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Questions tagged [mathematical-statistics]

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

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Question regarding equation in the multinominal distrubution of the exponential family

So this question is regarding to some notes i read in the book Machine learning and pattern recognition in chapter 2 in the section exponential family. First they state: So then they derive this ...
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2answers
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Which tests can be used for variables that are non-normal, but have homogenous variances?

I am trying to compare the effect of two treatments (planting distance) on the growth of plants (multiple species), using the variable growth rate in diameter for multiple years. I have 4 sets of ...
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0answers
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Distribute n balls among k bags [closed]

How many ways can we distribute n balls among k bags if (1) the balls and bags are distinguishable (e.g. numbered). (2) the bags are distinguishable; the balls are not. (3) balls and bags are ...
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How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?

Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion. How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
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1answer
73 views

Probability Question with k branches

Suppose my knowledge/ignorance of the number of branches of a certain store is given by the following probability law: P(k branches) = (1 − p)p^k where 0 < p < 1 and k = 0, 1, 2, 3, . . . If I ...
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0answers
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French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
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0answers
31 views

Calculating the probability of exceeding a given return

I have calculated the return of an investment with cost and benefit data as 80% with a 4.6% coefficient of variation. However, I need to carry the analysis further by calculation the probability of ...
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0answers
23 views

Underlying Poisson Distribution in Cox Regression with censoring

My question stems from this post: Does Cox Regression have an underlying Poisson distribution? Cox regression with no censoring can be interpreted as Poisson regression. Can we interpret Cox ...
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1answer
64 views

If all trimmed means are equal does this imply equal distributions?

I am trying to prove the following: Given that $\forall \alpha\in [0,1]$: $$\int_{F_S^{-1}(\alpha)}^{\infty}xf_S(x)\,dx = \int_{F_0^{-1}(\alpha)}^{\infty}yf_0(y)\,dy$$ where $F_S^{-1}(\alpha)$ and $...
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0answers
10 views

Renormalizing a distribution to reduce variance

I have a predictive model $M$ that generates an empirical predictive distribution $P_M$ via a set of samples. I cannot change the predictive model. I can evaluate the predictive performance using ...
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1answer
32 views

Combinatorics with colored balls [closed]

Seven blue and four red balls are to be arranged in order. How many ways can this be done if (1) The blue balls are distinguishable (e.g. numbered) as are the red balls. (2) Blue balls are ...
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2answers
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Normal Quantile Function With a lower bound not equal to infinity

I was recently at a statistics competition and a question came up as follows: They drew a normal distribution with $\mu=7$ and the area between the values $7.75$ and $8.25$ equal to $0.12$. No other ...
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0answers
18 views

What is local stationarity and how to detect it in time series

What is local stationarity time series and how to detect? and what is the difference between it and between global stationarity?
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0answers
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Does the size of original dataset influence measurment error in bootstrap? [closed]

As above. So for example would the measurment error of model's r-squared be higher if the bootstrap sample (1000) was drawn from original sample of 20 observation than if the bootstrap sample (1000) ...
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4answers
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How do I intuitively understand that independence is always symmetric?

Independence between two events, $A$ and $B$, is a symmetric relation, that is, if $P(A \mid B) = P(A)$, then $P(B \mid A) = P(B)$. The proof is very simple and can be found at the ProofWiki. ...
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0answers
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Consistency vs. Asymptotic Efficiency of estimator

I'm thinking about the relationship between an asymptotically efficient estimator and a consistent estimator, and I'd like to make sure that my thinking is correct. An estimator is asymptotically ...
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0answers
27 views

Can a Z-Test of Two Sample means in Excel be used to answer two separate hypotheses

Can a Z-Test of Two Sample means in Excel be used to answer two separate hypotheses. 1) Hypothesized Mean Difference=0 (Two-Tailed) (NULL vs Alternate, ho = ha) 2) Treatment > Control (One-Tailed) ...
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2answers
59 views

Prove covariance between sample value and sample mean [closed]

I'm trying to figure out how to prove that: $$\mathbb{Cov}(Y_i, \bar{Y}) = \frac{\sigma^2}{n},$$ and then use it to show that sample variance is an unbiased estimator.
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0answers
22 views

Asymptotically unbiased estimator vs consistent estimator [duplicate]

I'm wondering if there is a difference between an asymptotically unbiased estimator and a consistent estimator. For asymptotically unbiased estimators, the expected value of the estimator converges ...
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0answers
22 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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1answer
61 views

A different proof for KL divergence non-negativity

KL divergence's non-negativity can be proved in many ways. One could use the inequality $\log x \leq x - 1$ as a main step in the proof, another one could leverage the property of concave of the ...
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1answer
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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
28 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
44 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 $...
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0answers
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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
108 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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0answers
54 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 ...
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3answers
184 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
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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
75 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
49 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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1answer
126 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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0answers
27 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
31 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 ...
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1answer
64 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
19 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 ...
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1answer
40 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 ...
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0answers
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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
50 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
18 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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1answer
30 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 ...
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1answer
29 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
234 views

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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0answers
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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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0answers
14 views

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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2answers
48 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
37 views

Future of statistical methods in image segmentation? [closed]

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 ...
2
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1answer
62 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
16 views

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 ...