Questions tagged [mathematical-statistics]

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

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Linear regression model excel

Suppose you fit the regression model E(Y) = b0 + b1X1 + b2X2. Further, suppose that the p-value associated with coefficient b2 is 0.312. What does this tell us?
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Deriving the basis functions for natural cubic spline

Im looking through Section 5.2.1 of Elements of statistical learning, and am confused by this section I understand that, from this answer , the definition of $d_k(X)$ is important in order to derive ...
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Statistic to take account of sensitivity of temperature values

I have a unique problem. I am looking to quantify the difference between two temperature value of different components. I am looking at three statistic to see that difference Absolute mean difference ...
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1answer
960 views

Asymptotic Distribution of the Wald Test Statistic

I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to. For clarity, the binary ...
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Why does the linear regression of “x on y” intersect with the linear regression of “y on x” at (x̄, ȳ)?

Problem: The following table below shows the marks scored by seven students on two different mathematics tests. $$\begin{array}{c|c|c|} \text{Test 1 (x)} & 15 & 23 & 25 & 30 & 34 &...
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Correlation Coefficient Squared is Less Than or Equal to One

Problem Statement: Let $Y_1$ and $Y_2$ be jointly distributed random variables with finite variances. Let $\rho$ denote the correlation coefficient of $Y_1$ and $Y_2.$ Using the inequality $$[E(Y_1Y_2)...
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Linear transformation *not* to use to scale mean and SD

I'm working through the Exercises in Regression and Other Stories. Exercise 3.6 on Linear transformations asks you to provide a formula to rescale a variable with mean = 35 and SD = 10 to have mean = ...
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119 views

What is Factorial Design in the context of Linear regression

I have been trying to understand the concept behind factorial design and its importance in connection to linear regression. I will be glad if somebody can give a clear and tone down explanation in ...
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Is there a commonly-accepted/used notion of parametric statistical model equivalence?

To fix notation, let a set of possible data $X$ and a set of admissible parameter values $\Theta$ be given. Let $\mathscr P(X)$ be the set of probability distributions on $X$. A parametric ...
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Independence of groups within a single predictor variable

Say I have two variables: 'Sale Price' of a house, and 'Overall Quality' which is an ordinal variable with values between 1 and 10 (1=lowest quality, 10=highest quality). I want to perform an F test ...
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Confidence intervals/ellipses when variables are correlated

I was looking to create a confidence ellipse for my X and Y variable in order to identify potential outliers. I'm new to this area so my understanding and use of this method may be wrong (please ...
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349 views

References for use of symplectic geometry in statistics?

I have heard that many problems in Mathematical Statistics can be stated and solved in terms of Symplectic Geometry. Unfortunately this was a pretty vague statement and I am interested in something ...
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How to calculate the variance and expected value of a random variable with density function f(x) in R

was wondering how to calculate the expected value and variance of some function f(x). For the Expected value $\mu,$ I integrated x*f(x) and I'm confident that is correct, but I'm confused about how to ...
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Adjusting OverAll Time Performance using sectional Times

Assume I have two runners, Runner_1 is the average runner and Runner_2 is Usain Bolt. We let the two runners compete in 3 different races on a 100m distance. We assume that the benchmark time for 50m ...
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Which statement for statistical test is accurate, regarding significance?

Statistical tests between 2 mean shows us whether the difference between sample means is significant given the inherent sample variance. Statistical tests between 2 mean shows us whether the ...
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1answer
545 views

What are the units of entropy of a normal distribution?

I have a random process that follows a normal distribution. It's parameters are mean = 35 units, std.dev. = 8 units. I've seen from the wiki entry for the normal distribution that there is a formula ...
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1answer
36 views

Bayesian vs Frequentist Prediction Methods and Frequencey Garantees

After reading Larry Wassermans blog on the difference between Bayesian and Frequentist inference I started to appreciate that frequency guarantees can be desirable regardless of the inference method ...
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Correlation Coefficient for Hypergeometric-type Distribution

Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls $(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement). Let $Y_1,Y_2,...
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1answer
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Boostrap Confidence Interval Calculation

So, I have a sample dataset of size 500, and I've bootstrapped it 1000 times and took the mean of each bootstrap sample. So now, I have a list of means a.k.a ...
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Why “sorting” is needed for simple random sampling [closed]

Tutorials demonstrating simple random sampling, first list the full number of population members (i.e., sampling frame) in a column and then assign a random number from $0$ to $1$ from a uniform ...
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36 views

Calculate mean from range

Suppose we have data points like How can we calculate mean from this dataset
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24 views

Statistical test for Tweedie distribution

Is there any statistical test to test whether a highly skewed dataset follows the Tweedie distribution? I am asking this question because I have a highly skewed target variable and I am not sure it is ...
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1answer
155 views

convolution and deconvolution of random variables of different dimensions

Preliminary: Let's say we have $Y=X+Z$ ($Y$ is data, $X$ is latent variable and $Z$ is noise), where the random variables are all in $\mathbb{R}$. Then an inverse Fourier transform leads to \begin{...
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Difference between supervised machine learning and design of experiments?

I'm an experimental physicist by training and have used standard statistical methods to analyze data, and the design of experiments (DOE) framework to develop models of systems by varying inputs and ...
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236 views

If I have a vector of $N$ correlated probabilities. How can I turn them into binary $0,1$ without destroying the correlation?

My ultimate goal is to be able to have a way to generate a vector of size $N$ of correlated Bernoulli random variables. One way I am doing this is to use the Gaussian Coupla approach. However, the ...
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what is the difference between Lasso and Logistic regression with L1 penalty?

