values of $\theta$ that map to a given $g(\theta) = \eta$.  Then they note  $$ \sup_{\eta} L^*(\eta \mid X) = \sup_{\eta} \sup_{\{\theta: g(\theta) = \eta\}} L(\theta \mid X) = \sup_{\theta} L(\theta \mid X). $$  The first equality just applies the definition of the MLE of $\eta$. However, the next inequality confuses me. They write,     The second inequality follows because the iterated maximization is equal to the unconditional maximization over $\theta$...   Can someone justify this claim or provide some intuition if the claim is supposed to obviously follow from definitions?
Show that the locus formed by z in the equation $z^3 + iz = 1$  never crosses the co-ordinate axes in the Argand plane. Further show that $$|z| = \sqrt{\dfrac{-\Im(z)}{2\Re(z)\Im(z) +1}}$$  Concept: $z=x+iy$ $z^3=(x+iy)^3=x^3-3xy^2+i(3x^2y-y^3)$ $iz=-y+ix$ Combined equation is $x^3-3xy^2-y-1+i(3x^2y-y^3+x)=0$ How do we proceed from here onward? How do I easily switch between windows rather than applications in GNOME Shell? Novel where pyramids "grow" and change Earth's magnetic field? I read this in (most likely) the 1980s or 1990s. The protagonists found what they believed were alien spacecraft; over time, they &quot;morphed&quot; into Mesoamerican-style pyramids (stepped, not smooth like Egyptian pyramids). Each time they &quot;grew&quot;, an area where electrical equipment either failed or was unreliable grew; the pyramid itself became a new magnetic pole. It turned out that there were a dozen of these objects, spaced to cover the entire world from the corners of an icosahedron; the final morph of the pyramids caused the fields of electrical &quot;disablement&quot; to cover the entire world. I'm looking for a true 360 degree camera, so that's both vertical and horizontal 360 degrees. I've seen some clip-ons for the iPhone, but am really not wanting to by an iPhone 4S for this.  What I would like to see: - decent resolution: at least 1920x1080. - and 30fps  I can't settle for an alternative that's not vertically 360 degrees (or at least much more than 180 degrees).  I've seen the ball camera, which isn't out yet, but would be really well suited for my case, given it does video as well. How can I verify a Google authentication access token?  I need to somehow query Google and ask: Is [given access token] valid for the [example@example.com] Google account?  Short version: It's clear how an access token supplied through the  can be used to then request data from a range of Google services. It is not clear how to check if a given access token is valid for a given Google account. I'd like to know how.  Long version: I'm developing an API that uses token-based authentication. A token will be returned upon provision of a valid username+password or upon provision of a third-party token from any one of N verifiable services.  One of the third-party services will be Google, allowing a user to authenticate against my service using their Google account. This will later be extended to include Yahoo accounts, trusted OpenID providers and so on.  Schematic example of Google-based access:    The 'API' entity is under my full control. The 'public interface' entity is any web- or desktop-based app. Some public interfaces are under my control, others will not be and others still I may never even know about.  Therefore I cannot trust the token supplied to the API in step 3. This will be supplied along with the corresponding Google account email address.  I need to somehow query Google and ask: Is this access token valid for example@example.com?  In this case, example@example.com is the Google account unique identifier - the email address someone uses to log in to their Google account. This cannot be assumed to be a Gmail address - someone can have a Google account without having a Gmail account.  The Google documentation clearly states how, with an access token, data can be retrieved from a number of Google services. Nothing seems to state how you can check if a given access token is valid in the first place.  Update The token is valid for N Google services. I can't try a token against a Google service as means of verifying it as I won't know which subset of all Google's services a given user actually uses.  Furthermore, I'll never be using the Google authentication access token to access any Google services, merely as a means of verifying a supposed Google user actually is who they say they are. If there is another way of doing this I'm happy to try. Get Last Friday of Month in Java Do plugin files have to follow a specific convention to be "picked up" by WordPress? array package: problem with >{declarations} in preamble: Extra alignment tab has been changed to \cr I have a document that uses both the longtable and the array package. At first I ascribed the problem to an interaction of both. After testing, it seems array is the only culprit.  