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Questions tagged [mathematical-statistics]

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

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1answer
101 views

What is the physical intuition behind the equality $\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar x) x_i$?

Suppose that $x_1,\ldots,x_n$ are real numbers, and let $\bar x$ denote the average $\frac{\sum_i x_i}n.$ I know how to prove on paper that the equality $$\sum_i (x_i - \bar x)^2 = \sum_i (x_i - \bar ...
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2answers
79 views

Could you prove the theorem?

The theorem 1 is the result in Emil Bjornson's paper (PILOT-BASED BAYESIAN CHANNEL NORM ESTIMATION IN RAYLEIGH FADING MULTI-ANTENNA SYSTEMS). I want to know the proof omitted. Please, help me.
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0answers
9 views

How do you derive a risk function from a loss function given a normal distribution?

Let X1....X25 be a random sample from a normal distribution N(p,1) such that the domain of p stretches from negative infinity to positive infinity. Let y = X-bar (I.e. the sample mean) Let loss ...
3
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1answer
85 views
+50

Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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1answer
29 views

Covariance of Random Proportions in Multinomial Counts

In Agresti's Categorical Data Analysis Second Edition, at Section 14.1.4, there is a proof of the Asymptotic Normality of Functions of Multinomial Counts. It is stated that for a vector of responses $...
3
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1answer
213 views

what does it mean by more “efficient” estimator

When comparing two estimators, say $T_1$ and $T_2$, what does it mean by saying $T_1$ is more efficient than $T_2$? Could someone give an easy but very concrete example? Also I have another ...
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0answers
12 views

Determining variance of UMVUE

Let $X_1,...,X_n$ be iid with pdf given by $f(x;\theta)=\frac{log\theta}{\theta^{x-1}}I(x>1)$. My task is to determine if the $\mu=E[X]=1+\frac{1}{log\theta}$ can be estimated efficiently, i.e. if ...
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0answers
25 views

Independent Study Statistics/Probability Grad Level [duplicate]

I am trying to decide on topics for my independent study this semester. I am a Pre-Doctoral Mathematics student, so looking for a more math based text rather than engineering based (which I have found ...
2
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1answer
305 views

Conway–Maxwell–Poisson (CMP) distribution and exponential family

So I have a question here about the CMP distribution: My understanding is that $b(\theta)$ is only a function of $\theta$ but why is $v$ able to be included in that function, would $v$ not be a ...
1
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1answer
304 views

Multiple sufficient statistics and the factorization theorem

Suppose you are using the factorization theorem to find a sufficient statistic. Let us say that we have a negative sign in front of $T(x)$. How do you know whether or not to "absorb" a negative sign ...
2
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0answers
27 views

Is this a proper use of the Karlin-Rubin UMP test theorem?

For iid $X_1,...,X_n$ and the unknown parameter $\theta>1$, suppose that the likelihood function of a particular sample is given by: $$L(x;\theta)=log(\theta)^n\theta^{{n-\sum_{{i=1}}^nx_i}} I(x_{(...
2
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1answer
68 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
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1answer
19 views

Asking for inspiration

I am a teacher. I want to create a system where my students will take exams online and I will predict their chances of getting a certain marks in the final exam. Is there any mathematical model where ...
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0answers
9 views

Need help explaining the steps of a voluntary response [on hold]

what are the steps of a voluntary response (generally speaking)? What I mean by steps is that it can be used for other scenarios and broad so to speak. I tried reading and going over my Stats book and ...
1
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1answer
344 views

Conditional independence: conditioning on an empty set of random variables

Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is ...
6
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1answer
135 views

Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
3
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0answers
38 views

(When not assuming differentiability) what is the definition of Fisher information?

Here we assume (for simplicity) that the parameter $\theta$ is one-dimensional. When one has sufficient regularity and can push in partial derivatives to and pull out partial derivatives from the ...
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0answers
33 views

Is this the only way to determine if a parameter can be estimated efficiently?

