Statement: if $\hat{\theta}$ is the MLE of some parameter $\theta$ then asymptotically $\hat{\theta} \sim \mathcal{N}(\theta, I_\theta^{-1}),$ where $I_\theta$ is the fisher information matrix.   My question is, what if $\hat{\theta}$ is a biased estimator of $\theta$? Surely $\mu$ will not be $\theta$ then.
I can follow the proofs in which the asymptotic normal-distribution of a maximum likelihood estimator $\tilde{\theta}_n$ is derived.   however, does this already imply that the maximum likelihood estimates are asymptotically unbiased, i.e. do we have $$E(\tilde{\theta}_n) \to \theta \text{ as } n \to \infty?$$  Since I am aware that in general it is not true that convergence in distribution implies convergence in moments, an explanation would be nice.  The result is often stated e.g.  as "...it means that the bias of the maximum likelihood estimator is equal to zero up to the order $n^{-1/2}$")  1.) Are  some kind of regularity conditions from the mle theory used to establish this result?  2.) Or is a $\sqrt{n}$-convergence of an estimator (to a normal distribution) in general already enough to establish convergence of its moments?  Note: the Wikipedia article mentions Cox, David R.; Snell, E. Joyce (1968).  , as a source where the order of the bias is derived (formula (12) or (20)).   However in this paper I can't follow the arguments 100%, since their Taylor approximation of $L'(\widehat{\beta})$ is lacking the remainder term. What is the argument used here to discard it completely?
How to prove that homomorphism from field to ring is injective or zero? Possible Duplicate:          $f:K\rightarrow R$ is a homomorphism from field to ring. To prove is that it is injective or zero. A uniformly continuous function maps bounded set to bounded sets I am trying to prove the following:  If $A\subset\mathbb R$ is bounded and $\,f:A\to \mathbb R\,$ is uniformly continuous, then $f[A]$ is bounded.  Could you check my proof?  Let $A \subseteq [-K,K]\subseteq \mathbb R$. Let $\varepsilon = 1$. If $f$ is uniformly continuous there is $\delta$ wuth $|x-y| &lt;\delta$ imply that $|f(x)-f(y)|&lt;1$ for all $x,y\in A$. Let $a \in A$. Then because $A$ is bounded there is a finite number of balls $B(a_n,\delta)$ that cover $A$. Let the number be $N$. Then $f(a)-N \le f(x) \le f(a) + N$ for all $x\in A$. Is it possible to run Microsoft Office 2010? How do I change my username? How do I redirect in a class sub-classing the EventSubscriber class? Find the slope of tangent line to the curve at the point $(1, \pi/2)$  the equation is  $$\sin(xy) = x$$  The right answer was = Slope is infinite   My answer was taking the derivative and I end up in this equation   $$ (1-y\cos(xy))/ x\cos(xy)  $$  I substitute with the points and the calculator showed an error, Does the error means infinite ? Handling data in a PHP JSON Object Trends data from Twitter Search API in JSON.  Grabbing the file using:  $jsonurl = "http://search.twitter.com/trends.json"; $json = file_get_contents($jsonurl,0,null,null); $json_output = json_decode($json);   How do I work with data from this object. As an array? Only really need to extract data from the [name] values.  JSON object contains:  stdClass Object (     [trends] =&gt; Array         (             [0] =&gt; stdClass Object                 (                     [name] =&gt; Vote                     [url] =&gt; http://search.twitter.com/search?q=Vote                 )              [1] =&gt; stdClass Object                 (                     [name] =&gt; Halloween                     [url] =&gt; http://search.twitter.com/search?q=Halloween                 )              [2] =&gt; stdClass Object                 (                     [name] =&gt; Starbucks                     [url] =&gt; http://search.twitter.com/search?q=Starbucks                 )              [3] =&gt; stdClass Object                 (                     [name] =&gt; #flylady                     [url] =&gt; http://search.twitter.com/search?q=%23flylady                 )              [4] =&gt; stdClass Object                 (                     [name] =&gt; #votereport                     [url] =&gt; http://search.twitter.com/search?q=%23votereport                 )              [5] =&gt; stdClass Object                 (                     [name] =&gt; Election Day                     [url] =&gt; http://search.twitter.com/search?q=%22Election+Day%22                 )              [6] =&gt; stdClass Object                 (                     [name] =&gt; #PubCon                     [url] =&gt; http://search.twitter.com/search?q=%23PubCon                 )              [7] =&gt; stdClass Object                 (                     [name] =&gt; #defrag08                     [url] =&gt; http://search.twitter.com/search?q=%23defrag08                 )              [8] =&gt; stdClass Object                 (                     [name] =&gt; Melbourne Cup                     [url] =&gt; http://search.twitter.com/search?q=%22Melbourne+Cup%22                 )              [9] =&gt; stdClass Object                 (                     [name] =&gt; Cheney                     [url] =&gt; http://search.twitter.com/search?q=Cheney                 )          )      [as_of] =&gt; Mon, 03 Nov 2008 21:49:36 +0000 ) I need help running python in ubuntu  I can run python without a problem in my terminal but when I create a .py file I can't get it to run. Windows seems to have a length limit on file names when trying to delete, though it won't prevent those files from being created.  Our build process creates a number of temporary files (many build off of a WSDL) that run afoul of this limit.  Our ant script is somehow able to delete them when doing a clean, but sometimes I need to delete the workarea directory (where all the temp files go) without actually doing a full clean from ant.  This is the same errors , but the answers there don't really work for me as I'm dealing with a directory, not a file, and I don't always know what specific files or subdirectories are causing the problem.  And I'm trying to avoid any manual process (other than triggering a single command) to actually delete them.  