Questions tagged [mathematical-statistics]

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

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F-test and Wald test in time series

Assume we have the time series $X_{1,t}$ and $X_{2,t}$ where $X_{i,t}=(x_{i,1},...,x_{i,T})$. We build two models as (Assume the time series are weakly stationary): \begin{align} X_{1,t}=\sum_{\...
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This data is related to smog AQI. I want to know which method is applicable here to know the forecasting. Is it seasonal exponential smoothing method?

Mathe The daily average of Air Quality Index in Lahore is plotted in a time series graph
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what is the derivative function f(x+y) with respect to x? [migrated]

I have the digamma function of alpha + beta digamma(alpha+beta), I want to calculate the derivative this function with respect to alpha. I need it as a derivative of inverted beta-liouville function ...
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Predict future values of a variable from historical data

Thanks in advance I have a problem statement at hand that requires me to predict the lead time for a supplier-material combination. the historical lead times are erratic and continuous in nature and ...
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What is the justification of using the first-order Taylor expansion in the proof of asymptotic normality of MLEs?

Proofs of asymptotic normality that I have seen involve taking the first-order taylor expansion of the score function (derivative of the log-likelihood) about the MLE estimate evaluated at the true ...
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For every x, y ∈ IN = {1, . . . , N} define K(x, y) = min{x, y}. Prove that K is a valid kernel

I need to prove K is a valid Kernel. x and y are positive integers, and K is the minimum of x and y. I know I should prove K positive definite. But May I see how.
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Compute conditional probabilities in Bayesian hypothesis testing

I would like to solve a problem by the Bayesian approach. I have measurement samples $\texttt{y}_{1}$ and $\texttt{y}_{2}$, they are Gaussian independent with (scalar) means $\theta_{i}$; $i=1,2$ and ...
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Random Walks Question [closed]

I am trying to solve this question by using the reflection principles. Let a>c>0 and b>0. A is the set of all paths of a random walk which end at c in their final n’th step. B is the set of ...
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How to use machine learning to derive a mathematical function for image boundary boxes

Say I have several images taken from a car dashcam where I have boundary box around the car with corners A (x1,y1), B (x2,y1) ,C (x2, y2) ,D (x1, y2) and a boundary box around it’s associated number ...
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Comparing two cross-correlation functions with one another

Say for example I calculated two sets of cross-correlation matrices, one that compared time series A1 and A2 for 10 test subjects and another that compared time series B1 and B2 for the same 10 test ...
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What is a random variable in ADAM optimizer?

Look at the definition of ADAM optimizer, from the original paper of Kingma and Ba (paper): See algorithm $1$ for pseudo-code of our proposed algorithm Adam. Let $f(\theta)$ be a noisy objective ...
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Compute the Maximum A Posteriori (MAP) estimate of θ

How can I compute the Maximum A Posteriori (MAP) estimate of $\theta$ with those informations: a discrete random variable y with values in {1, 2, . . . , N} has a Binomial distribution depending on ...
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Sum of probabilities of all samples gives the total volume?

I am working on a churn problem (binary classification - whether a customer will churn or not). Now using logistic regression, I get the probability whether the customer will churn or not. Can I add ...
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How does the following term happen when using the Wald Test on two-sample binomials?

I'm having trouble understanding one of the steps of the Wald test used in two-sample binomial proportions. More specifically, when $X_1, ..., X_{n_1}$ are iid Binomial$(1, p_1$), $Y_1, ..., Y_{n_2}$ ...
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Comparing the spread of data from differently-sized populations

I have what seems like a simple but fundamental question. Some posts on CV seem to be along similar lines, but not quite address what I'm trying to understand (for instance Comparing percentages from ...
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"Median" version of L-moments

The L-moments are useful as robust summary statistics for various probability distributions, similar to the moments but only requiring the mean of the distribution to exist. Each L-moment is a linear ...
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Proof that p-value in OLS regression is symmetric

OLS regression is not symmetric, meaning that it produces different relationships if you flip the dependent and independent variables; however, it would seem odd if the p-values were different and ...
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Given a number of four-person families, how many people need to be selected so 50% of the time, half of them are in familes with another selected?

Doing some worldbuilding and was thinking about how connected my world is. I had thoughts on dealing with how I could calculate pedigree collapse and such, but wanted to use a simpler model to think ...
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Monty Hall Gone Crazy [closed]

Can Someone help me out with these ?Answer along with solution will be much appreciated Q1) Imagine that now host have the following instructions. Put a prize behind a random door. Let the guest guess ...
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1 answer
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How to calculate that a value falls within a range given another value historically

I have a forecasts for a specific variable and historical actual and forecast values for the same variable. Let's say that my forecast for the next time period is X, what method can I use to calculate ...
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2 votes
1 answer
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Sample size to assess grade deflation

My friend is a teacher at a University. In her course there is only 26 students, 13 of them fails, so she got deflation grade for this class. She believes that 26 students are not a good sample size ...
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is the $Var(\hat{\beta_1}|X)$ same as the variance of $Var(\hat{\beta_1})$ in simple linear regression?

Here are two captures from different videos on youtube, one obtains $Var(\hat{\beta_1})$, the other obtains $Var(\hat{\beta_1}|X)$ but they yields the same equations when the second capture further ...
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Given win ratio and team size, how to derive global player rank?

