New answers tagged

0 votes

Bayesian Posterior distribution for binomial distribution with uniform prior

To answer your last question first - the way you have written it, the $p$ is the same in the two distributions, so they share the same prior. Your calculation is indeed correct; you've found the ...
  • 33.9k
1 vote

Sequential Probability Ratio Test (Property Proof)

SPRT terminates at the $N$-th trial where $$N:=\min\left\{n\in \mathbb Z^{> 0}\big | \sum_{i=1}^n z_i~\geq \ln A ~\lor~\leq \ln B\right\} .\tag 1$$ Then it is needed to prove $$ \mathbb P\{N<\...
  • 1,082
1 vote

Understanding multiple linear regression residuals

Under OLS the residual is orthogonal to every column in the design matrix To understand this issue, it is worth understanding the concept of the column space of the design matrix $\mathbf{x}$. This ...
  • 102k
0 votes

Understanding multiple linear regression residuals

Assuming $X$ is full rank, you can see it in terms of the hat matrix $H = X(X^\top X)^{-1}X^\top$. This matrix is an orthogonal projection matrix onto the column space of $X$. The estimate of $y$ is $\...
2 votes

Simple Linear Regression Question: How does correlation between X and Y affect MSE?

In simple linear regression, the squared correlation between the $X$ variable and $Y$ variable is equal to the $R^2$. Also, $R^2$ has an equivalent representation involving MSE: $$ R^2=1-\dfrac{MSE}{...
  • 35.7k
1 vote

Understanding multiple linear regression residuals

Its been a long time since I've had to do some linear algebra, so forgive me if I've forgotten some of it. Let's begin by assuming that $X$ is full rank and $\operatorname{rnk}(X)=p<n$. Minimizing ...
1 vote

When the sample mean converges to the population mean, does the probability that the sample mean is equal to the population mean tend to 0?

It's possible set it up so that $\hat{\bar{y}}_N$ equals $\bar y_N$ at most finitely often. Make all the $y$ binary 0/1 and choose $n$ coprime to $N$,eg by taking $n$ to be the smallest prime greater ...
0 votes

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

First of all, I have to highlight that you used log10, which is unusual (although it doesn't affect anything) and the natural log is preferred and usually assumed ...
  • 662
2 votes
Accepted

Prove E[Y|X] = f(X)

$\mathbb{E}[Y|X] = \mathbb{E}[f(X)+\epsilon|X] = \mathbb{E}[f(X)|X]+\mathbb{E}[\epsilon|X] = \mathbb{E}[f(X)|X]+\mathbb{E}[\epsilon] = f(X)+0$
  • 2,316
2 votes

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

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 t-test and the CIs? If you want to compare 2 ...
  • 68.8k
0 votes

If mean is so sensitive, why use it in the first place?

We use the mean more than the median because it is additive, in two senses. (I am surprised that in 11 years, no one has really said this!) If data on a population is broken down into data about men ...
  • 2,157
0 votes
Accepted

hypothetical statistical test - type I and type II errors

You don't say it explicitly, but assume that the random number between 0 and 1 has a uniform distribution.Let's be more realistic and assume that you hope that it's uniform, but not sure, so you take ...
  • 662
0 votes

What is the probability of multiple events occurring together?

Your problem seems to be ill-formulated. Your original formulation suggests that the employee left, and that you are computing the probability $p(X)$ that the event causing the employee to leave was $...
0 votes

Does there exist any causal explainability tool for NNs?

I am afraid that the question you are asking doesn't have an answer that would be satisfactory for you. Causal inference us used for finding causal relations in noisy data. A neural network is a ...
  • 121k
1 vote

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

Even if the gof is amazing, is it not theoritically "wrong" to use non-left-truncated distributions for this kind of things? If it is the case that some simple parametric distribution fits ...
1 vote

How to show for positive Borel functions $g, \int_{a}^{b} g(x)dF(x) = \int_{a}^{b} g(x)f(x)dx$

Albeit the concerns re the ambiguities as shown in the comments deserve clarification, these sort of problems are standard measure theoretic exercises. So, I am leaving below a general brief ...
  • 1,082
1 vote

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

I assume that retention left-truncates the severity distribution and deductible does so for both the frequency and severity distributions (even in a less predictable way that the former). If an ...
  • 662
1 vote

Is this a known measure of "effective degrees of freedom" in regression?

