Questions tagged [mathematical-statistics]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
10 views

How is Nadayara Watson KDE proof

I was looking at Wikipedia article on Nadayara-Watson Kernel regression section, in the proof part they state But I'm having trouble understand why: Turns to just yi. Sorry I'm missing something so ...
0
votes
0answers
7 views

Clarification of line in proof of consistency theorem (Vapnik)

In Vapnik's Statistical Learning Theory (1998 edition) on pages 89-92, he proves a "key theorem of learning theory" that states the conditions for when: "the following two statements ...
0
votes
0answers
12 views

why in real gases we have potential energy? [closed]

I read that in ideal gases, we dont have any potential energy. But I also read what in real gases we have potential energy(positive Ep). Can someone explain why in real gases we have potential energy?
0
votes
0answers
16 views

Explain why the $\log(\mu/(1+\mu))$ is a appropriate link function for generalized linear model of Geometric distribution?

Given the geometric distribution $P(Y=y)=(1-\pi)^y\pi$, we can get the canonical link function is $$g(\mu)=\log(\frac{\mu}{1+\mu})$$ where $\mu=E(Y)$. $$\textbf{ Question: Explain why this link ...
0
votes
0answers
11 views

Expectation of Gaussian vector times its length

Suppose $\boldsymbol{x} \in \mathbb{R}^3$ such that $$\boldsymbol{x} \sim MVN(\mu, \sigma^2I_3)$$ I'm trying to figure out the expectation $\mathbb{E}[\|\boldsymbol{x}\|_2 \boldsymbol{x}]$. I'm aware ...
1
vote
0answers
32 views

Represent rejection region using p-value

Let's say we have some level $\alpha$ test (more precisely, family of tests parametrized by $\alpha \in (0,1)$), $\delta(\mathbf{x},\alpha)$. This means $$\sup_{\theta \in \Theta_0} P_\theta(\mathbf{X}...
-3
votes
1answer
39 views

What is a convincing and useful real application of the Central Limit Theorem?

I understand what the central limit theorem is about (ie. that the shape of the distribution of the sample means of any population (each calculated by taking 30+ random measurements from the ...
0
votes
0answers
6 views

How are the kernel density estimators and the tree structure related in Tree Parzen Estimators?

I'm still a bit confused about a few details of TPEs. So this should be a follow-up to this discussion: What is the "tree" structure in Tree Parzen Estimators? See the original TPE paper ...
1
vote
1answer
24 views

Shapiro-Wilk vs Anderson Darling test for normality

What makes the Shapiro-Wilk test have the best power for a given significance when compared to Anderson-Darling test and other tests.
1
vote
0answers
18 views

How to prove that trangle inequality is satisfied in Hausdorff distance [migrated]

I'm working on a problem my teacher asked me to check if I was interested, which is 'how to prove that Hausdorff Distance is strictly a distance function'. More specifically, how to prove that $$D_H(A,...
0
votes
0answers
16 views

Validity of a joint distribution and the associated Likelihoods

I want to check the validity of a code that produces a 2D contour between 2 parameters with their associated Likelihood (normalized). Here's an example below : As you can see, I have plotted, for 3 ...
0
votes
0answers
14 views

To what degree should one trust significance of different analysis of the same data set done by different groups? [duplicate]

Suppose there is a data set $D$. There are 20 groups of researchers independently. Each group will publish a few results on this data set $D$. Suppose each of them find about 3 tests significant($5\%$ ...
0
votes
0answers
25 views

Total Time on Test (TTT) Statistic

Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ denote an ordered sample of size $n$ from a life distribution. Let $T_n$ be the total time spent on test by the $n$ sample units until the failure of the longest ...
0
votes
1answer
43 views

linear model estimator proof

For a linear model, the noncentered form is $Y_i=\beta_0+\beta_1x_{i1}+...+\beta_kx_{ik}+\epsilon_i$ ---(a) the centered form is $Y_i=\alpha+\beta_1(x_{i1}-\bar{x}_1)+...+\beta_k(x_{ik}-\bar{x}_k)+\...
0
votes
0answers
13 views

Is multicollinearity acceptable where one variable is the anomaly between, or sum of, other variables?

