Questions tagged [mathematical-statistics]

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

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27 views

calculating CDF of kth order statistic

I have recently started probability and statistics on my own. Pls help in understanding below. $x_{(k)}$ is kth smallest random variable from sample of n iids ($x_1 \to x_n$) For calculating CDF of ...
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1answer
22 views

Can the mean of a frequency distribution be regarded as

Can the mean of a frequency distribution be regarded as a weighted mean with relative frequencies as weights ?
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12 views

Generate synthetic data given AUC for multivariate X

I need to simulate data for a fixed AUC and for multivariate X. I came across the link below which explains for the univariate x. Generate synthetic data given AUC. Has anyone had any ideas or ...
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1answer
51 views

Joint distribution from multivariate Normal distribution

Consider two random variables $\{Y_1, Y_2\}$, which follow a joint multivariate normal distribution: $Y = [Y_1, Y_2]^T,$ \begin{equation} Y_1 \sim \mathcal{N}(\mu_1,\sigma_1),\; Y_2\sim\mathcal{N}(\...
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0answers
23 views

Bayesian inference: all at once vs one at a time

Suppose that I have a prior on a parameter $\theta$ and update this prior in light of the realisation of $n$ random variables. It seems plausible that it is equivalent to update the prior $n$ times, ...
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8 views

How to generate a random positive definite matrix in R of a given dimension [closed]

I am writing a function in which the input is a correlation matrix and I need to generate a random positive definite matrix of the same dimensions. I am unsure what the code is for this?
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1answer
30 views

Days as dummy variables

How should you treat "days" as a variable from a statistical perspective. The "days" variable can be defined as an integer describing the day of the year as follows: days = 1,2,3,4....
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7 views

Who would you estimate the total user bade of a social network (Twitter, Faceboo,etc.)? [closed]

I need help with this, I don't know how to start, I don't remember any form for a estimator that estimates the total population
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31 views

Why is data normalization important in Monte Carlo simulation? [closed]

I'm trying to understand why data normalization is important in a Monte Carlo simulation using an approximation equation - since that approach can predict arbitrary shapes and functions with random ...
1
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1answer
23 views

Beta-Binomial conjugate proof

Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely: Is the integral being evaluated or not? How does the entire expression reduce to a ...
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1answer
39 views

Prove that for any 2 events A and B , $P(A) + P(B) - 1 ≤ P(AB) ≤ P(A) ≤ P(A\cup B) ≤ P(A) + P(B)$

I want to prove 𝑃(𝐴∩𝐵)⩾𝑃(𝐴)+𝑃(𝐵)−1. How can I simplify the following proof? 𝑃(𝐴∩𝐵)=𝑃(𝐴)−𝑃(𝐴∩𝐵𝖼)=𝑃(𝐴)−1+𝑃(𝐴𝖼∪𝐵)=𝑃(𝐴)−1+𝑃(𝐴𝖼)+𝑃(𝐵)−𝑃(𝐴𝖼∩𝐵)=𝑃(𝐴)+𝑃(𝐵)+𝑃(𝐴𝖼∩𝐵𝖼)−1⩾�...
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10 views

derivation of lasso minimal solution for diagonal X

I am interested in deriving $\beta_j=0$'s solution. Let $L(\beta)=\sum_i(y_i-\beta_i)^2+\lambda\sum_i|\beta_i|$ where I have assume predictors are of the form identity matrix and intercept is set to $...
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1answer
16 views

Why does the GAN paper make the assumption that $G(Z) = X$?

There exists crucial assumption in one (several) proof of the GAN paper that doesn't really make sense to me. Let $X \sim p_{data}$ be the random variable associated with the input data, and $Z \sim ...
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1answer
16 views

Most powerful test of size zero for $\theta$ given random sample from $U(0, \theta)$

I found a couple of questions on this site (Most powerful test of simple vs. simple in $\mathrm{Unif}[0, \theta]$ and UMP for $U(0,\theta)$ (simple x simple hypothesis)) that are similar to my problem,...
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25 views

Propose a code on R? [closed]

$1$-Objectives : We provide for every student a dataplay. i.e., a vector $(X_1;...;X_{500})\in\mathbb{R}^{500}$. This vector is the realization of a 500-sample $(X_1;...;X_{500})$ of a random variable ...
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2answers
83 views