This quantity seems to reduce to "Welch-Satterthwaite degrees of freedom" is the limit of variance of $x,y$ going to 0
2 votes
Accepted

Logistic regression: What are use cases for logistic regressions where $n \neq 1$, i.e., $n >1$?

Assuming that your $n$ is the number of cases of each type, you end up looking at proportions of each successful rather than $0-1$ and you weight by $n$. Stealing from https://stackoverflow.com/...
  • 32.4k
0 votes
Accepted

Can I take the p-value from a probability table for a modified z-score, just like a z-score?

It is worth noting that, with modern computers, we never (ever) have to use z-scores for sample statistics instead of t-scores even for large sample sizes, for which these two scores are quite close. ...
  • 662
1 vote
Accepted

How to show that $X_n + Y_n \to X + Y$ holds in the $L^1$ norm?

These are standard exercises. So, let me leave behind the ingredients that OP can utilise to formally construct the proofs. Let $(\Omega, \mathfrak A, \mathbf P) $ be the probability measure space. ...
  • 1,082
0 votes
Accepted

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

I'm trying to convert @Glen_b's comments into an answer as I understood them. The two insights that I missed were: Higher population values of a scale parameter mean that the asymptotic variance of ...
  • 26.4k
1 vote

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

Observation $1$. Let $(X, \mathfrak A) $ be a measurable space and let $A\subset X. $ Then $\mathbf 1_A$ is $\mathfrak A$-measurable if and only if $A\in \mathfrak A. $ The proof is straightforward ...
  • 1,082
2 votes
Accepted

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

center variables can help address the problem of multicollinearity problem when the regression includes interactions with polynomial terms. That isn't quite correct. Centering predictor variables can ...
  • 68.8k
0 votes
Accepted

Calculate the variance of a distribution analytically

You appear to be asking for the marginal distribution of $X$ where $(X,Y)$ has a uniform distribution on a sheared unit square. (The unit of measurement is the base of the square.) Because the ...
  • 297k
0 votes
Accepted

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

Because the two models are ostensibly the same, just with different sourcing of the variables, I would just plot the predicted probabilities against one another and look for large discrepancies. You ...
4 votes
Accepted

Gaussian fourth-moment formulas?

I have never seen an closed form expression for this. Probably because it is quite ugly. I have worked with a similar expression before, and I'd be happy to see if my expression is stands up to yours. ...
  • 291
0 votes

Standard error logic

This could be where the p-value comes in. In the first case, the estimated parameter is $10$ standard errors away from zero. This will result in a small p-value. In the second case, the estimated ...
  • 35.7k
0 votes

Binomial to Poisson Approximation

Going by Prof G E P Box's quote "in statistics no model is perfect but some are useful", modeling data using Probability Distributions also fits this quote very well, one can use any ...
0 votes

Calculate the variance of a distribution analytically

I think your h(x) is a so called mixture distribution. In your example, there are actually three distributions of h(x), either h1(x)=ax, h2(x)=c, or h3(x)=-x+w, with h1(x) and h3(x) both being uniform,...
  • 83
0 votes
Accepted

Binomial to Poisson Approximation

That should be $np\geq 5$ and $n(1-p) \geq 5$ for any approximation to be taken on Binomial Distribution. The approximation can be attributed to Central Limit Theorem. This type of approximation is ...
0 votes

Calculate the variance of a distribution analytically

Your distribution seems to be a Trapezoidal distribution; analytical expressions for its different modes can be found on the following page: https://en.wikipedia.org/wiki/Trapezoidal_distribution Hope ...
3 votes
Accepted

Structural Causal Models with cycles

Most of the current causal literature restricts itself to acyclic SCMs, but there has recently been a lot of research advancing the theory of cyclic causal systems. Although one of the first ...
  • 8,359
0 votes

Mathematical knowledge needed for learning upper level statistics

As a start to study Statistics mathematically and rigorously, I would recommend you to read: Shao J - Mathematical Statistics It is a very good book to begin your journey.
6 votes

When is a statistic not a statistic?

