# Tag Info

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The first formula is the population standard deviation and the second formula is the the sample standard deviation. The second formula is also related to the unbiased estimator of the variance - see wikipedia for further details. I suppose (here) in the UK they don't make the distinction between sample and population at high school. They certainly don't ...

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The dichotomy between the cases $d < 3$ and $d \geq 3$ for the admissibility of the MLE of the mean of a $d$-dimensional multivariate normal random variable is certainly shocking. There is another very famous example in probability and statistics in which there is a dichotomy between the $d < 3$ and $d \geq 3$ cases. This is the recurrence of a simple ...

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To define the two terms without using too much technical language: An estimator is consistent if, as the sample size increases, the estimator "converges" to the true value of the parameter being estimated. To be slightly more precise - consistency means that, as the sample size increases, the sampling distribution of the estimator becomes increasingly ...

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You can find everything here. However, here is a brief answer. Let $\mu$ and $\sigma^2$ be the mean and the variance of interest; you wish to estimate $\sigma^2$ based on a sample of size $n$. Now, let us say you use the following estimator: $S^2 = \frac{1}{n} \sum_{i=1}^n (X_{i} - \bar{X})^2$, where $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ is the ...

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You aren't strictly taking the "mean" of the likelihood, because the Likelihood function isn't a probability distribution over x. It isn't even a probability distribution anyway, but assuming you have a likelihood function that you can normalize into a PDF then it would be the probability of $Y$ not of $X$. This is a likelihood weighted average of $X$. I ...

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Unbiased estimates are typical in introductory statistics courses because they are: 1) classic, 2) easy to analyze mathematically. The Cramer-Rao lower bound is one of the main tools for 2). Away from unbiased estimates there is possible improvement. The bias-variance trade off is an important concept in statistics for understanding how biased estimates ...

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