Questions tagged [mathematical-statistics]

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

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Calculating a probability of a function of a random variable in a limit

I am reading the paper Luo et al.: Parameterized Explainer for Graph Neural Network and trying to understand the mathematical reasoning after Equation (4) on page 5: How do they calculate $\...
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How to work with an aggregate value set for multiple items

I have a natural disasters dataset from EMDAT that records the impact of a disaster as an aggregate (Total) against locations as follows: Total affected Locations 10000 locationA locationB locationC ...
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Formular of my lmer() Model

I know there is already a lot of information out there, e.g. How to write mathematical formula describing my lmer model?. My lmer() Model is pretty similar, but ...
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relationship of Standard error and t test

I know that t-value can be calculated by $(sample mean-population mean)/sem$ and that the standard error (SE) means the t value is calculated according to the sample size. What I do not understand ...
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Show that $\mathbb{P}(A):=\sum_{n=1}^{\infty} \alpha_{k} \mathbb{P}_{n}(A)$ is a probability on $(\Omega, \mathscr{A})$ [closed]

For $n \in \mathbb{N}$, let $\mathbb{P}_{n}$ be a probability on $(\Omega, \mathscr{A})$. Consider a sequence of real numbers $\left(\alpha_{n}\right)_{n \in \mathbb{N}}$ satisfying $\alpha_{n} \geq 0$...
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How do we know that the loss function for Generative Adversarial Networks (GAN) converges?

I'm working on loss function Generative Adversarial Network (GAN) and I wondered: what ensures the existence of the loss function $V(D,G)$? Is it proven that $V(D, G) = \mathbb{E}_{x \sim p_{data}(x)}[...
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What is $Cov(\mu(X), X\beta^{*})$? The projection of a nonlinear function onto the linear span of the features $X$

Consider a data generating mechanism of $Y_i = \mu(X_i)+ \epsilon_i$, $\epsilon_i \sim N(0,1)$. $\mu(X)$ is a nonlinear function. Suppose $X$ is also a random variable with $E[X]=0$ and $Var[X] = \...
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How to assess the most important features and their statistical significance without a supervised learning model

There is a task that doesn't appear to make much sense to me in a textbook. I have a dataset of categorical variables (the features and target are categorical) . I am asked to determine what are the ...
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Probabilistic Machine Learning: Product of gaussian pdfs of samples is equal to gaussian pdf of sample mean

I'm currently reading the book Probabilistic Machine Learning: An Introduction by Kevin P. Murphy and I'm stuck on the derivation of a formula in section 3.3.4 (Example: inferring an unknown vector) ...
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Restricted Boltzmann Machine derivation of conditional independencies

In restricted boltzmann machines the formula $P(H|V)=\prod_{j}P(H_j|V)$ holds. One can derive the formula the following way: $P(H=h|V=v)=\frac{P(V=v,H=h)}{P(V=v)}=\frac{P(V=v,H=h)}{\sum_{H}P(H=h,V=v)}...
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Is the consistency of plug-in estimators needed?

In generalized least squares, the regression parameter estimate is a minimizer of the following quadratic form, $$ \hat{\beta} = \text{argmin}_\beta(y - X\beta)^T\Omega(\theta)^{-1}(y - X\beta) $$ ...
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Properties of cumulative sum of a random walk?

A random walk can be generated by computing the cumulative sum of a list of random numbers. ...
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Looking for a predictive prior for what I thought to be Bayesian Analysis

I need a prior with 2 parameters location (μ) and scale (σ) (a transformation of mean and standard deviation). My distribution is over discrete values, equally spaced with distance $c$ between ...
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difference in p-value in simple vs multiple linear regression

Suppose we have have 3 variables: Y , X , Z. then we have : Simple Regression : $ Y = aX+b$ Multiple Regression : $Y = aX + bZ + c$ I know that in simple regression we can have a low p-value(showing ...
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Create a synthetic dataset from the measures of spread

