# Tagged Questions

The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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### probability of rank of a number

Suppose I have 10 numbers. I want to perform two experiments. First experiment: I pick one of the numbers and compute the probability of being rank from 1 to 10 of that number, i.e, what is the ...
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### Find a 95% upper confidence bound for lognormal dispersion

Let $X_1,\ldots,X_n$ be a sample from a population $X$. Assume that $X$ has a $lognormal$ distribution $(2,\sigma^{2})$, with $\sigma^{2}$ an unknow parameter. Find, for the variance of the population,...
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### When deciding the Hypothesis, do I, in some way, define the rejection region?

This has always troubled me a bit. When I choose my hypothesis, do I define in some way the rejection region [RR], or, do I do that by choosing the test statistic I want to use? By fixing the ...
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### sufficient statistics to estimate the unknown parameters

I am a beginner in statistical inference and am learning sufficient statistics. As far as I know the distributions conditional on the sufficient statistics doesn't depend on the unknown parameters. I ...
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### Properties of minimizing statistical distance

I have some data, thus I have its empirical distribution. I want to use a theoretical distribution to fit my data. For example, I observe my data is likely to be distributed as Pareto, so I use Pareto ...
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### find the maximum where gamma is attained

I have this problem and I am not figuring the beginning
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### the posterior under n observations computed as the posterior given a single observation [closed]

I have this problem: Show that, if $p(x|\theta)$ is an exponential family model and $q(\theta)$ its natural conjugate prior, the posterior $Π(\theta|x_{1:n})$ under $n$ observations can be computed ...
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### Looking for a good, rigorous book on Statistical Inference.

I'd like to brush up on my statistics, but most books are either overly "colourful" or just plain shallow, and certainly far from a Bourbaki-esque style of exposition. Is there any "Graduate Texts in ...
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### Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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### Missing approximation to get the Maximum A Posteriori (MAP) estimator of event times with a sparse prior

Assume that a signal $y$ is a noisy perturbation of time-shifted copies of a given waveform $f(t)$ defined on K time bins $\{ 0, \cdots, K-1 \}$: \forall t \in \{1, \cdots, T\}...
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### Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
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### How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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### Maximum a Posteriori (MAP) Estimator of Time Shifts with Poisson Process Prior

Assume that a signal $y$ is a noisy superposition of time-shifted copies of a given waveform $f(t)$ on a finite time interval $[0, T]$: y(t) = \sum_{i=1}^{n} f(t - \tau_j) + \...
I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\$ denote the squared ...