Exchangeability and joint distribution

The definition of an exchangeabilty for a finite sequence says that, if we have random variables $X_1,\ldots,X_n$, then for each permutation $\pi: \{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$, the joint ...
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0answers
9 views

even after multiple questions how is variance helpful over standard deviation in building a theory [closed]

How is variance as a concept preferred over standard deviation to build theory.? Can I have two scenarios with the same variance values
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1answer
34 views

Find distribution of the transformation $Y'(I-P_1)Y$ given distribution of $𝑌_{nx1}$ ~ 𝑁$(𝜇1, 𝜎^2𝐼)$

I am currently self learning econometrics from A Course in Econometrics of Goldberger and I'm attempting a problem sheet from the course given by my uni. The problem is the following: Let $𝑌_{nx1}$ ~�...
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1answer
16 views

Can I trick an ANN into learning individual values of an array from the average?

It is very easy to have an ANN that can compute the average of an input vector, but is it possible to have an ANN that can generate the output vector given just an average of the output vector as the ...
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1answer
24 views

Ancillary function of a random vector, which is independent of change of origin and scale

Let $(X_1,\ldots,X_n)$ be a random vector, whose distribution involves unknown: location parameter $\mu$ and a scale parameter $\sigma>0$. It follows, that any measurable function $f(X_1,\ldots,X_n)...
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20 views

Proof help: Log Inequalities [closed]

I have been working on a lemma and I understand every component of the proof but I do not understand how "the proof follows". Essential I need help piecing the components of the proof ...
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1answer
7 views

High correlation between linear regression residuals MSE and dependent variable's variance

I am building a linear regression model for 5000 gene expressions, and each gene has its own independent variables. So in total, there are 5000 linear models and for each model, I have 360 samples and ...
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0answers
20 views

Given 11 different integers from 1 to 20. Prove that at least two of them are exactly 5 apart. pigeonhole principle [closed]

Given 11 different integers from 1 to 20. Prove that at least two of them are exactly 5 apart.
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0answers
20 views

Poisson Distribution vs Poisson Process [duplicate]

I need to understand the relation between both these concepts with an example. I have an good understanding of Poisson distribution and it's applications. But I am having difficulties linking it to ...
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1answer
19 views

Lemma KL-Divergence (Differential Privacy)

I am studying differential privacy and I got stuck again in proof of a lemma. Which is: "$D_{\infty}^\delta(Y||Z) \leq \epsilon$ if and only if there exists a random variable $Y'$ such that $\...
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0answers
7 views

Not able to capture spikes in SARIMAX model for future forecast

Data has the spike in 2013 and 2017. To capture this effect in 2021, dummy column is introduced with 1 as spike in a month otherwise 0. Below is the output and the model used, Why the model is not ...
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16 views

what is the best sampling procedure to use? (stratified)

all, i am a little confused on what sampling method to use. i am doing a classification problem where i am trying to classify a person as having cancer or not. there is apparently huge variance in ...
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0answers
19 views

Interpretation of the ratio of two pdfs evaluated at a certain point?

What is the interpretation of the ratio of two pdfs evaluated at a certain point? Is that a statistical distance? Is there any applications of it? I only know one application of differential privacy, ...
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0answers
13 views

What to choose for null hypothesis and alternate hypothesis? [duplicate]

In this article which explains types of t-tests for “Independent Two-Sample t-test” example: We can confirm that the t-statistic is again less than the t-critical value so we fail to reject the null ...
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0answers
43 views

Why can't we accurately compute covariance matrix in high dimensions?

I am reading pg 651 of Elements of Statistical Learning,where is says: "The simplest form of regularization assumes that the features are independent within each class, that is, the within-class ...
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1answer
189 views

Mann-Whitney test for a large sample

I'm not a statistician, so pardon me for being naive on this subject. I'm trying to understand if there's any statistically significant difference in the medians of 2 groups. Here are some of the ...
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0answers
10 views

Understanding Generalized Canonical Correlation Analysis (GCCA)

I am trying to learn a single embedding from multiple views using Generalized Canonical Correlation Analysis (GCCA) using the below formula. I am confused in the below formula regarding what is to be ...
6
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1answer
265 views

How to combine standard errors for correlated variables

I was wondering what would be the formula for calculating the standard error of a quantity (A) that is the ratio of 2 quantities (A = B/C) if B and C are correlated? According to page 2 of http://www....
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12 views

How to reasonably use z-score time series with goal-setting? [closed]

I have several business groups that periodically report on their individual primary metrics. The metrics differ in units of measure - some are numerically on the order of 10^6, others on 10^2, and so ...
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0answers
13 views

How to calculate Average Spend from a Spending percentile value

I have a dataset which contains percentile values of all households spending more than 200 $ on food in a ZIP Code. Sample Dataset: ...
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0answers
15 views

How can I solve for posterior probabilities when the evidence only is informative about a proportion of the CDF?