I am new to coding and statistics. I wanna do classification on a data set and i wanna know if Lasso and Logistic regression with L1 penalty are different?
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Is the pooled variance equal to the variance of the centered populations?

Suppose two data arrays of length $n$ with variance $\sigma^2$ and mean $\mu$. Is the pooled variance $\sigma_P^2 = \frac{(n_1-1)\sigma_1^2 + (n_2-1)\sigma_2^2}{(n_1-1) + (n_2-1)}$ equal to the ...
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1answer
51 views

Analyzing the statistical independence of a Bernouli sequence

For my research, I need to check whether a sequence of Bernoulli trails (1 as success and 0 as failure) is a Bernoulli process and are statistically independent. As per the following discussion, and ...
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How to compare two estimates?

I have determined the volume of multiple objects via two different methods. I would like to express, per object, something like the percent error: $\delta = \frac{|V - V_{approx}|}{V} \cdot 100$ ...
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How do I adjust the corona virus situation for demand forecasting? [closed]

A lot of business have been hit by the current pandemic. If I want to do a demand forecast for say next 3 months, how do I adjust the corona virus situation in my model?. Which method should I use? ...
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These questions regarding probability distribution are confusing to solve. Need your help [closed]

Consider a race with 140 participants. How many possible outcomes are there for the top three positions of gold, silver, and bronze? (4 ...
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1answer
40 views

How do you test for independence of a process?

I'm working on a research project for one of my professors. He wants to know whether a variable that takes on different values over a large period of time (say 1,000,000 different values) is i.i.d. ...
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42 views

Understanding statistics numbers vs. counters

Suppose I analyze text and find that there are: Word count: 3465 Number of lines: 130 Number of paragraphs: 5 Are these actually statistics of the text? I am not sure because statistics involves "...
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Statistical literature on task prioritisation problems

I am lookig for statistical papers on task prioritisation problems. In particular I am looking for solutions to the following problem or slight variations thereof: You have a set of tasks, each with a ...
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Probability from two Cohort's Rates

I have a retention rate for two variants grouped to cohorts of users based on their sign-up dates (11-2019 cohort will include all customers signed up during Nov. 2019). **Var A** Cohort Total Month0 ...
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Derivation of Neuhaus, Jewell(1993)

I wish to ask a derivation problem in Neuhaus, Jewell(1993) - "A geometric approach to assess bias due to omitted covariates in generalized linear models" The statistical True model dealt in ...
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How to find value of omega in CUSUM?

I am trying to understand CUSUM or Cumulative Sum Control Chart. In the example on above linked Wikipedia page, the formula for high CUSUM value indicating positive ...
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What's the common factored form of presenting Rao's score statistic?

Rao's score statistic is defined as $$S(\theta_0) = \frac{U(\theta_0)^2}{I(\theta_0)}$$ where $U(\theta_0)$ is the score function and $I(\theta_0)$ is the Fisher information. When one formulates $S$ ...
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1answer
27 views

Regarding modelling longitudinal variables using Two stage mixed effects modeling

I have a question about the basic understating of key statistical methodology. I came across the idea about two stage modelling to incorporate longitudinal predictors. Lets say there is a continuous ...
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25 views

Is non-normality of data an issue for large samples?

I am aware that due to the CLT, when analysing and working with means, non-normality of data isn't an issue if the sample size is large. However, I am reading Multivariate Analysis by Hair which ...
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1answer
50 views

Most Efficient Estimator and Uniformly minimum variance unbiased estimator

I am studying Estimation theory from "Introduction to theory of statistics" by "Mood and Graybill". After completing I thought I understood UMVUE (uniformly minimum variance ...
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Is this the only way to determine if a parameter can be estimated efficiently?

I am tasked with determining if a particular parameter can be estimated efficiently. Given that an efficient estimator is an unbiased estimator which achieves the Cramer-Rao lower-bound, is the only ...
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Does Fisher scoring always outperform Newton optimization?

My understanding is that Fisher scoring has several advantages over Newton raphson optimization such as Computational efficiency: if certain conditions are met (example:During MLE estimation, if link ...
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Confidence/Prediction Interval on binary/multiclass problems

I've recently fine-tuned a deep learning framework/model BERT for a sentiment classification task. I'd like to look at the confidence/prediction interval of the predicted sentiment scores (class 1 and ...
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Is curse of dimensionality more harmful for distance based models? [duplicate]

I want to understand what exactly is curse of dimensionality and how does it effect the model performance. Does the the concept apply to all the models?. Is it equally bad for distance based models ...
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Confused about the Mann-Whitney $U$ test. Does it test distribution equality (pdf) or just mean/median equality?

I am rather confused about the Mann Whitney test, many statements I read state it tests for distribution equality between two populations and some state it tests for means/median/central tendency only....
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25 views

What's is the equivalent, by switching dydx to dxdy, of the following conditional probability integral of two continuous random variables?

Given that $X$ and $Y$ are two continuous random variables and where $f_{Y|x}(x, y)$ is the conditional density function of the two variables when $\{x<y, 0<x<\infty, 0<y<\infty\}$, ...
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How to determine if results of two classification models are statistically significant without knowing individual predictions?

I have two classification models that I want to compare to see if one is "statistically" an improvement over the other. Both use the same training and test set and are only ran once for each ...
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245 views

minimum number of rolls necessary to determine how many sides a die has

For example, say you have a black box that has a number of n-sided dice in it. You have 4-sided dice, 6-sided, 8, 10, 12, 20 and so on. The die sides are all A except for one side that says B, e.g. ...

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