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Questions tagged [mathematical-statistics]

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

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Uncorrelated errors with the regressor in a reduced form VAR

I have a reduced form VAR $$\begin{equation} y_t = c_o + A y_{t-1} + \epsilon_t \end{equation}$$ Where, $y_t \in \mathbb{R}^2$, $A$ is a $2$X$2$ matrix and $$\begin{equation} E(\epsilon_t \...
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13 views

General techniques for coupling a set of random variables with mutual dependence

Disclaimer: the usage of coupling is in the title is not of the usual definition in probability theory. Suppose I have a set of random variables $\{X_1, X_2, \dots, X_n\}$, indexed by time $t$, and ...
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2answers
22 views

How can I predict the value after a point with a short time of data?

I have a customer's online data. I have data such as the number of items purchased by the customer, the number and number of keyword queries for the customer, the age of the customer, the residential ...
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2answers
57 views

Why does minimizing absolute value and squares of residuals in a regression give different answers?

We are minimizing either the $1$ norm of the residuals, least absolute value, or the $2$ norm, least squares. Least absolute value: $$\min_\beta||y - x \beta||_1$$ Least squares: $$\min_\beta||y -...
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15 views

Matching Pursuit & Boosting: Exponential Convergence?

In the paper by Bühlmann BOOSTING FOR HIGH-DIMENSIONAL LINEAR MODELS, he introduces an algorithm, called componentwise linear least squares, and relates it to the Matching Pursuit algorithm by Mallat &...
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33 views

Main differences (problems and mathematics) between traditional statistics and high dimensional statistics

High dimensional statistics seems to be hot nowadays. What are the main differences, in terms of questions and problems it tries to solve, as well as the mathematical tools used, between "traditional"...
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17 views

Quantifying time series predictions

After building a time-series prediction model, Test Data: ...
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1answer
66 views

Check if log-likelihood function is correctly derived

This question is a continuation of this one. By guesswork, I found out that $\vec{\theta}=(5.2,5.3,1.0)=$ $(A,B,C)$ was a good guess that made my model $$y_i=A\sin\left(\frac{x_i}{B}\right)+C\...
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10 views

What are some higher level statistical methods used to measure the impacts of an advertising campaign (handling seasonality, time trends, etc.)?

I am trying to help my firm utilize better metrics for future growth. What variables are needed for the stochastic frontier production functions to measure if a firm is minimizing costs or maximizing ...
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89 views

How to find the likelihood given data?

I have a textfile with the two columns $$\mathbf{x}=(x_1,...,x_i)$$ $$\mathbf{y}=(y_1,...,y_i)$$ I want to use the following model for the data $$y_i=A\sin\left(\frac{x_i}{B}\right)+C\epsilon_i,$$...
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20 views

Statistics for Market research [duplicate]

I'm looking for some sources in order to learn about Market Research (a statistical approach with R/Python). You know, the kind of things you must know if you wanna work for Nielsen Kantar, etc.
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What is the intuition behind taking the sum of square roots, squared

In a recent publication, the authors report the following transformation when aggregating across three different scales: Cognitive style level was used as a control variable and captured as ...
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12 views

Fluctuation in mean squared error

I am minimizing mean square error to find vector $f$, with some constraints, and my objective function is \begin{equation} E = f^{T}(R_{aa}-R_{ya}^{T}R_{yy}^{-1}R_{ya}^{})f^{} \end{equation} where $...
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18 views

Proof of variance of ridge estimate with only one predictor!

Let's consider Ridge with only one predictor (extreme and simple case). I would like to proof that $V(B_r)=\sigma^2/(1+\lambda)$, so its variance it less than OLS variance, that is $V(B_{OLS})=\sigma^...
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159 views

Testing whether $X\sim\mathsf N(0,1)$ against the alternative that $f(x) =\frac{2}{\Gamma(1/4)}\text{exp}(−x^4)\text{ }I_{(-\infty,\infty)}(x)$

Consider the most powerful test of the null hypothesis that $X$ is a standard normal random variable against the alternative that $X$ is a random variable having pdf $$f(x) =\frac{2}{\Gamma(...
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Did Deborah Mayo refute Birnbaum's proof of the likelihood principle?

