# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### Posterior Distribution using a Normal Likelihood and Laplace Prior

I have the working out below but is this correct. I just want the posterior distribution of when mu=0 given x. What I have tried is setting mu=0 after rewriting the first pdf with the summation ...
1 vote
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### How does Huber compute the $\operatorname{var}(s_n)/E[s_n]^2$ and $\operatorname{var}(d_n)/E[d_n]^2$?

(N.B. I am cross posting this question from math stackexchange since after x days I have still not received any responses.) How does Huber in book 'Robust statistical procedures' in chapter 1 compute ...
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### How to deal with negative loss during deep learning training, when using negative log likelihood loss? [duplicate]

I have a question regarding how to deal with negative loss during the training of my deep learning model. I've observed many instances of negative loss generated by the data points as outlined below. ...
1 vote
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### Probability RV is min among several iid RVs [duplicate]

This question is inspired by this programming question. Suppose I have 3 RVs which are independent. $$X \sim N(25.5, 2.5)$$ $$Y \sim N(25.2, 3.5)$$ $$Z \sim N(24.9, 1.7)$$ I want to know what is the ...
266 views

### What is the cross product of two probability distributions $P \times Q$?

In many papers on machine learning and statistics, I encounter the following notation Let $P$ be a distribution, and let $Q$ be another distribution. Then the author creates an object $P \times Q$ ...
1 vote
57 views

### Ensemble mean of a fraction

I want to compute the ensemble mean of the term: $\frac{Y^2}{X}$ Both $X$ and $Y$ are random variables that are not independent. I want to compute $E[\frac{Y^2}{X}]$. I proceed as follows, (Using the ...
29 views

### How to obtain the local unbiased condition for an estimator from global unbiased condition?

A standard problem in classical statistics is to find a good estimator that minimizes a given cost function under certain conditions. Normally we want to require the estimator $\hat\theta$ to be ...
92 views

### Finding Sample Range of Fisher's z-distribution via Approximating Hypergeometric $\,_2F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$

Recently, I have encountered Hypergeometric function $\,_2 F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$ in the context of order statistics. In particular, I am trying to evaluate an ...
62 views

### Probability distribution of measurements and Parameters of measurements

I am new to statistics and recently learned about ISO guidelines for Accuracy & Precision and Uncertainty & Error. I have tried to plot a graph for what I have learnt including all the ...
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### Is there a law or theorem related to occurrence of an event with highest probability in a population with infinite size?

Assume, we have a key that appears in either of the three rooms randomly (red room, blue room, and green room). We have the following probability distribution: ...
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### In Bland Altman Analysis, does it make sense to have bias as a primary endpoint, but 95% LoAs as a secondary endpoint?

In the context of Bland Altman analysis, is it justifiable to take a mean difference as a primary endpoint, but the 95% limits of agreement as a secondary endpoint? My understanding is that in B&A ... 1 vote
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### Unifying multiple sample data (meta-analysis)

I would like to unify multiple sample data (of different sizes) into some "unified sample" to evaluate its collective variance. Is this something statisticians do? I thought of unifying ...
355 views

### Covariance between sample mean and sample variance

I am trying to figure out the covariance between sample mean and sample variance from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between ...
1 vote
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### Probability generating function [closed]

X and Y are independent random variables having poisson distribution with parameters a and b respectively. By using probability generating function, prove that X + Y have a poisson distribution and ...
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### Intuition of Influence Function and Score function: $E[IF(X)S_{\beta}(X; \theta_0)]$

Question I find a theorem regarding influence function and score function \begin{align*} E\left\{IF(Z) S_\beta\left(Z, \theta_0\right)\right\}&=1\\ E\left\{IF(Z) S_\eta^T\left(Z, \theta_0\right)\...
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### Expected absolute deviation greater than standard Laplace

Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that \begin{equation*} E[|x-a|] \geq |a|+...
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### Minimizing cross entropy over a restricted domain?

Suppose $f(x;q)$ is the true distribution. The support of the random variable $X$ is $\Omega$. Suppose, I am interested in a particular subset of $\Xi \subset \Omega$. I would like to minimize the ...
16 views

### Mixture of Conditional Random Variables by Sampling

I am struggling to put my transformation of data into mathematical contexts. My goal is to define a mapping that transforms the original data into some awkwardly mixed data. In my simulation study, I ...
14 views

### In Sutton & Barto, what is meaning of weight given to actual final return after termination in the Delta return? (equation 12.3)

The equation 12.3, in the book RL An Intro by Sutton & Barto, is shown below with its "diagramatic" explanation in figure 12.2: Once the episode has terminated at time T, what's the ...
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### Why does the reparameterization trick work when some components are still stochastic? [duplicate]

I am trying to understand the reparameterization trick. I got some intuition while looking at this popular question, but I still feel largely confused. I am putting my understanding and doubts here ...
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### Natural Exponential family with polynomial partition function

Consider a natural exponential family: \begin{equation} f(x|\theta) = h(x) e^{ x \theta -A(\theta) } \end{equation} We assume that we are supported on $\mathbb{R}$ (i.e., $h(x)>0$ for \$x \in \...
1 vote