Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
6,389
questions
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5 views
Convergence to 0 in probability for non-iid random variables
Assume $U_k$ are correlated standard normal random variables.
Let $R_k := a_k U_k^2$, with $a_k > 0$ and $\sum_{k=1}^{\infty} a_k < \infty$.
How can we prove that $S_p:= \frac{1}{p}\sum_{k=1}^{p}...
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0answers
6 views
Show that (binary) cross-entropy is linear w.r.t labels
I am reading up on mixup augmentation. The original idea is that labels are one-hot encoded and then the loss function is just $\ell(\hat{x}, \hat{y}) $where
$$\hat{x} = \lambda x_1 + (1 − \lambda)...
1
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1answer
30 views
Ridge regression and distribution of estimate?
When OLS overfits observed data, does it give skewed distribution of estimates?
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0answers
7 views
Should there be a transpose in calculating the parameters of SVM
I am trying to code SVM from scratch using a small toy problem that involves five support vector values. In the code below, there are 5 support vectors arbitrary chosen and denoted by the variables <...
0
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1answer
22 views
Cramer-Rao Lower Bound for Poisson Adjusted MLE
For X$_i$ iid Poisson($\theta$), let y($\theta$)=$\theta$e$^{-\theta}$ be the function when X=1. Since, normally, the MLE of $\theta$ for a Poisson is $\bar{x}$ = $\frac{1}{n}$$\sum_{i=1}^{n} X_i$ , I ...
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7 views
Not sure how to start analyzing my dataset- could use any wisdom
I am new to statistics, and while I am not asking for anyone to do anything for me, I would appreciate and use any statistics wisdom that I could get. I was entrusted with some undergraduate research ...
0
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0answers
20 views
How accurately would the approval rating polls predict the election outcome? [closed]
How accurately would the approval rating polls predict the election outcome?
When Trump consistently had a 10% margin of likely voters against him wouldn't this sufficiently prove that he lost fair ...
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0answers
3 views
How can I find the optimal time to update data mart?
I am assigned to find out the optimal or some reasonable time when to update a table in the data mart I am working on. The data that I need to update in the data mart are collected from air ...
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0answers
14 views
Bootstrap bias-corrected percentiles
Following [1] p. 185-186 the $BC_a$ (bootstrap corrected and accelerated) confidence intervals are given by:
$$
(\hat{\theta}^{*(\alpha_1)},\hat{\theta}^{*(\alpha_2)}),
$$
where $\hat{\theta}^{*(\...
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0answers
19 views
Point estimation of parameters
Let $\theta$ be a parameter with values in $\Theta$ that should be estimated by some given data $X$.
The corresponding estimate is denoted as $\hat\theta = \hat\theta(X)$.
Several times I read that ...
3
votes
1answer
81 views
Expected value of (continous) exponential distribution proof/derivation
I started with the following exponential distribution:
$$ f_{exp}(x;\lambda) = \lambda\, e^{-x\lambda} \quad \forall\, x \in \mathbb{R}^+ $$
I know from internal courseslides and wikipedia that the ...
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5 views
Appropriate ways of analyzing the influence of two independent variables on a reaction time?
I have the following experimental data. The dependent variable is a reaction time which has been measured on n = 30 test subjects. The reaction time was measured six times per test subject and the ...
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0answers
13 views
What are the minimum and maximum values for the Type-I and Type-II error probabilities of a hypothesis test?
My professor asked this question in a video lecture recently and said it should be something we could "derive" on our own. However, I'm having a hard time wrapping my head around the ...
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0answers
6 views
Noise estimation from multiple observations of 2 dimensional data [closed]
I have a n x 5 x k matrix. n is the number of observations (same 5 x k matrix recorded n times). 5 is the number of classes with k parameters. So, n observations of 5 classes with k parameters each. ...
0
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1answer
19 views
The Degree of Freedom of the Pearson's X^2-test
everyone.
The degree of freedom for a Person's X^2 test is (I-1)*(J-1) and it is obtained by (IJ-1) -([I-1]+[J-1]) =(I-1)(J-1).
I am just wondering why it is (IJ-1) -([I-1]+[J-1])??
What is (IJ-1) ...
0
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1answer
22 views
Proof of B1by OLS in a multiple regression model with dummies
Given the model Y=B0 + B1C + B2A + e, where C=(0,1) and A=(0,1), how can I derive the formula for the estimator of B1 by OLS? I already know the formula for the general case in which A and C are not ...
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0answers
4 views
Statistical test to examine multiple attributes' effect on an outcome
I want to compare different attributes of a population and see if one is more indicative of an outcome measurement than the other.
In a general case, suppose I want to examine $k$ attributes. Each ...