This is somewhat related to my previous question here: An example where the likelihood principle *really* matters? Apparently, Deborah Mayo published a paper in Statistical Science refuting Birnbaum'...
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singularity at the origin of a particular function

what does it mean to say that a particular function is singular at the origin? e.g., it is said that the SCAD penalty is singular at the origin does not have continuous second order derivative. then ...
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1answer
32 views

Survival Analysis - Relationship between CDF and survival function

I am trying to teach myself survival analysis and I followed this youtube video. The video states that given: $ F_x(t) = \Pr[T_x \leq t ]$ is the future probability of the life for a person who has ...
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22 views

Projection in AR model

I am currently reading the Brookwell and Davis Book and cuurently read about the PACF. On page 98 they derive the PACF for the AR(1) model $$ X(t)=0.9X(t-1)+Z(t) $$ and say that the orthogonal ...
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1answer
58 views

Densest subgraph : density of an intersection

I found an exercice in my textbook and I can't find the answer : $G = (V;E)$ an undirected graph. $H_1 = (V_1;E_1)$ and $H_2 = (V_2;E_2)$ are two densest subgraphs in G, i.e., for any subgraph $H = (...
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25 views

Question about CLT proof

I'm working through the CLT proof on Wikipedia trying to get a better intuition, and it made me wonder what an individual distribution looks like after dropping the o(t^2/n) terms from the Taylor ...
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2answers
35 views

One-sided hypothesis test (error rates)

Can someone please help me understand how the expressions after the equals (marked in red) are arrived at? I don't quite understand the very last one where the c.d.f. $\phi$ comes into the picture. My ...
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19 views

Finding the score for an event of interest

I would like some inputs in a problem I am trying to solve, which I don't know if I should go all the way to supervised learning or general statistics: I have data behavior data about how customers ...
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Statistical concept of fitting individuals to predefined slots based on suitability

Is there a statistics oncept where I can use where I can fit individuals/items in a group of available slots. The slot doesn't always match the person's skill/preference. Example 1: All kids in a ...
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13 views

Degrees of Freedom in Backward elimination

OK. I just had an exam and realized I made a mistake on a question and wanted to seek some guidance. The problem was essentially: "Use backward elimination to choose a set of predictors to predict ...
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3answers
431 views

Is there a constraint on the sum of the type-I & type II error probabilities?

Is it true that if $H_0$ and $H_a$ are complementary hypotheses of the Binomial trial, i.e., the negation of $H_0$ is $H_a$ then the type-I error $\alpha$ plus type-II error $\beta$ equals 1? Or is ...
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1answer
68 views

XGBoost tree “Value” output: [duplicate]

Using the following R code I obtain a decision tree using the agaricus dataset: ...
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1answer
27 views

Unsure which data to use for ANOVA test

I have determined that a one-way ANOVA test of variances will work best for my data. However, the nature of my data makes me very concerned about making any claims based on significance or not. My ...
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82 views

Finding the UMP size $0.1$ test with $U\sim\text{Unif}(\theta, \theta+1)$

$U$ is a uniform $(\theta, \theta+1)$ random variable, where where $\theta\in(−0.5,0.5)$. Consider testing $$H_0:\theta\geq0$$ $$H_1: \theta \lt0$$ (a) Give the test function of a ...
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11 views

Misleading two-way ANOVA table interpretation

How can the hypothetical cases below (if possible) be explained? Assuming randomness, normality, homoscedasticity, etc 1-Overall effect (Fbetween) is significant Individual factors are NOT ...
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1answer
42 views

Type 1 and Type 2 errors trade-off

Reducing Type 1 error will always result in increasing the Type 2 error This statement is false. I understand the definitions of Type 1 and Type 2 errors. What I understand is that there is, in fact, ...
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28 views

Understanding and alternative derivation of the back propagation rule for ANN

I'm quite new to machine learning, and as far as I understand the back-propagation rule is an algorithm the allow to compute the gradient of the cost function defined when training a ANN. First ...
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1answer
77 views

Relation between t-value and correlation coefficient r

Page 6 of this article shows that following holds for an "independent samples t-test". I was wondering how the equation to the Right of equal sign would change if we consider a "paired samples t-...
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1answer
45 views

Taylor expansion for random variables

Let $X_n$ and $Y_n$ be random variables such that $X_n-Y_n\overset{p}{\longrightarrow}0$ as $n\rightarrow\infty$. Let $f(.)$ is a differentiable function. Is the following correct? $f(X_n) = f(Y_n) + ...
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1answer
35 views

Possible to do some kind of statistical analysis without N?

