Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
6,847
questions
0
votes
0answers
7 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
16 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
12 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
26 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
22 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
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0answers
18 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
23 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
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1answer
23 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|} \...
1
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0answers
40 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
32 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
38 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
13 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 ...
0
votes
2answers
43 views
Normal distribution - finding X value given probability P(-x < X < x) = 0.95
I am learning statistics by myself online and I just encountered a problem that I am not able to solve.
X~NormalDistributed(10, 4) so that I need to find P(-x<X-10<x) = 0.95. It resulted to the ...
2
votes
0answers
36 views
probability of P(A|B∪C) lower than individual probabilities P(A|B) and P(A|C), how is that possible?
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
19 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
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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
125 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
5 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
32 views
Does sample of random variable have same distribution as the random variable? [closed]
If Y = Binomial distribution ($B_{p, n}$).
Can anyone tell me what will be a sample of Y?
And, is the sample a random variable with same distribution as of Y?
0
votes
0answers
19 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 ...
0
votes
0answers
7 views
statistical way of evaluating multiple devices [closed]
I have 12 devices, and we evaluate their performance by running some tests (could not compare results among different runs)
however, not every time we run all the devices, for example
...
1
vote
0answers
35 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
60 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
40 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
32 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
17 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
286 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
49 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
26 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)...
0
votes
0answers
12 views
Expected value of log-likelihood and KL divergence
Background:
Let $x_t = Ax_{t-1} + w_t$ be a discrete linear time invariant system where:
$x_t \in \mathbb{R}^d$ for all time samples $t$ corresponds to the state vector
$A\in \mathbb{R}^{d\times d}$ ...
1
vote
0answers
36 views
consistency of mle of double exponential distribution ( not advanced)
Let $y_i\sim DE(\mu, \sigma), $ $i=1,2,...,n, \ i.i.d.$
Where DE represents the double exponential distribution. The the MLE of \sigma is:
$\hat\sigma = \frac{1}{n} \sum_{i=1}^{n}|y_i-med(y_i)|$, ...
0
votes
0answers
14 views
Finding a statistical method to summarizes set of points [closed]
I want you to help me find a statistical method that summarizes these points in a single point. The idea from each of these points is to find how close they are to one. Therefore, I would just like to ...
2
votes
1answer
40 views
Variance of the median
For large $N$ the sample median is approximately normally distributed with mean $μ$ and variance $π/2N$. The efficiency for large $N$ is thus $2/π≈0.64$
Can somebody explain this for me?
Where does ...
0
votes
0answers
34 views
Standard deviation of population and sample [closed]
I have a value of population's standard deviation (say σ) and I have sample of size N and SD of sample (say s). Population and sample supposed to be normally-distributed.
Using χ² distribution I can ...
0
votes
0answers
21 views
Statistics question [closed]
An athletic teams consists of 2 Level 1 students, 5 Level 2 students and 3 Level 3 students. The team members are ranked according to their performance after a race. Assuming no 2 team members run the ...
0
votes
0answers
40 views
How to optimize a function with respect to a distribution, in the context of variational inference
Context: I am learning about variational inference. The reference I am following is linked at the end of this post.
Goal: I want to learn how a marginal variational distribution $q_k$ is optimal in ...
1
vote
2answers
82 views
Derive the distribution of the fraction ${\sum{x_i}}\Big/{\sqrt{\sum{x_i^2}}}$ [closed]
Suppose ${X_i}$ follows the standard normal distributon N(0,1), what is the distribution of $\frac{\sum{x_i}}{\sqrt{\sum{x_i^2}}}$
0
votes
1answer
23 views
How to check for the accuracy of prediction
I am doing a personal project to see how well does FIFA potential player stats predict the actual overall stat after 3 years.
Meaning, if a player has a potential of 85 in 2015, how accurate should I ...
1
vote
1answer
21 views
Forming a consistent estimator for the area under the regression line
I am trying to solve the following problem:
Take the following simple linear regression model, where $x_i \in \mathbb R$:
$y_i=\beta_0 + x_i \beta_1 + \epsilon_i$
Given that:
$\mathbb E[\epsilon_i]=...
0
votes
0answers
14 views
Correlation Feature Selection
I have a huge dataset with a lot of features (approx. 1050) and a target variable. The features can be broken up into groups (7 groups to be exact).
My initial approach to feature selection was to ...
2
votes
0answers
25 views
Intuition of using p(x) (true distribution probability) in KL Divergence definition
We all know that $D(p||q) = \sum_x p(x)log\frac{p(x)}{q(x)}$ and it is used to quantify the difference between the true distribution p and the observed distribution q. However, I do not get the ...
0
votes
1answer
32 views
What does the skewness, when converted to the units of the data, represent?
I have calculated the skewness of my data. I was wondering what the skewness, when converted to the units of the data, represents in non-mathematical (biological?) terms? For instance, I know that the ...
5
votes
1answer
77 views
Deriving the limiting distribution of a sum of Pareto distributed variables
For a series of independent and identical Pareto distributed variables $X_i$ with $\alpha > 2$, their sum $S_n = \sum_{i=1}^{n} X_i$ has a normal distribution as limiting distribution for $n\to \...
0
votes
1answer
24 views
Conditional gamma distribution derivation
Suppose we specify the gamma pdf in the following format:
$f(x) = \lambda e^{-\lambda x} \frac{(\lambda x)^{n - 1}}{(n - 1)!}$
Further suppose we want the distribution of a $\text{gamma}(\lambda = 1,n ...
2
votes
1answer
51 views
Variance of the ridge regression estimator
I have some concerns about the image below (note that $\mathbf W_{\lambda} = (\mathbf X^\top \mathbf X + \lambda \mathbf I)^{-1} \mathbf X^\top \mathbf X$):
My main concern is that this derivation of ...
1
vote
1answer
28 views
Expected value of the ridge regression estimator
I am trying to understand this derivation:
I think everything except the last equality is fairly simple, but I do not understand the last equality. Is there an error here?
I appreciate any help.
0
votes
0answers
14 views
The relationship between granger causality and correlation
In one of my experiments I got that whenever I have a significant granger causality, I have a high correlation. I was wondering if this is true in general. In particular, what confuses me is that when ...
0
votes
1answer
37 views
Step in proof of poisson probability
for the case when $s < t$
Let $N(t)$ be the amount of arrivals occurring at time period t in a Poisson process.
When $N(t)$ is known, the number of events by time $s$ can then be shown to follow a ...