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Questions tagged [mathematical-statistics]

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

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Bayesian hierarchical design

Hope someone can give me a hint!!!! Bayesian hierarchical design Some people think that in the Bayesian statistical model, the design can be distributed to the unknowns in an endless design. For ...
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P-value on statistical tests [duplicate]

If I have a p-value of 0.01 on a statistical test, does that mean the probability of my hypothesis being wrong is 1%? If not, why not?
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Equivalence of gamma and inverse gamma:Does choosing a prior lead to possibly different evaluations?

Suppose we had an expressions that was proportional to $$\frac{1}{\sigma^{3}}\exp(\frac{-\beta}{\sigma^{2}})$$ My question is, can you choose different possible parametrization for a prior ...
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Regression Line Calculation | SQRT [on hold]

This is either a typo or I am missing something big. The answer provided is 484 Pounds but I get 4.7 pounds.
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1answer
49 views

Type I and Type II errors in Hypothesis Testing

I am confused about the last highlighted sentence regarding finding a subset S, for which BOTH Type I and Type II error probabilities are 0. For $P_\theta S$, which is the probability with which we ...
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1answer
19 views

Small calculation in Max entropy

I am wondering where the -1 comes from in the derivatives with respect to the probability. For example, these notes. For the basic example with the uniform distribution. We want to minimze $[-\int ...
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what is the explicit form of average marginal effects of Probit model

I came across a question about the AME estimator and its asymptotic distribution. Firstly, I want a definition of AME in the expression of a mathematical expression. and see how could I get the ...
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The uniform distribution is U(0,θ) belongs to scale family [on hold]

If x follow uniform distribution is U(0,θ) since X=θ*Z where Z follow uniform(0,1) Then X ~ U(0,θ) belongs to scale family. How can I prove X ~ U(0,θ) belongs to scale in more formal way? Thanks
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most powerful test between two density function

I need to derive the MP level α test: let $f(x)_0=(1/√2π)~~ exp(−x^2/2)~~ and~~ f(x)_1=(1/2)exp(−|x|)$ $H_0:f(x)=f(x)_0 ~~ vs~~~ f(x)=(1/2)f(x)_0+(1/2)f(x)_1$ below what I did, but I can not ...
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1answer
27 views

Estimating the variance of a function of MLE

Say I have the following likelihood : $$ l(\alpha, \lambda) = n(\log \alpha + \log \lambda) + (\alpha -1 )\sum x_i - \lambda \sum x_i^\alpha $$ which is that of Weibull distribution. The question ...
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I have this as an assignment question and can not figure it out, need some help please [on hold]

a)The mean weight of King Penguins found in an Antarctic colony last year was 15.4 kg. In a sample of 35 penguins same time this year in the same colony, the mean penguin weight is 14.6 kg. Assume the ...
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1answer
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Which kernel is this

If $x \sim N(0,\sigma^{2})$ and $\pi(\sigma)=\frac{1}{\sigma}exp(\frac{-a}{\sigma^{2}})$ then the posterior would be proportional to As, $f(x|\sigma)$ proportional to $\frac{1}{\sigma}exp(\frac{-...
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1answer
32 views

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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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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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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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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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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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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how to calculate confidence interval of mean and std with estimated CI of shape and scale in gamma distribution?

I have used maximum likelihood estimation to obtain the estimations of gamma distribution parameters shape and scale, but how to use this estimated value and confidence interval to calculate ...
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Quantifying time series predictions

After building a time-series prediction model, Test Data: ...
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1answer
61 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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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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1answer
85 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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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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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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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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2answers
154 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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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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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
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Densest subgraph : densitity of an intersection

I found an exercice in my textbook and I can't find the awnser : $G = (V;E)$ 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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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
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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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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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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
411 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
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XGBoost tree “Value” output: [duplicate]

Using the following R code I obtain a decision tree using the agaricus dataset: ...
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1answer
25 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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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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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
35 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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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
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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
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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
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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
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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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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 = ...