2
votes
1answer
25 views
Second moment conditional multivariate gaussian distribution
What is the second moment of the conditional distribution of one multivariate normal variable given another multivariate normal variable? I know the formula for the second central moment (the variance)...
0
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0answers
33 views
Elements of Statistical Learning Exercise 5.1 [closed]
I need help with Exercise 5.14 from the ESL Textbook:
Derive the constraints on $\alpha_j$ in the thin-plate spline expansion (5.39) to guarantee that the penalty $J(f)$ is finite.
$$
f(x) = \beta_0 + ...
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0answers
6 views
How to generate a table of my sample of specific cases in Rstudio? [closed]
I'm using Rstudio for sampling. My data has $392$ cases and I need a sample of $20$. I'm required to use my last $4$ digits of my student number to generate the random case numbers I will use for my ...
1
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1answer
45 views
Finding the P-value from T-test
Suppose $X_1,…,X_n$ are modeled as normally distributed with mean $μ$, and have T statistic $1.4$.
What is the p-value for testing $H_0:μ=0$ against $H_a:μ<0$?
How can I find the p-value if I am ...
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0answers
13 views
Unsure of how $E_*\{\hat f^*(x;h) - f(x;h)\}^2$ is being explicitly evaluated in a paper on bootstrapping for density estimation
I'm reading a 1989 paper by Charles Taylor 'Bootstrap Choice of the Smoothing Parameter in Kernel Density Estimation' and I'm not sure how he arrived at a certain result.
Let $X_1,X_2,\dots,X_n$ be a ...
5
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1answer
107 views
Bernouilli ML parameter estimation from indirect observations
Suppose I have a coin with a probability $p$ for heads and $1-p$ for tails.
My aim is to estimate $p$ using the max likelihood criterion.
I flip the coin several times but cannot directly observe the ...
0
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2answers
26 views
Best test to use for investigating if the difference between means is significant?
I'm studying the length of stay for patients in a hospital. I have a sample of n=4533 observations. Each of these observations are assigned to an admin group numbered between 1 and 8, based on the ...
3
votes
1answer
82 views
+50
Inference in Dirichlet process mixtures via collapsed Gibbs sampling
I need to cluster some data $\{x1, \ldots, x_n\}$ through a Dirichlet process mixture model.
Consider the following Dirichlet process mixture model, in which the base measure is a $NIW(\mu_0, \...
0
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0answers
9 views
How to account for Confidence in likert scale questionaires?
So I'm want to setup a table that is able to take measures of various metriks (likert 0-10) that come
from a questionaire.
I now want to factor in the confidence of each measure, which should have ...
0
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0answers
7 views
Adding noise to ARIMA forecasts
I'm learning about ARIMA forecasts and I applied it on one of the example data I have. When I used model on the train set, it is giving me peaks, valleys and other irregularities. But on the test data,...
0
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0answers
13 views
Can embeddings function as control variables?
Suppose I am building a linear regression model
$$y = X + Z + \epsilon$$
where $X$ are covariates, $Z$ is a confounder, and $\epsilon$ is noise. $Z$ is a categorical variable with an enormous number ...
1
vote
1answer
56 views
Problem with Bayes theorem and bigram probabilities
I'm working with Bayes’ Theorem, but I can't fix the numbers, and I don't know why. I have a very simple set of sequential events, grouping them into bigrams (sequential groups of two events):
$$ABBAB ...
0
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0answers
10 views
How can I use Boxplots to find patterns in each box data?
I have represented my dataset in Boxplot 1 in order to check if there is patterns in meals consumption within each age group.
What I can see, is that there is a kind of patterns in each age group, ...
0
votes
1answer
18 views
Using r-squared values to correlate two values that can't be measured at the same time
I have data A and B that can be measured at the same time and have an R^2 value of 0.91. I also have data B and C that can be measured at the same time and have and R^2 value of 0.89. Is there a way ...
0
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0answers
12 views
Best Time Series Analysis Book for Self Learning? [duplicate]
I am an M.Stat student and I was interested in Time Series but the course won't be available at my University until the semester after I graduate so I wanted to self teach the topic. I have some ...
0
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0answers
37 views
How can I find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\theta}_{n}^{(2)}}$ (in terms of $\tau_1$, $\tau_2$ and $\alpha$)?
Given that $n^{\alpha}[\hat{\theta}_{n}^{(1)}−\theta_0] \xrightarrow{L} \tau_1 H$ and $n^{\alpha}[\hat{\theta}_{n}^{(2)}−\theta_0] \xrightarrow{L} \tau_2 H$, find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\...
-1
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0answers
50 views
Density function [closed]
Consider the density
\begin{align}
f(x)&=(2\pi)^{-n/2}e^{a}b\\
a &= \frac{-\sum_{i=1}^nx_i^{2}}{2}\\
b &=1 + \prod_{i=1}^n x_i e^{\frac{-x_i^{2}}{2}}
\end{align}
on $\mathbb{R^{n}}$ :
Is ...