I have certain measures of spread (p50, p66, p75, p80, p90, p95, p98, p99, p99.9, p99.99, p100, Median, Mean, Min, Max)as my attributes. Can I regenerate a data distribution from these? Use case: I ...
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Is $f(\widehat{x})$ ($\widehat{x}$ is a random sample of $x$) equivalent to $\widehat{f(x)}$ (a random sample of $f(x)$)? [closed]

Assuming $f(x)$ is an injective function, is sampling $\widehat{x}$ from the distribution of $x$ then calculating $f(\widehat{x})$ equivalent to directly sampling from the distribution of $f(x)$? Note ...
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Can I fix one end in Gaussian Process Regression (GPR)?

I just learnt some basics about Gaussian process regression and how to do it for X-Y data via the GPy package. I wonder whether I can fix one end of the regression curve (or in other words I'm 100% ...
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Nonparametric Order Statistics - Does this Exist?

I was reading about Order Statistics (https://en.wikipedia.org/wiki/Order_statistic) : Apparently, if we have a sample with "k" number of elements (e.g. x1, x2, ...xk) and assume a ...
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Why sample size is not a part of sufficient statistic?

Following simple example from Wikipedia's definition of sufficient statistic with Bernoulli distribution with parameter $\theta$, where sufficient statistic is a sum of successes $$T(X_n)=\sum_{i=1}^n ...
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Python : How to Generate positive number with a given standard deviation and mean within 0-1? [closed]

I want to generate a list (or array) of all positive numbers, and all the numbers need to be within 0-1. I've seen one code, ...
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Can there be negative values for doses in one-stage dose-response meta-analysis?

I am trying to perform a one-stage dose-response meta-analysis as described by Crippa et al. https://pubmed.ncbi.nlm.nih.gov/29742975/ However, I am assessing the change in alcohol intake over time as ...
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Computing Variability Independent of mean; differences in calculation methods

This question is mostly about another post here on stats.stackexchange. In general, VIM (Variability Independent of the Mean) is usually calculated, as the name suggests, to calculate the variability ...
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How to include limited independent variables in linear regression models?

I want to determine the impact of competition on quality using linear regression. Competition is represented by the Hirschmann-Index (HHI), which is an indicator for market concentration. It can take ...
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How to derive a formula for the sampling distribution of the coefficient of variation and the skewness

Assume that $T$ has an Erlang distribution: $$\displaystyle f \left(t \, | \, k \right)=\frac{\lambda ^{k }~t ^{k -1}~e^{-\lambda ~t }}{\left(k -1\right)!}$$ and $K$ has a geometric distribution $$\...
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A question about the expected value of real-valued random variables

Given two not independent r.v. $X,Y$ such that $\mathbf{E}[X]=1$, ¿Is it true that $\mathbf{E}[XY]\approx \mathbf{E}[Y]$? And in that case, ¿can we stablish the rate of approximation? I think that ...
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Central limit theorem with sorted samples

I recently encountered a strange situation while dealing with sampling : Let's suppose I have X1 ... Xn random samples drawn from a population. Then I sort every samples and I make the sum of every ...
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For a N x 2 matrix, is there a difference between R squared and NS efficiency?

I came across a similar question in the forum yet I failed to understand any practical difference between the two coefficients. To me it seems like it's pure semantics when a prediction is a ...
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difference between stratification and blocking in experimental design

What is the difference between stratification and blocking in experimental design(or randomized control trial)? Blocking is to obtain homogeneity in a controlled factor. However, I can block more than ...
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How to calculate the most likely time spent per customer category based on this data set?

I know how many customers of each customer-size group each sales representative handles on an annual basis. Is it possible to calculate the likely time/effort required by each customer size based on ...
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For a block randomized experiment how many possible unique assignments are there?

