Questions tagged [mathematical-statistics]

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

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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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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)...
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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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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 <...
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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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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 ...
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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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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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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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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 ...
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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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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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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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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. ...
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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) ...
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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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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 ...
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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)...
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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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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 ...
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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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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 ...
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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 ...
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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 ...
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1answer
82 views
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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, \...
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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 ...
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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,...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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{\...
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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 ...
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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 ...
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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(...
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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
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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 $\boldsymbol X$. ...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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≤...
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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 ...
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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 ...
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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 ...
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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 ...

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