Question: How do I derive my global rank among players, given my win ratio and the size of the teams in randomly matched games? The game is Call of Duty: Modern Warfare (2019). win ratio: 2.73 total ...
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In Bayesian hierarchical models, what is the difference between an Empirical Bayesian approach to parametrising priors vs using flat hyperpriors

Say I have a simple hierarchical model, where: $y_{g,i} = \beta_g x_{g,i} + e_{g,i}$ where $g$ represents the group, $i$ represents the individual within the group, and $e$ is the error. So the ...
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How to derive this conditional expectation (Multivariate normal distribution..?) [duplicate]

Problem is as follows: $Consider\,an\,iid\,sample\,of\,\{X_i,\,i=1,\,2,\,\cdots, n\},\,where\, X_i \sim \\ N( \begin{pmatrix} \rho \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{...
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Conditional Independence Testing using Fisher

I would like to prove the conditional independence of two categorical variables ("a","b") given a third categorical variable "z". I need some help in explaining the ...
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Finite population correction for the variance

Just when I thought I was starting to understand Bessel's correction, I noticed that it is not valid when the sample size equals the population size and so likely not valid for sample sizes close to ...
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In standard normal distribution table how i deal with value of Z greater than or equal to 5

How can i deal with sample distribution table when Z is greater or equal to 5? For example:(-2.04 <z<-5.96) Sample distribution table value for -2.04 will be 0.2018 then what will be the sample ...
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Why do the standard errors I get on the statsmodels regresssion coefficients look incorrect?

The regression readouts from statsmodels.OLS provide standard errors. These aren't matching the ones I get analytically when applying the following formula: sqrt [ ...
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An extremely basic question about statistics

We do statistics on data which we assume to be outcome of random trials. The question however, is -- are we assuming that the outcome can be represented as a random variable? The outcome of an ...
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If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$

If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$ I have no idea what to address with the conditional expectation part. Thank you for any comments, someone has ...
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Integral of the product of two gaussian

Here is the big picture of my problem: In the image below, X and Y represent 2 independent gaussian distributions. So the circles are the representation of the resultant bivariate gaussian. This ...
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Product of kernels vs Composition of kernels

According to Wikipedia there are two main operations between two kernels: product and composition. They look almost identical to me and I cannot figure out what's the intuition between these different ...
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I have quarterly data for my dependent variable while annual data for independent variables

Can I ask if I can use quarterly data for timeseries and convert the dependent variable to an annual data to use for multiple regression? Is that possible? If yes, can you provide an article to ...
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Investigating significant difference in composition between years

I want to investigate if there is a difference in the trophic composition of fisheries landing data between years. I have several years of data and I have the raw weight of each trophic group for each ...
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Assess team win rate by combining individual win rates?

Suppose I have 5 players on each team in a game where each player selects a character (League of legends, Valorant, etc.). I am considering the overall win rates of the characters, and the unique win ...
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Choosing the correct statistical test to compare different methods on the same population

In my study I have a single population of 8 participants. I am interested in looking at differences in behavioural data between each subject (the behavioural data being collected under the same ...
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Proof of the Student t-test for independent samples drawn from the same normal distribution when $\mu \neq 0$

I'm following the proof in Cramer's book Mathematical Methods of Statistics, $\S 29.4$. There it is assumed that we have two independent samples $x_1,\ldots, x_{n_1}$ and $y_1,\ldots,y_{n_2}$ drawn ...
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Why do we need a binary tree for this way of computing the two-sample Kolmogorov-Smirnov statistic?

In this paper, Detecting Change in Data Streams, section 5 Algorithms, the authors show a way to compute the two-sample Kolmogorov-Smirnov statistic over intervals and initial segments in $O(log(m_{1}+...
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How do I find the quartiles of a skew normal distribution parametric model of some data?

I have to program this in some environment so I won't be able to access other softwares. Let's say I've got some 50 numbers that is more or less skewed to one side. If we are to assume that it is like ...
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On the likelihood assessment of two extremely rare events

I've been pondering the concept of subjectivity to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my question over trivial hypothetical examples below. I did ...
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Appropriate to replace -inf with 0 after log transform? [duplicate]

I have a dataset of customers and their purchase data. Meaning, for each customer id, I have variables indicating number of unique products they bought, Number of online orders they placed, How many ...
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Sufficient condition for $ \sigma_{X}^{2} \leq \sigma_{Y}^{2}$

Suppose $X$ and $Y$ are random variables whose expected values are $\mu_X$ and $\mu_Y$, and variances are $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$, respectively. Also, we suppose $F_x$ and $F_Y$ are the ...
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Estimate a parameter that is not in a PDF, makes sense?

I have been reading some posts about the proof of the invariance of MLE because I did not fully understood the proof given in Statistical inference by Casella Berger in the page 320. My doubt is the ...
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1 answer
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Covariance between the first elements of the samples

There are two non-independent samples $a_1, a_2, ... ,a_n$ and $b_1, b_2, ..., b_n$. In Python I can calculate with np.cov($\{a_1, ..., a_n\},\{b_1, ..., b_n\}$ ) сovariance between these samples. But ...
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Integration of the sum of gaussians over a line [closed]

suppose we have two independent gaussian distributions $X$ and $Y$ . let $Z = X+Y$ which will be a gaussian too. the goal is to compute : $ \int_{L} P(X+Y=c) dXdY$ Where L is the line : $X+Y=c$. $P(X+...
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6 votes
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Product of two independent Student distributions

What is the product of two independent student t distributions? In which case does this product product result in another t distribution?
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3 answers
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Distribution change of variables

I'm working on distributions for a physics problem and I am quite stumped on how to proceed properly. The problem is as follows: I start at the point $(0,0)$ and go a distance $\eta$ in x direction $(\...
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Propensity of customer to buy a new product?

I am tasked with deciding whether a new product should be developed. I have data of customer transactions of previous products and wish to build a model which determines whether customers will have a ...
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2 votes
1 answer
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Why not link features instead of selecting them - Clustering

Currently, I am working on customer segmentation using their purchase data. I plan to use clustering techniques. So, my data has below info for each customer (9 features and 1 id field) Now I am ...
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