I want to find the Bayesian solution to a game where one has to infer the probability of a latent variable being within a certain range conditional on the outcomes of a Bernoulli experiment. I first ...
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0answers
16 views

How to prove that in a linear model with an interaction term, centering the covariates doesn't impact the estimate of interaction coefficient? [duplicate]

So if the model is $y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_{12} x_{i1} x_{i2} + \epsilon_i$, how can it be proved that the estimate and standard error for $\beta_{12}$ the same if we ...
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0answers
3 views

What test to use when checking significance between two accuracy and perplexity values for language models trained on the same dataset?

I created different language models on the same dataset and I want to check if one model performs significantly better than the other, by comparing accuracy and perplexity. I have the following: Model ...
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0answers
37 views

Piecewise exponential baseline hazard in R without using inbuilt function

I am trying to write a function in R that would fit a piecewise exponential baseline hazard for an arbitrary number of knots/intervals (e.g., 4 knots) assuming a gamma frailty. I'm really hoping ...
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1answer
25 views

Bayesian estimation Prior adaptation [closed]

I have a dataset of 1 dimensional 20points as prior information, so assuming prior distribution to be Gaussian distribution we can easily find its variance and mean. Now we will use this prior finding ...
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0answers
22 views

How to construct the likelihood function of compound Poisson process?

Since in the compound Poisson process (CPP), the jumps occur according to the Poisson process with intensity $\lambda(t)$. The jumps size is iid random variables and itself independent of the Poisson ...
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0answers
10 views

Alternatives to Luces Choice Rule?

I'm currently working with a cognitive model (quite simple) which estimates probabilities of choosing an item out of a set of different items. This is done with Luces Choice rule. However, I'm not ...
2
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1answer
114 views

When does $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exist?

Consider a univariate stochastic process (time series) $X_t$. I am interested in conditions under which $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists. For example if $X_t$ is a stationary process ...
3
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1answer
47 views

Likelihood Ratio and Sufficient Statistics

I am not very experienced with statistics, so I apologize if this is an incredibly basic question. A book I am reading (Examples and Problems in Mathematical Statistics - Zacks) makes the following ...
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0answers
11 views

What to do if some conditions of DoE are hard or impossible to run?

Assume we have 3 factors with 2 levels each. If I understood it correctly, for small number of factors we can consider a full factorial design, in this case would be 2^3 full factorial design. What ...
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0answers
13 views

How would you work out the total number of possible plays in a game of dominoes?

Looking at a standard double-six dominoes set that has 28 tiles 0:0 through 6:6 how would we calculate (or estimate) the number of possible moves in a single game? For this so far I've come up with: ...
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1answer
58 views

Find best unbiased estimator for $\theta$ when $X_i\sim U(-\theta,\theta)$

I am having an issue finding a best unbiased estimator for $\theta$. Any help is appreciated. Let $X_1, ..., X_n$ be a random sample from a population with pdf: $f(x\mid\theta)=\frac{1}{2\theta}$ $-\...
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0answers
25 views

Error terms and Residual in Time Series

I started learning Time Series Forecasting and the below questions keep my head confused a lot. Many papers mention that the forecast errors have to be normally distributed and what about the white ...
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0answers
10 views

Combine Two Covariance Matrix [duplicate]

I have two sets of data and each data set has its own mean, standard deviation, and covariance matrix. Can I combine the covariance matrices from each data set to make the covariance matrix for the ...
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0answers
17 views

How to determine the conditions under whicht a time series is weakly stationary

Let $A, B$ be two uncorrelated random variables with mean $\mu_{A}, \mu_{B}$ and variances $\sigma_{A}^{2}, \sigma_{B}^{2},$ respectively. For some $0 \leq \theta \leq \frac{\pi}{2},$ set $X_{t}=A \...

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