I am tasked with determining if a particular parameter can be estimated efficiently. Given that an efficient estimator is an unbiased estimator which achieves the Cramer-Rao lower-bound, is the only ...
1
vote
1answer
343 views

Inequality on variance of sum

I want to prove that $$\operatorname{Var}\left(\sum\limits_{i=1}^m{X_i}\right) \leq m\sum\limits_{i=1}^m{\operatorname{Var}(X_i)} \,. \>$$ A too complicated proof is to write $$ a_{ij}=\sqrt {Cov(...
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0answers
14 views

Is the sum of two dependent sub-gaussian variables X and Y still follows sub-gaussian distribution?

I am trying to prove the random vector with dependent sub-gaussian coordinates is also a sub-gaussian random vector. The related question has been asked in https://math.stackexchange.com/q/3072363/...
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0answers
25 views

Why do we take `(Bias) ^2` in total error in a model? [duplicate]

I was recently studying some book and few blogs and come to note that : Total error = Bias^2 +Variance + irreducible error Also, I know that these are the errors ...
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0answers
15 views

Proof that a poisson regression with 2 categorical predictors is the same as each observed mean

I have a poisson regression with 2 categorical predictors to predict\estimate the sales of product $p_i$ on location $j$: $\hat{X}_j(p_i) = e^{\gamma_0 + \alpha_i + \beta_j + \mu_{i,j}}$ It appears ...
2
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1answer
57 views
2
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1answer
40 views

Two approaches for finding a MLE in a binomial setting

I'm learning towards an exam in mathematical statistics and I came across the following question. I was wondering if the second approach of solving the question is legitimate. If both are correct, is ...
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0answers
17 views

Clarification on a formula [on hold]

I am starting to take my first classes on statistics. At 0.47 seconds of this video - https://www.youtube.com/watch?v=a8abc9SgVUM, a formula is applied to predict a student's score on a particular ...
1
vote
1answer
340 views

Is it possible to have a validation error less than train error for a while followed by the reverse behaviour?

I am solving for a regression (using tensorflow's DNNRegressor) problem. When I sampled out 20% data (randomly) and divided it further into train-eval (90-10%, random but mutually exclusive), I ...
7
votes
1answer
97 views

Convergence of the Matérn covariance function to the squared exponential

The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ...
3
votes
1answer
141 views

Large deviations proof question

Below is part of the proof of large deviations result. K is cumulant generating function. Can anyone explain how the last step follows? This is page 157 of McCullagh's "Tensor Methods in Statistics"
8
votes
2answers
48k views

What is the difference between “factors” and “covariate” in terms of ANCOVA? [duplicate]

I am a bit confused on the term "covariate". It seems like the term can mean two different things. In ANCOVA, the term is used for the third variable that is not directly related to the experiment. ...
0
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0answers
24 views

Derive the estimator for the integrated squared bias $\int \left(\operatorname{E}\hat{f} - f\right)^2 $

This problem is found in p. 77 of Wand & Jones' (1995) book. If you are familiar with nonparametric estimation you may skip this introduction. Suppose we want to minimize the integrated squared ...
0
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1answer
50 views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
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2answers
75 views

Existence of $E(X^2)$ when $X$ has the pdf $f(x)= \frac{1}{(2+x^2)^{3/2}}$

In a competitive exam, I came across an objective question which says Let $X$ be a continuous random variable with the probability density function $$f(x)= \frac{1}{(2+x^2)^{3/2}}\quad,\,-\infty&...
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0answers
7 views

slope of a linear function in semi-log plot [migrated]

I have a decreasing linear function. So,the slope of this function df/dt <0. Now, if we plot this function in a semi-log plot with log(t) in horizontal axis and y in vertical axis, can we say that ...
4
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1answer
166 views

Example for which the CLT holds but the LLN doesn`t

I am currently thinking about the relationship between the law of large numbers and the central limit theorem and I was wondering whether someone can give me an example of a familiy of random ...
2
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1answer
38 views

How to read checkresiduals graphics in R?

