Questions tagged [mathematical-statistics]

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

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Modelling losses in insurance - why nobody seems to talk about left-truncated distribution?

In the (re)insurance field, it is quite common to simulate losses via a Monte Carlo approach and a stochastic generator of both frequency and severity individually (independency between both being ...
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Computing the inverse Beta cdf in R [closed]

In MATLAB, the function betainv(P, A, B) computes the beta CDF inverse with parameters A and B specified for the corresponding probabilities in P. Is there a similar function in R that can do the same?...
1 vote
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Does there exist any causal explainability tool for NNs?

Say I have a neural network which has 1,000,000 parameters built using 100 features and I wish to understand the underlying data-generating process for how the model arrived at each prediction. Simply ...
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3 votes
1 answer
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Logistic regression: What are use cases for logistic regressions where $n \neq 1$, i.e., $n >1$? [duplicate]

NOTE: This question relates to Binomial logistic regression. Thus, the $n$ in the title, refers to the parameter of the Binomial distribution. Most real-world use cases of logistic regression ...
0 votes
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7 views

Entropy discribe sample with less data bias? [closed]

Say if we work in youtube.we want to analysis user prefrence for movie. If user A only watches two thrill movie,then entropy would be 1*log(1)=0. If user B watches two movie,a comedy and aa thrill,...
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1 vote
1 answer
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Is this a known measure of "effective degrees of freedom" in regression?

Suppose $x,y$ are I.I.D random variables in $d$ dimensions. Consider the following quantity: $$R=\frac{(E\|x\|^2)^2}{E\langle x, y\rangle^2}$$ equivalently for empirical distribution with data matrix $...
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How to show for positive Borel functions $g, \int_{a}^{b} g(x)dF(x) = \int_{a}^{b} g(x)f(x)dx$

$\textbf{Problem:}$ Let $\mu$ be an absolutely continous probability distribution with distribution function $F(x)$ and density $f(x)$. For any Borel set $\Lambda$ and $a<b$, let $\nu\left(\Lambda \...
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How do I calculate the expected values if the correlation coefficients are -1/0/+1? [closed]

How do I calculate the expected values if the correlation coefficients are -1/0/+1? I know that -1 means backwards correlation and so on but I just need an example.
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1 vote
1 answer
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Can I take the p-value from a probability table for a modified z-score, just like a z-score?

I have a question regarding the modified Z-score, can I use this score in a Z-probability table, i.e. the p-value from a modified z-score? where: z-score = (xi - μ)/σ (the z-score probability can be ...
0 votes
0 answers
23 views

How can I back transform a log data to interpret t-test and get original CI?

I have data that is not normally distributed. I can log-transform it to be normally distributed, and then perform a t-test and get confidence intervals (CI). But how do I interpret the results of the ...
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1 vote
1 answer
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How to show that $X_n + Y_n \to X + Y$ holds in the $L^1$ norm?

Let $X_n$ and $Y_n$ be sequences of random variables. Show that $X_n + Y_n \to X + Y$ (1) $X_nY_n \to XY$ (2) If $\mathbb{P}(X=0) = 0, \; \frac{Y_n}{X_n} \to \frac{Y}{X}$ (3) are true for convergence ...
2 votes
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+100

Value of $E[X^T A X B X^T CX]$ for Gaussian matrices $X$?

Suppose $X$ is an $m\times n$ random matrix where rows $x$ are I.I.D. samples of $n$-dimensional Gaussian, and $A,B,C$ are matrices of appropriate dimension. What is the value of the following matrix? ...
1 vote
0 answers
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How to derive the F statistic from likelihood ratio test

I am trying to test whether sigma1 is different from sigma 2 (2 ind samples that follow a normal dist) and I have no idea how to use the likelihood ratio to derive the F statistic that tests this.
-2 votes
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Show that for a squared error loss function L(^θ, θ) = (^θ - θ)^2, the mean of the posterior distribution is the best estimate for the parameter θ [closed]

Show that for a squared error loss function L(^θ, θ) = (^θ - θ)^2, the mean of the posterior distribution is the best estimate for the parameter θ. I'm literally lost on this question, if anyone could ...
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Compare distributions after transforming from continuous to discrete [closed]

I have two variables: X and Y. X variable (varies between 0 and 300) is continuous and Y with same range of variation but is discrete and have 3 classes (0-100, 100-200 and 200-300), which is pretty ...
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Standard errors from zero vs. actual standard error value

