Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
6,862
questions
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
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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
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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 $\mu=E(Y)$.
$$\textbf{ Question: Explain why this link ...
0
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0answers
11 views
Expectation of Gaussian vector times its length
Suppose $\boldsymbol{x} \in \mathbb{R}^3$ such that
$$\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
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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
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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
$$D_H(A,...
0
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0answers
16 views
Validity of a joint distribution and the associated Likelihoods
I want to check the validity of a code that produces a 2D contour between 2 parameters with their associated Likelihood (normalized).
Here's an example below :
As you can see, I have plotted, for 3 ...
0
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0answers
14 views
To what degree should one trust significance of different analysis of the same data set done by different groups? [duplicate]
Suppose there is a data set $D$. There are 20 groups of researchers independently.
Each group will publish a few results on this data set $D$. Suppose each of them find about 3 tests significant($5\%$ ...
0
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0answers
25 views
Total Time on Test (TTT) Statistic
Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ denote an ordered sample of size $n$ from a life distribution. Let $T_n$ be the total time spent on test by the $n$ sample units until the failure of the longest ...
0
votes
1answer
43 views
linear model estimator proof
For a linear model, the noncentered form is
$Y_i=\beta_0+\beta_1x_{i1}+...+\beta_kx_{ik}+\epsilon_i$ ---(a)
the centered form is
$Y_i=\alpha+\beta_1(x_{i1}-\bar{x}_1)+...+\beta_k(x_{ik}-\bar{x}_k)+\...
0
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0answers
13 views
Is multicollinearity acceptable where one variable is the anomaly between, or sum of, other variables?
I'm using the variance inflation factor (VIF) to calculate multi-collinearity between the variables I'd like to include within a regression model. I was reading this article: https://...
0
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0answers
13 views
Isolating two variables with log [closed]
I have the following equation:
$$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
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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
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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
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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
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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
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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
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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
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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
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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
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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)...