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Questions tagged [mathematical-statistics]

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

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Find the same components between two distributions

GMM assumes that a distribution consists of multiple Gaussian Distributions. Is it possible to find the same components between the two distribution? For example, there are two distribution D1 and D2. ...
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Monte Carlo simulation from autocorrelated empirical distribution

So I'm trying to perform a Monte Carlo simulation an empirical distribution of log-return time series data. However, Ljung-Box test indicates that there is significant autocorrelation at multiple lags....
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Why does the cumulative distribution function for discrete random variable right continuous?

If the random variable is discrete, then the cumulative value should also be discrete because the variable can only take on discrete values, right?
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Introductory books on bayesian statistics with focus on normal distribution

I am searching for introductory books on bayesian statistics. Which Focus on normal distribution (Most of the books I came across through this answer focus on binomial distribution) Practical ...
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1answer
40 views

Distribution of a one realization of a stochastic process [closed]

Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this ...
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1answer
33 views

Question about prediction: if a variable can predict another variable [closed]

I have a question about the prediction. This question stopped me for a whole afternoon, still do not have an idea on how to solve it. Would you please give me a clue. The question is: ...
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1answer
17 views

How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
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1answer
28 views

Calculating variance of process with time-varying variance

This is a question stemming off a previous post I had regarding calculating portfolio volatility. For a portfolio consisting of multiple assets, I understand that there are multiple ways to calculate ...
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1answer
34 views

Calculating portfolio volatility from portfolio returns vs. from covariance matrix

I'm having trouble understanding the difference in calculating portfolio volatility via the portfolio returns vs. via the covariance matrix. To be more specific: I understand that on the individual ...
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1answer
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Statistics and Probability. Please show solution [closed]

Miss Romero noted that the mean scores of a random sample of 15 grade 8 students who had taken a special test were 80.5. If the standard deviation of the scores was 3.1 and the sample came from an ...
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25 views

Log Likelihood and parameter space

I'm taking a 3rd year statistical theory class in which we're introducing the likelihood function with a bit more mathematical rigor. Under the discussion for maximizing the likelihood function $\...
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0answers
7 views

Specifying the bandwidth parameter for Local Whittle estimation

In every Local Whittle function I have seen I have to set bandwith parameter usually denoted m such that m = floor(1+T^delta). I am interested in the delta, how do I choose what value to put in the ...
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43 views

How to compare two dispersion measures in a specific probability density function?

I am very interested in the properties of a specific probability density function (pdf) proposed in this article. It seems to me that this pdf is a very general one to capture the properties of wide ...
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2answers
107 views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
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10 views

Dependency of variable P on multiple factors, and factors interactions

In my data, I have a variable/vector P which is dependent on variables X, Y, Z, K and gender. The factors X, Y, Z are interacting. What kind of statistical tool/method I can use to explore dependency ...
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1answer
55 views

Algorithm for modified knapsack problem

We have n-players in a game. We have a population of players we can choose from. Each player score is a normally distributed random variable and each player has a cost to add to the team. We are ...
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1answer
58 views

Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
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1answer
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Accounting Student taking Master in Stat, need your help regarding taking subjects

I had a Bachelor in Accounting and now I'm doing a Masters in Statistics. This is my first semester and I have to Choose at least 3 out of 6 offered subjects for this semester. These subjects Are: ...
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1answer
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AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...
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2answers
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OLS with ARMA errors, tip or two

I need a tip or two. I am performing OLS with dynamic factors (4x1 factors each representing a PANEL of 24 series, hence 4 time series). My OLS has autocorrelation in the error so I want to use OLS ...
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0answers
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Principle coordinates analysis - visualization and assigning text to graph in r [closed]

I've recently run a principle coordinates analysis in r with ~ 200 subjects and 17 binary variables. I've plotted my points successfully, the challenge I'm facing is that when I plot the text with the ...
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1answer
37 views

Use linear projection without constant to obtain the linear projection with constant

We know that the linear projection of $y$ on $x_0$ $x_1$, $x_2$, . . . $x_K$ always exists and is unique: $$L(y|x)= \gamma_0 x_0 + \gamma_1 x_1 + ... + \gamma_k x_k = x\gamma$$ where $x = (x_0, x_1, ...
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In a manuscript, is an incomplete beta function a useful simplification of a sum, if the argument ends up negative?

I'm working on a manuscript which I intend to submit to a statistics journal. I have a sum which I "simplify" to the regularized incomplete beta function $I_{-t}(a,b)$, where $t$ is a positive real ...
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0answers
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Distribution of a square of a random variable [duplicate]

If X is normally distributed with mean mu and variance sigma^2, then what can I know about the distribution of X^2 ?
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0answers
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Structural models and relationship (statistical associations)

I'm starting to study econometrics from the Wooldrige's book. But some doubts arise regarding the structural model, its definition and its comprehension in general. Let's look at the author's opinion. ...
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Linear Regression Expected Value [duplicate]

Whiles reading An Introduction To Statistical Learning under linear regression (Chapter 3), I found: $$E(Y - \hat{Y})^2 = |f(X) - \hat{f}(X)|^2 + Var(ε)$$ where $E(Y - \hat{Y})^2$ represents the ...
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2answers
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Is $(1-L)Y_t=\epsilon_t \to Y_t=(1-L)^{-1} \epsilon_t$ mathematically correct?

