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Questions tagged [mathematical-statistics]

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

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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 ...
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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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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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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 ...
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1answer
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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 $...
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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 ...
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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_{(...
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1answer
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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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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 ...
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2answers
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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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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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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 ...
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1answer
57 views

Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?

I understand the intuition behind the question but I'm trying to prove it to myself with math.
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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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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 ...
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1answer
96 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 ...
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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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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 ...
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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 ...
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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, ...
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2answers
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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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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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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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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1answer
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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 ...
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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. ...
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1answer
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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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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 ...
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How AAE is calculated for curve fitting and what does it's value indicate?

I have experimental results which are fitted to different equations, the fit is assessed by R^2 and AAE and I don't know how to interpret AAE values.
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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 ...
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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 ...
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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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Fitting hedge fund returns to theoretical probability distributions

I know that a lot of work has been done characterizing the first four moments of monthly hedge fund returns across a variety of fund types and strategies, and that work indicates that the higher ...
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1answer
64 views

Basic calculations with Order Statistics

I've come across the following problem, and I am tempted to delve into order statistics to solve this. I would greatly appreciate any help! Suppose you draw 6 independent samples from a continuous ...
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1answer
59 views

Why is correlation formula the way it is? Or Say how it formed? [duplicate]

I am really being confused by why the correlation formula is called the correlation of two variables $X$ and $Y$. Also how is it derived? The part where we divide covariance by product of standard ...
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Find supremum of Type II error in Neyman-Pearson framework

Let $X_1,\dots,X_n$ be an iid sample from an $N(\theta,1)$ distribution. We want to test $H_0:\:\theta=0$ against the alternative $H_1\:\theta \neq 0$ using the test statistic $$T_n(X_1,\dots,X_n) = \...
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175 views

Are all symmetric matrices with diagonal elements 1 and other values between -1 and 1 correlation matrices?

A question for the statisticians and other math lovers: Are all symmetric matrices with diagonal elements 1 and other values between $-1$ and 1 correlation matrices?
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Shotgun sequence statistics

I trying to understand the statistics behind shotgun sequencing. Below is an excerpt from a book. Can anyone tell what does the below bold sentence mean? I did not understand the interval range and ...
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Given sample variance and sample size, test two variance which one holds

If I am given the sample size and the sample variances of a Normal population, how can I test the hypothesis: H0: $\sigma^2=\sigma_0^2$, H1: $\sigma^2=\sigma_1^2$
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How to prove variance of OLS estimator in matrix form?

I am reading Wooldridge's Introductory Econometrics (2000), don't judge me, old version = cheap second hand book, and in the page P94 Theorem 3.2 of Multiple Regression Analysis, it says that: $$ Var(...
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1answer
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nonexistence of the expected value of heavy-tails distributions

I would be much grateful if you could help me with this. Is the nonexistence of the expected values of Cauchy distribution because of the heavy tails? How can we prove this using calculus or even ...
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Calculating CI's using bootstrapping on the holdout test dataset

I’m trying to calculate 95% confidence intervals for the sensitivity and specificity of a decision model that I’m building. I’ve split my dataset into 90/10 train and test sets. I’ve used the 90% ...
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Calculate contrasts equivalent to effects of mixed ANOVA

I have a 2x2x2 mixed factorial design with one between-subjects factor (C) and two within-subjects factors (A and B). I analyzed the data with a mixed ANOVA and now want to calculate t-test ...
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1answer
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Finding expression of $n$-th derivative, when $n$ is large

For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
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Likelihood of Gamma Distribution

I'm studying bayesian stats on my own and came across the following problem on Coursera. Can anyone help me understand how to work through this? $x_{i}\stackrel {iid}{\sim} Beta(\alpha,\beta), i=1.......
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R^square for a pre-determined linear regression

I would like to produce the R^square goodness-of-fit statistics for a predictive model. I have the base data (10, 000 number of x-values) which are the true values given by an analytic/deterministic ...
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A/B test : Is there a way interpreting factor “A” group is greater than factor “B” group?

Plz.. understand my lack of English skills.. T.T I have a problem dealing with statistical hypothesis testing. To be specific, we have two special delivery method like (new) : method "A", (old) : ...
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About the power spectrum and confidence upper limit

For now, I have a coupled system with 5 variables and use the Runge-Kutta method to integrate. ...