Here's a minimal example.  \documentclass{article} \usepackage{array} \begin{document}  \begin{tabular}{p{0.4\textwidth}&gt;{\raggedright}p{0.4\textwidth}} 1 &amp; a \\ 2 &amp; b \\ 3 &amp; c \\ \end{tabular} \end{document}   It produces the error  ! Extra alignment tab has been changed to \cr. What are the criteria for questions to be selected for Hot Network Questions? Extra blue line appearing in the top bar (Firefox only) Whenever we click on the Inbox or the Achievements in the top bar, an extra blue line appears near by them. This is reproducible on Stack Overflow and Anime &amp; Manga(as I use only those) but not on Meta Stack Overflow. Here is a snapshot of that.    I'm using Firefox 25.0.1 on Win7 Pro.  Edit: Just to be double sure, I restarted the FF and still I'm able to see it. Here is another snapshot, with the 2 distinct sites. What is the meaning of this "let there be light" joke? Expanding code in etoolbox's \ifblank I am trying to use the etoolbox \ifblank (and its cousin \notblank), but I want to expand any macros in its test statement.  For example:  \documentclass{article} \usepackage{etoolbox}  \begin{document}  \newtoggle{longformt} \newcommand{\longform}{\toggletrue{longformt}} \newcommand{\iflong}[2]{\iftoggle{longformt}{#1}{#2}}  \newcommand{\quoteifnotblank}[1]{%     \notblank{#1}{``#1''}{}% }  % Try commenting this line. \longform  This should be completely blank if the form is short: \quoteifnotblank{\iflong{Some long quoted text}{}}  \end{document}   The problem here is that \notblank sees its argument of \iflong{Some long quoted text}{} as a string, not as code to be expanded, so the quotes always appear.  How do I expand arguments before they are passed to \notblank? How to reverse name and first name? With biblatex and verbose-trad2 style, I would like to reverse the order of the last name and the first name for inline references.  The default behavior is:     John DOE, Title, place : publisher, year   for inline references (in my case, in a footnote with footcite command) and:     DOE, John, Title, place : publisher, year   in the bibliography at the end of the document.  I would like to have both inline and final references in the following format (similar to the bibiography style).      DOE, John, Title, place : publisher, year   P-S : There are some close questions (like ) but only about the final bibliography (and no inline references). \DeclareNameAlias{sortname}{last-first} works only for the ending references (not inline ones). WPF Binding to variable / DependencyProperty How can I identify an anonymous inner class in a NotSerializableException Replacements for switch statement in Python? Why is LaTeX3 taking so long to come out? I feel as though I have been hearing about the LaTeX3 project for a very long time.   Why is it taking so long for it to come into fruition? Are there major technical difficulties, or is it purely a lack of developers that is causing a bottleneck? What are the basics of commands in Minecraft Java Edition? Sign of the vector (especially $g$) while calculating I'm a high school student and I'm studying Newton's second Law. While my teacher is calculating a net force of an object he always treat all vectors as positive numbers. I think what he meant is the magnitude of the vector which is always positive, but the problem is that he didn't use the magnitude symbol, for example like $||Fg||$. He said the acceleration due to gravity , g, is positive, but shouldn't it be negative since it is pointing down. This also cause a problem: every time when I calculate the net force I can't add all forces together, instead I need to choose to use minus sign when the vector is pointing down. So in general which way is correct? Top bar search doesn't pick up subdomains I wanted to switch to  in the handy dandy site switcher in the new top bar. Entering "cook" brings up nothing:     ... so I have to remember that the boutique title of the site is "Seasoned Advice":     ... in which case the site switcher also thinks that maybe I was thinking of heading over to  to see what's up. So the site switcher searches site names and descriptions, but not subdomains. This is unexpected behavior - I believe it should search all three.