Questions tagged [mathematical-statistics]

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

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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-...
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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 $\mathbb E[X_1] <\infty$. ...
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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 ?
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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 + \...
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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 $n$ ...
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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 ...
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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 $\hat \lambda = 1/\bar X$...
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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|} \...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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 ...
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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<\...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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. ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 $A\in \mathbb{R}^{d\times d}$ ...
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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: $\hat\sigma = \frac{1}{n} \sum_{i=1}^{n}|y_i-med(y_i)|$, ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\frac{\sum{x_i}}{\sqrt{\sum{x_i^2}}}$
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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 ...
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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]=...
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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 ...
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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 ...
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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 ...
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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 \...
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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
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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 ...
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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.
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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 ...
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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 ...

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