I need to check the residuals of two models in R so I can determine how bad or good are said models. First, I've started simulating an INAR(2) model and wanted to fit a more convenient model, then, ...
1
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1answer
33 views

Future of statistical methods in image segmentation? [on hold]

I was looking for a purely statistical method for image segmentation and found many, e.g. Hidden Markov Random Fields with EM algorithm. But it seems to me that these methods are nowadays completely ...
44
votes
7answers
19k views

How does the reparameterization trick for VAEs work and why is it important?

How does the reparameterization trick for variational autoencoders (VAE) work? Is there an intuitive and easy explanation without simplifying the underlying math? And why do we need the 'trick'?
0
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1answer
17 views

Simple family medical research

The study involved $20$ families, each of $\sim3$ people. For each patient continious parameter $X$ is measured. Each patient has diagnosis $Y=\{\text{ill},\text{healthy}\}=\{1,0\}$. Problem is to ...
0
votes
1answer
67 views

Can a kernel be used to define the cardinality of the union of 2 sets?

Given a space $\Omega $ with 2 sets $X\subseteq \Omega $ and $Y\subseteq \Omega $, how can I define a kernel $k(X,Y)=|X\cup Y|$? I know that I can define a kernel $k_1(X,Y)=|X\cap Y|$ by setting $k_1(...
1
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1answer
21 views

How to obtain the inverse of the variance covariance matrix of GLS (Random Effects Model)

In the standard GLS set up how do you find the inverse of the variance covariance matrix? $$y _ { i t } = \beta _ { 0 } + x _ { i t } ^ { \prime } \beta + \alpha _ { i } + u _ { i t } \hspace{35pt} u ...
3
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1answer
43 views

Is the sampling distribution of a complete sufficient statistic free from relevant subsets?

Let $T_{\theta}(\mathbf{x})$ be a complete, sufficient statistic $T_{\theta}: \Omega \mapsto \mathbb{R}$, where $T_{\theta}$ is indexed by the parameter $\theta \in \mathbb{R}^n$. Is it true that the ...
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0answers
16 views

Creating a contingency table for Chi-Square test

So I have this data I was wondering how to create a contingency table for it to be able to perform the Chi-square test. Please note that the two groups are not equal and every single cell contains a ...
4
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1answer
328 views

Why low rank expansions can exploit the redundancy that exist between different feature channels and filters?

I read Jaderberg et al., 2014 paper about Speeding up Convolutional Neural Network with Low Rank Expansions. In the introduction, it is written in bold font: Our key insight is to exploit the ...
2
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1answer
38 views

Independent and Identically distributed assumption in Maximum likelihood estimation

I was reading about Maximum likelihood estimation from various sources on the internet and I noticed that MLE makes an assumption about the data known as IID but I didn't completely understand why is ...
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0answers
17 views

Resources for prerequisites to Probablistic Machine Learning Models

I am a self-learner and have done several machine learning courses but diving into Bayesian or Probabilistic Graphical Models I feel like my prior knowledge is inadequate. I have done some Probability ...
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0answers
16 views

Test for average of independent, non-Identical binomial distributions

I have a box with $m$ coins, each with a probability $p_i$ for $1\leq i \leq m$ of flipping heads. As an experiment, I flip each coin $n_i$ times, recording the results. What is a statistical test I ...
0
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1answer
23 views

Margin of error for a specific question on a survey?

Background: We want to find the margin of error, at a 95% confidence interval on a particular question in a survey. If there are 10 questions in the survey, this question is towards the very end. ...
0
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1answer
27 views

Meaning terms O() “terms order at most” and o( ) “terms of smaller order than”

In the paper: "Risk aversion in the small and in the large" by John Pratt from 1964, a formula is derived for the approximation of the risk premium: rp ≈ 0.5*σ^2*r(x) The risk can be represented ...
3
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1answer
223 views

Estimator that is optimal under all sensible loss (evaluation) functions

Consider a probability distribution $D$ with a parameter $\theta$ and an i.i.d. sample $S$ from that distribution. I am interested in an estimator $\hat\theta(S)$ of $\theta$ that satisfies the ...
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1answer
22 views

Appropriate way to get Cross Validated AUC

I was thinking about cross-validation and how it is the most appropriate way to do it... Let's take the case of binary logistic regression where the goal is to calculate the AUC. Make the partition ...