Obviously a standard error value close to zero is more desirable. However, is there a rule of thumb for how many standard errors AWAY from zero is considered good or acceptable? Does this also change ...
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0 votes
1 answer
35 views

How does center variable help address multicollinearity problems for interaction with polynomial terms

I hope someone can help me better understand how center variables can help address the problem of multicollinearity problem when the regression includes interactions with polynomial terms. My ...
0 votes
0 answers
11 views

How do I determine a noise distribution? [closed]

Let’s say we have a set of instances $X={x_1, x_2, ..., x_n}$ whose distribution is unaware, but the mean and var values can be calculated. Now we add a noise generated from a certain distribution (...
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0 answers
11 views

Random Variables as Function of Correlation Coefficient [closed]

If we have that $X, Y\~N(0, 1)$ and that $Corr(X, Y)=\rho$, how can we write $Y$ as a function of $X?$
0 votes
1 answer
20 views

To check if the churn probability score from old and new model is similar

I have calculated the churn probability score for every customer id using glm model. So, I have a data frame with every customer id and its churn probability score. Ex cust_id 1 has a score of 0.11 ...
1 vote
1 answer
20 views

Standard error logic

Obviously the closer the standard error to zero the better. However, what if the values the in which the standard errors are from are extremely large and the standard errors are much higher than zero ...
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2 votes
2 answers
34 views

Binomial to Poisson Approximation

So, a little context. The image you see is from the GCE A-LEVEL syllabus where they have defined the conditions for approximating binomial to poisson. But why did they have mention that the ...
5 votes
1 answer
112 views

Gaussian fourth-moment formulas?

Suppose $X$ is an $m\times n$ random matrix where rows are I.I.D. samples of $n$-dimensional Gaussian, and $A$ is an $n\times n$ matrix. I'm looking for the value of the following quantities: $$E[X^T ...
1 vote
3 answers
86 views

Calculate the variance of a distribution analytically

I want to calculate the variance of a certain distribution. I have a rectangle that is getting shifted to the right (i.e. shear transformation). To obtain the distribution I am computing the value of ...
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1 vote
0 answers
24 views

Deriving distribution under change of variables between spaces of unequal dimension

For a function of random variables $T:\mathbb{R}^n \mapsto \mathbb{R}^m$ Wikipedia outlines how to handle three cases: $m = n = 1$ $m=n > 1$ $n>1 \land m=1$ There seems to be two missing cases:...
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For a continuous distribution, can I replace the mean by the median in two sample z-test?

For a continuous distribution, can I replace the mean by the median in two sample z-test? Assuming that populations are normally distributed. I feel like they can because both median and sample mean ...
1 vote
1 answer
42 views

Mathematical knowledge needed for learning upper level statistics

im a math student who wants to study statistics in depth for a master degree(maybe doctor degree). And I was wondering what kind of math knowledge might be needed for upper level statistics. I wanna ...
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0 votes
0 answers
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get a p-value which is greater than 1 in sign test [closed]

I was confused about calculating the p value for both single sampled sign tests. Suppose we have a sample size of 12 and 6 values are below our hypothesized median of $m_x$and 6 are above. And that $...
-1 votes
0 answers
9 views

Regression and Distribution of Inverses

We know that $\beta = (X^TX)^{-1}X^Ty,$ but when can we "distribute" the -1 in to get $(X^TX)^{-1}=X^{-1}X^{T^{-1}}$?
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1 answer
23 views

Derivation of the formula for the asymptotic relative efficiency of two estimators with different estimands

Background In their book, Huber & Ronchetti (pp. 2-3) compare the efficiency of the mean absolute deviation $d_n$ with the standard deviation $s_n$ with the following formula: $$ \operatorname{ARE}...
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4 votes
2 answers
21 views

Structural Causal Models with cycles

I this context I am referring to Structural Causal Models (SCM) such as presented here https://fabiandablander.com/r/Causal-Inference.html or here https://medium.data4sci.com/causal-inference-part-iv-...
0 votes
0 answers
22 views

Jenks Natural breaks - Interpreting Goodness of Variance Fit

I am trying to find breaks in a multiple continuous type variables. So, I tried the jenks natural breaks algorithm. Based on the code from here, I managed to find ...
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-1 votes
1 answer
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How to compare the mean to the mean of mean values of dummy variables?

