Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
4,085 questions
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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 ...
1
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1answer
13 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 ...
4
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0answers
54 views
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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12 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
15 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 ...
2
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1answer
36 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
22 views
Combinatorics with colored balls [on hold]
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 ...
1
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2answers
20 views
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
10 views
How to generate samples without replacement using mvrnorm in R? [on hold]
I'm using the mvrnorm function in R to generate positively correlated bivariate data with specified means, variances and correlation coefficient. I'm generating 50,...
0
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0answers
8 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?
1
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0answers
12 views
Does the size of original dataset influence measurment error in bootstrap? [on hold]
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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3answers
33 views
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
9 views
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
22 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)
...
1
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2answers
38 views
Prove covariance between sample value and sample mean [on hold]
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.
1
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0answers
21 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
16 views
How do you derive a risk function from a loss function given a normal distribution? [closed]
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 ...
0
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0answers
13 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 ...
2
votes
1answer
22 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
26 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 ...
1
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1answer
38 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 $...
0
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0answers
9 views
Need help explaining the steps of a voluntary response [closed]
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 ...
2
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0answers
30 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
76 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 ...
1
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0answers
36 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 ...
4
votes
2answers
115 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.
-1
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0answers
15 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/...
0
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0answers
16 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
62 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.
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 ...
-1
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0answers
17 views
Clarification on a formula [closed]
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 ...
7
votes
1answer
106 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 ...
0
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0answers
26 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 ...
0
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0answers
26 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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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 ...
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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2answers
76 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&...
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 ...
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 ...
0
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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 ...
3
votes
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 ...
0
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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 ...
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 ...
0
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1answer
24 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. ...
4
votes
1answer
100 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 ...
1
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0answers
17 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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0answers
9 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.
1
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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 ...
1
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1answer
35 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 ...