I'm using the variance inflation factor (VIF) to calculate multi-collinearity between the variables I'd like to include within a regression model. I was reading this article: https://...
0
votes
0answers
13 views

Isolating two variables with log [closed]

I have the following equation: $$a=\log\left(\frac{x}{\log(y)}\right)$$ I am trying to isolate the $x$ and $y$ variables to get just $x / y =\dots$ From my algebra though it seems like this might not ...
0
votes
0answers
21 views

Variance and mean of residuals after rescaling by the sample variance

Suppose there are n random variables, $\mathbf{X}_{1},\cdots,\mathbf{X}_{n}$. Define $\bar{\mathbf{X}}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{X}_{i}$ and $\mathbf{S}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{X}_{...
1
vote
1answer
31 views

Forecasting model with OLS

I am trying to estimate a model by using OLS (ordinary least squares) regression, I find that the default rate (my dependent variable) $y_t$ is non-stationary, therefore I take the difference to make ...
1
vote
1answer
40 views

Mathematically finding a threshold for deciding on rare word

I am implementing spell-correction facility as a preprocessing step of for a text classification project. For this reason I have to make a knowledge-base where I shall be putting all the words along ...
0
votes
0answers
17 views

How to demonstrate a Correlation Inequality? [closed]

If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X, y_i \in Y, z_i \in Z$, we have $0 < x_i < 1, 0 < y_i < 1, 0 < z_i < 1$. The correlation between Z, Y is greater ...
0
votes
0answers
26 views

Why do Quartile, Decile, Percentile have plus 1 in their formula? [duplicate]

I searched this on the internet, but barely found anything, and what I found was not satisfactory. Quartile divides the data into 4 equal parts. Qi = i * (n+1)/4 where i=1, 2, 3 Decile divides the ...
0
votes
0answers
9 views

Is there any formal mathematical analysis or statistical analysis for transformer or attention mechanism? [closed]

Attention is prevailing these years in many fields, but hardly can I find any formal reference to prove its high performance.
0
votes
0answers
10 views

Finding variance in restricted row in long process [closed]

I have a data imported from an excel file. I want to find variance of each 13 rows such that firtly calculate the variance of row [1-13] (1 and 13 inclusive) , after that row [2-14] , after that row[3-...
1
vote
0answers
21 views

Strong Law of Large Numbers for bootstrapping: Proving bootstrap sample mean converges a.s. to mean

Setup Let $X_1, X_2, \dots$ be an i.i.d. sequence of random variables with the common distribution function $F(\cdot) := \mathbb P(X_1 \leq \cdot)$ and finite first moment $\mathbb E[X_1] <\infty$. ...
0
votes
0answers
15 views

Proof Maximum likelihood Hypergeometric model [closed]

Could somebody explain me how to infer in the hyper geometric model, and show the proof to obtain rigorously the maximum likelihood estimator in this model ?
0
votes
0answers
30 views

Limiting behavior of the ridge regression estimator as $\lambda \to \infty$

I am a bit confused about a few aspects of the behavior of the ridge regression estimator as $\lambda \to \infty$ (see photos below). The facts that the bias is $- \lambda (\mathbf X^\top \mathbf X + \...
0
votes
0answers
30 views

What is the correlation between random variables after being multiplied by the same lower triangle matrix decomposed from a covariance matrix?

Assume $C_{n \times n}$ is a positive, symmetric and semi-definite covariance matrix, we know that the LU decomposition exists, i.e., $C_{n \times n}=L_{n \times n}U_{n \times n}$. Now $n$ ...
0
votes
0answers
20 views

Is it possible for a set of random variables to each be highly correlated with another variable, but not highly correlated with each other? [duplicate]

Let $X_1, ..., X_n$ and $Y$ be random variables. Is it possible for the $X_i$'s to all have a high magnitude of correlation (absolute value of Pearson's $r$) but not be strongly correlated with each ...
0
votes
0answers
25 views

Exponential distribution confidence interval

Question: I think I mostly understand (a) and (b), but I don't understand (c). My Work: Asymptotically, $2\big[ \ell(\hat \lambda) - \ell(\lambda) \big] \sim \chi_1^2$, where $\hat \lambda = 1/\bar X$...
1
vote
1answer
29 views

Identity for K-Means Clustering

The property (12.18) from here states that $$\frac{1}{|C_k|} \sum_{i, i' \in C_k} \sum_{j = 1}^{p} \left(x_{ij} - x_{i'j}\right)^2 = 2 \sum_{i \in C_k} \sum_{j = 1}^{p} \left(x_{ij} - \frac{1}{|C_k|} \...
2
votes
0answers
46 views

What is a confidence distribution?