I am trying to validate the calculations of a mean value for a collection of binary variable values. The validation process is where I compare x to y where: x is the mean value of percentage of ...
0 votes
0 answers
11 views

Statistical Test of the significance of difference between 2 ordered columns

I'm trying to come up with an appropriate statistical test for the significance of the difference between 2 ordered columns A = [a0, a1, ..., aN] and B = [b0, b1, ..., bN]. Each of the columns is of ...
5 votes
2 answers
2k views

When is a statistic not a statistic?

I have a fairly involved ``statistic'' that involves transformation of a set of samples $\mathbf{x}$ by a massive machine-learned matrix and subsequent nonlinear processing into said statistic $\...
0 votes
0 answers
14 views

Quantile Prediction of Cartesian Sum

I want to predict (or find exactly if it is possible) the quantiles of all pairs sum of two arrays. More clearly, Say I have two sets: {2,5,7,11,23,56} and {10,15,25,50,100,123,180} All pairs (...
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3 votes
0 answers
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+50

What kind of study design is this and what kind of statistical analysis should be suitable?

I plan to conduct a study in which respondents will be asked to solve total 6 puzzles together. Is there a specific study design (name) for this? The scenario would be: First, ...
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2 votes
1 answer
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How to determine interventional distributions from observational data?

How do we compute/query interventional distributions from observational data (i.e. without knowledge of the causal graph such as a Structural Causal Model (SCM))?
0 votes
0 answers
18 views

Questions about interventional distribution? [closed]

How do we compute/query interventional distributions from observational data without knowledge of the causal graph (e.g. Structural Causal Model (SCM))?
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0 answers
20 views

Casino odds - Independent events and theoretical probability [duplicate]

Let's say you are in a Casino and you are playing Craps. Your desired outcome is 1/36, (every time it is rolled). Every event is independent. If you get a hit (a 1/36 chance) you get paid 36 times ...
2 votes
1 answer
65 views

If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree on $\mathcal{B}$?

Let random variables X and Y be defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume their distribution functions $F_X$ and $F_Y$ agree for all $x \in \mathbb{R}$. How ...
1 vote
1 answer
41 views

Is $Y=Y(\omega) = \inf_{0 \leq t \leq 1}X_t(\omega) = 1_A(\omega)$ not measurable if $A \notin \mathcal{B}[0,1]?$

Consider the probability space $([0,1], \mathcal{B}[0,1],\lambda)$ where $\lambda$ is the Lebesgue measure. Let $A \subset [0,1] $ and define $ X_t(\omega) = \begin{cases} 0 \;\; if \; \; t = \omega \...
0 votes
0 answers
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Chapter 5 political stats [closed]

(5 points) Imagine that you are a researcher, and that as part of your research you need to gauge the favorability of free lunch programs within the county school districts. During your research, you ...
1 vote
0 answers
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Variance of inverse RV

I am trying to find variance of Y=1/(1+abs(x)) where X is Gaussian RV with mean m, var sigma^2To do that my initial step is to find MGF. can anyone give me ...
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0 votes
0 answers
15 views

about the derivative of the characteristic function of the Poisson distribution [closed]

I do not understand why differentiation of the above equation leads to the below equation. Can you please tell me?
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20 votes
8 answers
2k views

What are some good blogs for Mathematical Statistics and Machine Learning?

I am looking for blogs that focus on the mathematical theory of Statistics and Machine Learning, ideally at a research or "advanced" level. The blog doesn't have to be solely about these ...
2 votes
1 answer
17 views

I need help clarifying what (R1,..RN) is in this context

Let R be the space of all the N! permutations of (1,2, ..., N), and let R = (R1,...RN) be uniformly distributed over R. The thing i don't understand is what "R = (R1,...RN) be uniformly ...
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1 vote
1 answer
22 views

Backend process of lmer function [closed]

I am trying to learn mixed models using lmer function and I am able to use it. But I am not able to understand what is going on in the backend like how it is taking care of fixed effects and random ...
2 votes
0 answers
33 views

Resampling to handle class imbalance in logistic regression [duplicate]

I was wondering if anyone could help me understand resampling for class imbalance. From what I have learned, class imbalance is usually a small data problem where the less prevalent class usually ...
0 votes
0 answers
4 views

HMM estimation (Baum-Welch) with 2 independent state variables

I'm trying to estimate a hidden Markov auto-regressive model with two independent state variables. You can think of one as determining the levels for the mean-reversion and the other determining the ...

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