If I try deleting the directory from Explorer I get the error   Cannot delete [file name]: The file name you specified is not valid or too long.   Specify a different file name   Trying Remove-Item in powershell gives the following error:  Remove-Item : The specified path, file name, or both are too long. The fully qualified file name must be less than 260 characters, and the directory name must be less than 248 characters. At line:1 char:12 + Remove-Item  &lt;&lt;&lt;&lt; -force -Recurse &lt;directory&gt;   Does anyone know of any tools or easy ways to get around this delete error without having to manually find the problem files and move/rename them? Explain how (or if) a box full of photons would weigh more due to massless photons MySQL shutting down without warning, out of memory? Utopia of children brought up by robots I am looking for the title of a science-fiction book describing a colony (utopia) on another planet that has been established as a social experiment. The inhabitants had been brought up only by robots and therefore work perfectly as a society. I think there is no formal government and people work on what they are interested. If any catastrophe happens, they are able to form groups of specialists who react to the threat. Later there comes a group of outsiders who decide to join and live in the utopia. I am not sure, but author was probably Canadian. Which test should I use for ordinal data? In my experiment, I asked 50 participants to rank on a 5 point scale which voice they preferred. I summed each particpants' rating for each voice: the total for voice 1 is 95, the total for voice 2 is 186.  Am I correct in doing this? Or should I analyse the raw data?  Which test should I use for this data? Would a t-test work here? Alternatively I thought I could use a chi-square test.   I am interested in seeing if the preference for voice 2 is significant. I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing? We have noticed that when using nginx internal 301 and 302 handling, nginx will serve a small document body with the appropriate Location: ... header.  Something along the lines of (in html): 301 redirect - nginx.  As appropriate in the above behaviour, a content-type text/html and content-length header is also sent.  We do a lot of 302 and some 301 redirects, the above behaviour is wasted bandwidth in our opinion.  Any way to disable this behaviour?  One idea that crossed our mind was to set error_page 301 302 to an empty text file. We have not tested this yet, but I am assuming even with the above, the content-type and content-length (0) headers will be sent.  So, is there a clean way to send a "body-less" 301/302 redirect with nginx? How could I execute a script in Shell when a shortcut key is pressed.   Essentially what I need is when a shortcut key is pressed the script should read from a file and display that content in the terminal. Why is the charge naming convention wrong? I recently came to know about the Conventional Current vs. Electron Flow issue. Doing some search I found that the reason for this is that Benjamin Franklin made a mistake when naming positive and negative charges. There is even this little comic about that   My question is, how can a naming convention be wrong? Perhaps I don't understand what is the actual mistake here, I don't know. But I don't see how a naming convention could be wrong or right. There is no right and wrong about that. It could've been any other name, like charge A and charge B.  I'll appreciate anyone can help me understand what is wrong in all this. Add a link to the timeline of a post For the past year I have been quite active on SO and a bit on meta, but I had never heard of the timeline of a post until today in a comment on meta.  There was an  two years ago of this feature as "experimental". ( for that question).  This feature was , answer: this exists, but the link has to be built manually.  I think this could be quite interesting sometimes and I would like to use this, but I will not do so if I have to build the link each time by myself.  This feature has existed for two years and is still "experimental"? Who is using it? Who knows about it?  My feature request: Implement a link in the posts to their timeline. I am sure this nice feature will be used. Cannot set up multiple desktops (workspaces) Find joint density function of X and X+Y (exponential distribution) I tried finding the following density function:     Let $X$ and $Y$ be independent random variables, each having the exponential distribution with parameter $\lambda$. Find the joint density function of $X$ and $Z=X+Y$.   $\begin{aligned}f_{X,X+Y}=&amp;\frac{\partial}{\partial x\partial z}\mathbb P (X\leq x,Y\leq z-x)\\=&amp;\int_{-\infty}^x\int_{-\infty}^{z-x}f(u,v)\,\mathrm dv=\frac{\partial}{\partial z}\left[\frac{\partial}{\partial x}\int_0^x\int_0^{z-x}\lambda^2e^{-\lambda u-\lambda v}\,\mathrm dv\,\mathrm d u\right]\\ =&amp;\frac{\partial}{\partial z}\int_0^{z-x}\lambda^2e^{-\lambda x-\lambda v}\,\mathrm dv=\frac{\partial}{\partial z}\int_x^{z}\lambda^2e^{-\lambda x-\lambda(v-x)}\,\mathrm dv\\ =&amp;\lambda^2e^{-\lambda x-\lambda(z-x)}=\lambda^2e^{-\lambda z} \end{aligned}$  But I don't think this is correct, because $\mathbb P(X\leq x,X+Y\leq z)$ isn't equal to $\mathbb P(X\leq x,Y\leq z-x)$. But I can't think of anything else.  Could someone help me a little bit with this exercise?  I'm going to try the following:  $\mathbb P(X\leq x,Z\leq z)=\int_{-\infty}^x\int_{-\infty}^zf_X(u)f_Z(v)\,\mathrm du\,\mathrm dv$,  where I use the convolution formula. If I made no mistake, this gives  $f_{X+Y}(z)=z\lambda^2e^{-\lambda z}$, if $z&gt;0$.  Then calculating the joint density function:  $\begin{aligned}f_{X,X+Y}(x,z)=\frac{\partial}{\partial z}\left[\int_{-\infty}^z\lambda^3ve^{-\lambda x-\lambda v}\,\mathrm dv\right]. \end{aligned}$  Oh wait, this is also incorrect, because $X$ and $X+Y$ are dependent.  I really don't know how to proceed; could someone help?  I've also looked at this , but they mention the Jacobian, and my teacher has chosen to skip the paragraph about the Jacobian, so it wouldn't make sense if I had to use that.