In a block randomized experiment say of 4 individuals split into 2 even blocks with 1 individual in each block getting assigned to treatment and the other control, how many possible unique assignments ...
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Why not always use CI's from LRT since they don't require symmetry?

I'm confused on why anyone would appeal to asymptotic normality of mle, $$\hat{\theta} - \theta_0 \rightarrow^D N(0,I^{-1}(\theta))$$ When we can simply invert the likelihood ratio test $$L(\hat{\...
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A method to find the proportions within each combination of 3 categorical variables

I have three categorical variables that describe each item, and I know how many items I have for each combination of a pair of variables. I am looking to find out exactly how many items I have for ...
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Statistical inference question evaluating a joint probability mesure w.r.t. a distribution

Note: I have edited this question and delted my old question after following the guidlines to be more clear. I hope this new question is clear and well posed. I attempted this question but I got stuck ...
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Which regression model to chose?

Let's suppose I want to measure the impact of M&A in the hospital sector. I have a dataset containing hundreds of hospitals, a quality indicator and the type of ownership for each over a time ...
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Distribution of rainfall over locations per year [duplicate]

$5.2$ Suppose $X_{1}, X_{2}, \ldots$ are jointly continuous and independent, each distributed with marginal pdf $f(x)$, where each $X_{i}$ represents annual rainfall at a given location. (a) Find the ...
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Proving a result in maximum likelihood theory: accuracy of the quadratic approx. to loglikelihood

I need help proving a result shown in a paper. I am reading Assessing the Quadratic Approximation to the Log Likelihood Function in Nonnormal Linear Models by Salomon Minkin. The paper defines several ...
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Why does sum cov(Xi,Yi) = n cov(X,Y) [closed]

I am trying to understand how to find cov of two sample means.
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1 answer
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what statistical test should be used for testing number/proportion of people who agreed in 2 different years

I want to measure if the number of people who agreed on a likert scale (1 strongly disagree - 5 strongly agree), so responded with a 4 or 5 had a statistically significant increase from the previous ...
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Splitting data to study datsets based on in- and out-of-distribution

I found an interesting paper for which they evaluate the in and outdistribution on different splittings of the dataset. For this they propose four different in distribution settings. The two I'm ...
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Why do we calculate probability distribution on leaf nodes at Sum-Product Networks (SPN)?

I am new to Sum-Product Networks (SPN) and still trying to understand some concepts. I understand that we need to give inputs at the leaf nodes of SPN. But why do we have gaussian distribution and ...
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3 votes
1 answer
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What type of prediction model will be suitable in this case?

We have 100 subjects (A,B,C....) in total. We took 2000 tissue samples from each subject. For each subject, we evaluated the outcome(PASS or Fail), but we only evaluated the outcome on the subject ...
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1 answer
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Does RCTs imply causation?

Can statistics help to definitively deduce causality? RCTs require that you state the set of variables that you believe influence a given outcome of interest, a priori, therefore if one wants to ...
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Within/Between Multivariate Analysis

I am interested in analysing two dependent variables, let's say Alpha and Beta. I tested 40 participants, divided into two different treatment groups: Group A (n = 20) and Group B (n = 20) Within each ...
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Calculation of the standard deviation of the power in dB

I wanted to find out the stability of my system so for that i calculated standard deviation of power samples data which is in dB but I am not sure whether i have used correct formula or not. As far as ...
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1 vote
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asymptotic distribution of correlation coefficient in bivariate normal distribution

Suppose \begin{align} \begin{pmatrix} X_i\\ Y_i \end{pmatrix} \sim_{iid} N \begin{pmatrix} \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \begin{pmatrix} \sigma^2_1 & \rho \sigma_1 \sigma_2 \\ \...
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Comparing the probabilities of 2 different completely separate models that predict different outcomes

You have 2 separate models, one is a binary classifier that predicts whether a customer will pay off their credit card balance on time and another binary classifier (separate model) whether a customer ...
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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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