This answer is a theoretical supplement to Tim's more practical answer (+1). There is an axiomatic and mathematical side of statistics. Random variables are measurable functions on the outcome space $\...
  • 4,952
3 votes

Structural Causal Models with cycles

Well, the fundamental rule of causality is that causes must precede effects - that is a strict inequality in time. So it is not permissible to have $X_i(t)=f_i(X_j(t),\dots,U_i(t)),$ but then turn ...
1 vote

How to compare the mean to the mean of mean values of dummy variables?

Suppose you have v1 = 0 1 v2 = 1 1 0 0 0 0 Total mean = 3/8 meanv1 = 1/2 meanv2 = 1/3 mean(meanv1, meanv2) = (1/2 + 1/3) / 2 = 5/12 3/8 is not equal to 5/12 Only if v1 and v2 have the same number of ...
  • 83
2 votes

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

An Outsider's Tour of Reinforcement Learning by Ben Recht gives a short introduction into RL and draws connection to control theory.
20 votes
Accepted

When is a statistic not a statistic?

The statistic is defined as A statistic is a function $T (X^n )$ of the data. (Larry Wasserman All of Statistics, p. 137) A statistic (singular) or sample statistic is any quantity computed from ...
  • 121k
0 votes

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

Towards data science a collection of articles focussing on data science, machine learning, artificial intelligence and programming. It is written by various authors. The articles often focus on ...
2 votes
Accepted

How to determine interventional distributions from observational data?

In general, observational data is not sufficient to obtain the interventional distributions. You will "only" obtain the Markov equivalence class (e.g. with the ...
  • 8,359
2 votes

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

In the last couple of years I have warmed up to using geometry to understand deep learning models, and indeed various types of statistical models. While I recommend the book Geometric Deep Learning: ...
1 vote

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

This is neither really a blog nor just about statistics and many times very basic, but I found many good advices and ideas in there so I decided to add it as an answer https://chrisalbon.com/#...
4 votes

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

Observation $1.$ Let $\mathbf P_1, ~\mathbf P_2$ be two probability measures on $(\Omega, \mathcal F). $ Let $\mathcal P$ be a $\pi$-system such that $$\mathbf P_1(A) =\mathbf P_2(A), ~~~\forall A\in \...
  • 1,082
4 votes

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

https://statisticaloddsandends.wordpress.com/ reminds me of Gunderson blog, nicely written with code and clear explanations.
3 votes

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

ICLR recently introduced its Blog Track and its taken inspiration from some blogs like Bach's. Best thing is that it's peer-reviewed and contains diverse topics from diverse authors (often a group of ...
8 votes

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

Andrew Gelman: https://statmodeling.stat.columbia.edu. Gelman is a professor of statistics and political science at Columbia, and has co-authored several statistics books, including Bayesian Data ...
7 votes

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

Francis Bach's Machine Learning Research blog is an "easy to digest" introduction to some of his research works and related topics ("easy" as in easier than reading the original ...
1 vote

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

Your statement will make a bit more sense if you use more standard notation to differentiate vectors of values from sets. I would write it as follows: Let $\mathscr{R}$ be the space of all $N!$ ...
  • 102k
0 votes

A formal definition of a "measure of association"

I think about this a little differently than I used to, and will respond to myself here. kjetil's answer is still the best researched answer here and Ben's answer is thought provoking. The following ...
  • 4,952

Top 50 recent answers are included