# cardinal

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 Oct11 comment Variance of a marginal order distribution less than the variance of the full distribution? You can speak of taking the variance of a random variable and this is quite common. In fact, one of the nice things about the mathematical theory of expectation is that you can often find the variance of a random variable without having any explicit information about its distribution. :) Oct11 comment Order statistic expectations and the variance of the original distribution (+1) For this answer and the other similar one. By happenstance the examples you give in both were quite close to ones I had contrived, but did not have a chance to post. It's always a nice present to see myself thinking along the same lines as you have in the answers you post. :) Oct11 comment Variance of a marginal order distribution less than the variance of the full distribution? A word on notation: I have made some edits to this post. While everyone is entitled to their own approach to mathematical notation, it's often useful to have conventions. It is quite uncommon to denote a distribution by the letter $X$ or $Y$, and it is quite common to denote a random variable by those letters. Oct11 revised Variance of a marginal order distribution less than the variance of the full distribution? added 29 characters in body Oct11 comment Order statistic expectations and the variance of the original distribution Consider a $\mathrm{Ber}(p)$ random variable. Then $\mathbb E X - \mathrm X_{1n} = p - p^n$ and this is not monotonic in $p(1-p)$. Oct11 comment Order statistic expectations and the variance of the original distribution I fear this question may not be well-posed (yet). For example, some order statistics may have well-defined means, even when the mean (or variance) of the distribution is undefined. Second, the parameterization of a particular family of distributions may not be closely related to the variance. Are you, perhaps, interested in some restricted class of distributions, such as location-scale families or similar? More details would be welcome. Oct10 comment Is there a known MLE for the numerator df of a sample of F statistics? The $F$ distribution has some nice monotonicity properties associated with it. If you want, you can look at this question and answer. I'm not quite sure the paper there will be immediately relevant to your problem, though. Oct10 comment Is there always a maximizer for any MLE problem? (+1) Boundedness of the parameter space doesn't hold in a lot of simple cases, even. But, for practical purposes, we generally know our parameters are bounded. :) Oct10 comment Is there always a maximizer for any MLE problem? (+1) For the update. Note that in Gaussian mixture models both unbounded likelihood and multiple local maxima are present, in general. To make matters worse, the likelihood becomes unbounded at particularly pathological solutions. In general, multiple maxima may not be as bad of an issue. In some cases, these maxima converge to one another fast enough that picking any of them can still yield a reasonable (even, efficient) estimator of the parameter of interest asymptotically. Oct10 comment Is there always a maximizer for any MLE problem? So are you saying "quadratic functions with more than one peak" is a reference to, e.g., a Gaussian mixture model, perhaps? If so, an edit could probably clear up some confusion. Oct10 comment Is there always a maximizer for any MLE problem? (+1) Nice answer. As hinted to in my comments to the OP, this is the answer I was hoping would be posted (even the counterexample was carefully chosen with this in mind). :) Oct10 comment Is there always a maximizer for any MLE problem? I think I must be misunderstanding your stated example. What quadratic functions have more than one peak? Oct10 comment Is there always a maximizer for any MLE problem? Counterexample (convexity): Let $X_1,X_2,\ldots,X_n$ be iid $\mathcal N(0,\sigma^2)$. Though there is a unique MLE, neither the likelihood nor log-likelihood is convex in $\sigma^2$. Oct10 comment Is there always a maximizer for any MLE problem? Absent additional assumptions, the statement given regarding maxima is false. For example, if the parameter space is closed and bounded and the likelihood function is continuous in the parameters, then a maximum must exist. Absent either of these additional conditions, the result need not hold. Regarding convexity, it fails even in the most simple and common of examples. :) Oct10 comment Is there always a maximizer for any MLE problem? (+1) Are you sure there are not some qualifications that have gone unstated in your question? As it stands, the engineer's statement is false in so many different ways it's almost hard to know where to begin. :) Oct9 comment What is the expected gain from taking a winner's bet? Well, you're on the right track. In repeated play of even the simplest of games with an edge, maximizing expected value leads to a probability of ruin (i.e., going broke) of 1 in the long run. Strategies to optimize other sensible objective criteria do not have this (rather serious) flaw. Oct9 comment What is the expected gain from taking a winner's bet? While it's not immediately relevant to this particular game, it's worth mentioning that even in games in which the bettor has an edge, maximizing expectation is a very poor strategy to use in repeated play. Oct9 revised How to find a confidence interval for a contrast? added 26 characters in body Oct9 comment Finding expectation of reciprocal of sample mean This is a pretty standard example/exercise found in most intro math stats books. What text are you using for the class? Oct9 comment Finding expectation of reciprocal of sample mean Hint: $1/\bar{X} = n / S_n$ where $S_n = \sum_{k=1}^n X_k$. Now, what is the distribution of $S_n$? From that, can you find $\mathbb E (1/S_n)$? Use linearity to conclude.