So the thing is that I got some good data for my research project, but the problem is that the data I got only have percentages and standard errors, but no population (N). I would need to do some ...
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1answer
47 views

Transform X to get Y such that Y has a Uniform(0,1) distribution

A random variable $X$ has the PDF $f_X(x) = \frac{x - 1}{2}, \ 1 < x < 3$ Find a monotone function $u(x)$ such that the variable $Y = u(X)$ has the distribution $Uniform(0,1)$.
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24 views

Converting Cohen's d to r for two paired groups

I know that there is an exact formula for converting a Cohen's d effect size to a Pearson correlation coefficient ($r$) in the context of two 'independent' groups as discussed HERE (see p. 6): $r = ...
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1answer
56 views

Quantile Function

I have seen the definition of quantile function here, which is as follows (slightly modified): Let $X$ be a real-valued non-degenerate random variable with distribution function $F_X(x)=\mathbb{P}({X\...
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1answer
160 views

Where is my mistake in this definition of Bayes Factor?

From "The Bayesian Choice" by Christian P. Robert. The definition of the Bayes factor is given to be the ratio of the posterior probabilities of the null and the alternative hypothesis over the ratio ...
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1answer
45 views

Regression model when the dependent and independent variables show exponential distribution

As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution. Do I have to perform a ...
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3answers
375 views

An example where the likelihood principle *really* matters?

Is there an example where two different defensible tests with proportional likelihoods would lead one to markedly different (and equally defensible) inferences, for instance, where the p-values are ...
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1answer
27 views

How to prove the distribution of Generalised Instrumental Variables Estimator

$\hat { \beta } _ { GI V } = ( X ^ { \prime } Z ( Z ^ { \prime } Z ) ^ { - 1 } Z ^ { \prime } X ) ^ { - 1 } X ^ { \prime } Z ( Z ^ { \prime } Z ) ^ { - 1 } Z^ { \prime }y\hspace{35pt}(a)$ $(a)$ is ...
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0answers
62 views

Does this correlation make sense?

I'm trying to work on a ML project and I have a dataset and I'm trying to see if there is a correlation between some of the features in my dataset. The dataset contains inspection notes for car parts ...
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1answer
28 views

need explanation about the exponent parameter s in zipf distribution

I need to model the popularity of some requested files from a library with Zipf distribution and I want to simulate it in MATLAB. I don't know what's the effect of parameter s on my result. for ...
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Problem determining the expectation

I really have no clue how to solve subquestion D the following question. I have tried everything. I tried various methods, but I seem to constantly get stuck on the same thing. One of my attempts ...
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28 views

Relation between Expectations of sum of Bernoulli variables

Let $X_1, ... X_n$ and $X_1^+, ... X_n^+$ be two finite sequences of non-independent, non identically distributed Bernoulli variables such that $E[X_i^+] \geq E[X_i]$. If we define $S = \sum_{i =1}^n ...
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Proof for copula determined by correlation matrix

How can I proof that the copula of an elliptical distribution $El(\mu, \sigma^2, g_n)$ is fully determined by the generator function $g_n$ and the correlation matrix extracted from $\sigma^2$.
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3answers
76 views

Log-normal returns

Let $P_t$ denote a stock price distributed as $\operatorname{lognormal}(\mu , \sigma^2 )$. Suppose we construct simple returns $R_t=\frac{P_t-P_{t-1}}{P_{t-1}}$. My question is: What is the ...
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1answer
89 views

Estimating conditional probability with many samples

I am confused about the estimation of conditional probabilities. Suppose I want to predict a binary outcome variable $Y = 0,1$ given $n$ categorical features $X = (X_1, \ldots, X_n)$, i.e. to ...
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1answer
45 views

Asymptotic Distribution of function of $\bar{X}$

$X_i \stackrel{iid}{\sim} N(\mu,\sigma^2)$ i=1,2,...,n $Z_n=\sqrt{n}(\bar{X} - \mu)$ I believe the asymptotic distribution of $Z_n$ is $N(0,\sigma^2)$. So what would the asymptotic distribution of ...