0
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0answers
18 views
Estimating survival curves from Cox regression results
I can understand that it is possible to estimate survival curves directly from the results of a Cox regression.
The way it can be done, mathematically, is furthermore very nicely explained in this ...
-1
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0answers
11 views
find general solution of PDE [closed]
I'm a newbie for mathematics and now I'm learning PDE and stuck on that. Could anyone help me out to understand this elimination from PDE. The equation is similar to solve r + s -6t = e^(2x+y) + sin(...
0
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0answers
26 views
Solve Partial Differential Equation [migrated]
I'm a newbie for mathematics and now I'm learning PDE and stuck on that. Could anyone help me out to understand this elimination from PDE. The equation is similar to solve (D^2 -6DD'+9D'^2)u = ycosx
2
votes
0answers
40 views
Fisher information of $\rho$ in a symmetric normal $N_p(\mathbf 0,\Sigma)$ distribution
Suppose $\boldsymbol X\sim N_p(\mathbf0,\Sigma)$ where $\Sigma=(1-\rho)I_p+\rho\mathbf1\mathbf1'$ is positive definite. I am trying to find the Fisher information of $\rho$ based on $\boldsymbol X$. ...
2
votes
1answer
16 views
constant model elasticity demand
I need to be corrected if i have something wrong :
the elasticity of the demand is calculated : dq/dp if q is the quantity and p the price
so to calculate it i take two prices p1 and p2, if i have ...
0
votes
1answer
25 views
Question about one step in the derivation of the variance of the slope in a linear regression
After deriving the formula myself in an unnecessarily complicated way, I discovered this very old thread:
https://stats.stackexchange.com/a/89155/148856
There is however a step that I don't understand,...
0
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0answers
28 views
Simple sufficient statistic
Given a sample $\{X_1, ..., X_n \}$ with the following pdf
$$ f(x; \theta) = \begin{cases} 2\theta^{-2} x, & \mbox{for } 0 \leq x \leq \theta \\ 0, & \mbox{otherwise } \end{cases}$$
find a ...
1
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0answers
44 views
Resource of python statistics problems and their solutions
I am learning ML / DS, after learning Pandas, I found a lot of Pandas exercises to practice like this
https://github.com/guipsamora/pandas_exercises
Can anyone share such exercises with a solution for ...
0
votes
0answers
15 views
The difference between a standard error and a propagated error
I'm confused about the difference between the standard error and the propagated error. For example, say that I got two measurements which I want to average: $16.04 \pm 0.31$ and $15.72 \pm 0.28$. The ...
0
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0answers
11 views
Finding the statistical significance of a Matthews Correlation Coefficient (MCC) in binary classification (what is a good MCC)?
I have a very imbalanced dataset. Thus, I am using MCC to evaluate the performance of various ML algorithms. It appears that literature is entirely lacking in ways to evaluate how good an MCC score is....
1
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0answers
28 views
Markov Chain for Amino Acid Sequence
I was wondering if it is possible to apply Markov Chain to this amino acid sequence "ddvlsldeddddsdyncgednd" to find the probability of this sequence forming in this particular positional ...
1
vote
2answers
38 views
Finding a type 1 error
Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and the goal is to test if $H_0:λ≤1$ versus $H_a:λ≥2$. Suppose that the test statistic is $S=\sum_{i=1}^{10}T_i$, and rejection of the null occurs when $S≤...
0
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0answers
17 views
How to test at percentage if a distribution follows a poison distribution [closed]
The Investment department of the Mexican development agency randomly selected 100 SMEs and examined them for profitability on their investment. The following results were obtained:
Number of Profits ...
3
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0answers
100 views
+50
As $n \to \infty$, can we 'ignore' a matrix in an expectation that does not depend on $n$?
Let $A_n = \sum_{i=1}^n X_iX_i^T$ and $B = X_1X_1^T$ be random matrices of dimension $m\times m$. Note that the elements of $B$ are dependent on elements of $A$. Note that the vectors $X_i$ are iid ...
0
votes
1answer
44 views
Proof that $RSS=n*\sigma_{Y}^{2}*(1-r^2)$
I have given the function for a regression $Y_i=\beta_0+\beta_1x_1+\epsilon_i$ and I'm asked to show a proof that $RSS=n\sigma_Y^2(1-r^2)$ given $r=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}$, but I don't see ...
0
votes
2answers
31 views
Find relation between Categorical dependent variable and continuous independent variable
I have one sample group of data where I found the difference between two categorical data (this is my dependent variable) and continuous - numerical data (this is my independent variable).
I want to ...