I am going through this paper. I am unable to understand the definition of a confidence distribution. A statistic C is an exact confidence distribution for a real parameter ρ if: ρ → C(ρ; y) is a ...
0
votes
1answer
54 views

Drawing graph of variance using R [closed]

I am a self -learner and try to learn statistics with R ,but i encounter with a problem i could not handle it such that I want to produce a graph of the variance of a binomial distribution with a ...
2
votes
1answer
106 views

Random effects one-way layout analysis

I am analyzing the model $X_{ij} = \mu + \alpha_i + \epsilon_{ij}$ where $\alpha_i \sim N(0, \tau^2)$ (i.i.d), $\epsilon_{ij} \sim N(0, \sigma^2)$ (i.i.d), and they're each independent of each other. ...
0
votes
0answers
17 views

Maxmium likelihood for the random effects one-way layout

Question: I am hoping someone can help me figure this out, or at least tell me whether I'm on the right track or not. My attempt: The likelihood of $\mathbf X$ will be the product of the multivariate ...
3
votes
1answer
72 views

How is it possible to have $P(A|B \cup C)$ lower than both $P(A|B)$ and $P(A|C)$?

Let's say we investigate disease probability given two symptoms. Thus we have 3 variables: A) has disease B) has symptom 1 C) has symptom 2 We have following data: Now we want to see which symptom ...
0
votes
0answers
21 views

UMVUE of the following parameter

Suppose I have $\{X_i : 1\le i \le m\}$ which are i.i.d random variables having Poisson distribution with parameter $\lambda$ and let $N_i = \min\{k : X_k > p \text{ and } k \ge i\}$ where $p<\...
0
votes
0answers
12 views

pairwise proportion test within three categories in R

I would need help in order to statistically test if proportion between columns are significantly different from each other. Here is an example where I would like to test for each Brand, if Ref, Dog ...
1
vote
3answers
134 views

I don't completely understand the concept of PCA analysis [closed]

First of all, PCA analysis is not something I came across in my economics studies. But, recently, I wanted to make a PCA analysis of American GDP. I started to read about the fundamentals of PCA and ...
0
votes
0answers
7 views

How to interpret the variable importance varImp() when training a LASSO/Ridge regression using the library caret and method = "glmnet"?

I have trained an elastic net regularized model and left with my top two variables - both factors. • How can I interpret the importance of each one? • Should I train a new linear model including only ...
0
votes
0answers
20 views

Strange notation in beta parameter estimation of mixed models

Suppose we have the following mixed model: $y = \boldsymbol{X \beta} + \boldsymbol{Z \gamma} + \epsilon$ Where: $\mathbf{X}$ is the fixed effect design matrix. $\boldsymbol{Z}$ is the design matrix of ...
0
votes
0answers
17 views

Expectation of inverse of sum of i.i.d. positive variables [duplicate]

Description: There is an indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time ...
1
vote
0answers
37 views

Probability of drawing N elements from M elements with per element draw probability without replacement

I have a list of M elements with per element draw probability, summing to 1.0. Example (M=10): L = [0.3, 0.2, 0.01, 0.05, 0.02, 0.02, 0.2, 0.05, 0.05, 0.1] Now I ...
1
vote
2answers
64 views

Sufficient Statistic for $f(x,\theta)=\dfrac{2}{\theta^{2}} (\theta-x) \cdot 1_{(0,\theta)}(x), \;\forall \theta \in (0,\theta) $

Let $X_{1},\ldots, X_{n}$ be random variables independent and identically distributed; show that the following density function is in the exponential family and find the sufficient statistic for $\...
1
vote
1answer
41 views

Getting pearson corelation coefficient greater than 1

I was learning about the metrics which measure the relationship between the variables. I wrote a python code that can generate and calculate covariance and person correlation coefficient. ...
1
vote
0answers
33 views

Expectation of inverse of sum of iid random variables [closed]

Description: There is a indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time \...
0
votes
0answers
25 views

PCA: is linearity important?

I have a 6 dimensions dataset where I want to apply PCA to remove one dimension. I did a small analysis to check for relationships in my data and concluded that there is very low linear correlation ...
0
votes
1answer
19 views

Estimating Population mean given a sample, and the population size

For my study, I have population size known (in order of millions), and a sample data (about 30 sample size). I can estimate population mean using 95% confidence interval by t distribution or normal ...
4
votes
1answer
329 views

Should I make equally sized samples for the Mann-Whitney U test if originally I have unequal sample sizes

I have 2 groups with unequal sizes (control 70% and test 30%) and need to find out if there is a significant difference between these groups. I've read that MW test works fine with unequal sized ...
2
votes
1answer
51 views

Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution

Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function: $$ f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R} $$ Find the joint ...
1
vote
1answer
27 views

Assumption to apply the delta method

When proving the delta method of distributions in my textbook we make the following assumption: Let $X_{n}$ be a sequence of random variables. and: ${\sqrt{n}[X_n- c]\,\xrightarrow{D}\,\mathcal{N}(0,1)...

1
2 3 4 5
138