Consider an AR(1) model with coefficient $\rho =1$, i.e. a random walk, $Y_t=Y_{t-1}+\epsilon_t$. We usually say that since the AR polynomial has a unit root, it cannot be inverted and therefore the ...
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1answer
27 views

Using a generalised linear model for tennis winning probability

I have a dataset of historical tennis matches, one row per match. Each row has the ELO points of each player, and a calculated ELO difference. And of course a column to indicate if Player 1 won as a 1 ...
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0answers
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Stationarity of ADL(p, q) with heteroskedasticity

Suppose I have the model $$y_t = \alpha_0 + \alpha_1 y_{t-1} + ... + \alpha_p y_{t-p} + \beta_0 x_t + ... + \beta_q x_{t-q} + \epsilon_t,$$ where $\{x_t\}$ is a stationary process and $\epsilon_t$ has ...
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0answers
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Identification of mixture models with location shift

I am studying identification of mixture models of the type $$ F(x)=\sum_{j=1}^J \lambda_j G(x-\mu_j) $$ with $F,G$ denoting univariate CDFs, $G$ symmetric about $0$, $\mu_1<\mu_2<...<\mu_J$, $...
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1answer
44 views

Proof for simulation of NHPP by thinning

Background: I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
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Marketing Funnel Percentage Calculations

I implement a marketing funnel in python and I have an issue around the topic that I can't solve neither in statistical level. Lets say that we have the funnel A which contains ...
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Given a simple logit panel data model, the MLE estimators are inconsistent

Consider a binary choice model, $P \left( y _ { i t } = 1 | x _ { i t } , \alpha _ { i } \right) = F \left( x _ { i t } \beta _ { 0 } + \alpha _ { i } \right)$, $$F ( z ) = \frac { e ^ { z } } { 1 + e ...
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1answer
19 views

Sample size needed?

I need some help calculating the minimum sample size needed for my upcoming trial. I’m doing higher degree research on eHealth, so I have little knowledge about statistics and sample size calculation. ...
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What is the relationship between Online Learning and Statistical Learning?

Online Learning also known as Online Convex Opimization has famous algorithms like Follow-the-Leader and Online Gradient Descent (See OCO Book) Now stoastic programming has algorithms like Sample ...
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Uniform distribution on the simplex. - Thomas cover

I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...
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1answer
25 views

What happends if we standarise with a slightly shifted mean?

For deep neural networks or any ML algorithms , lets say the mean of my dataset is 5 and instead I normalize with 6 or 4, something slightly off. What affect does this have on learning?
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Using GARCH to model seasonal variance clustering

Question: is it a good idea to model variance clustering in a time series with GARCH when the clusters are seasonal? Details: Imagine that we are given a time series $(x_t)_{t\in \mathbb{N}}$ that ...
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1answer
106 views

In the superpopulation framework of causal inference, what is the necessity of assuming the outcomes, treatment and covariates are jointly iid?

In the textbook "Causal Inference for Statistics" by Rubin and Imbens, the following argument is made on pg. 39: "In part of this text we view our sample of size N as a random sample from an ...
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1answer
65 views

How to Identify dependent variable from my dataset?

I have a dataset which consists of 50 variables. I want to programmatically(using python) identify what is my dependent variable is. Are there any tests to separate the dependent variable from ...
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0answers
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Help with centered multiple linear models

For the multiple linear model in centered form prove that where is the L-2 norm, defined as Any help would be greatly appreciated. Perhaps links to papers, chapters, websites that could help me ...
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1answer
33 views

Algebra in Cantelli-Cheybyshev Inequality Proof

I am confused by the following (possibly simple) algebra in the proof of the Cantelli-Cheybyshev inequality. I am following Rohatgi and Saleh (2015, Section 3.4 Lemma 1) where we plug in $\phi(x)=(x+c)...
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1answer
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Get better confidence interval by drawing multiple samples

Imagine that I have population of size N and I draw a sample of size n from which I compute a proportion $\hat{p}$ of a categorical variable. And I want to get a confidence interval of 95%. So I ...
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1answer
19 views

relationship between correlation, covariiance and conditional distribution

What are the relationships between correlation and conditional distribution. For instance, given three dependent variables, X1, X2 and ...
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0answers
13 views

Statistical significance regression analysis [duplicate]

Imagine a regression with 3 independent variables. All variables have positive and statistically significant signs. When you add other variable, the coefficient of your main interest variable becomes ...
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1answer
33 views

Need help with Least Squares Estimator

I am interested in a One-Way ANOVA model:                           &...
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1answer
29 views

Why can we treat MGF in this way

For the standard proof that if $Z \sim N(0,1)$ than $Z^{2} \sim \chi^{2}_{1}$ We write: $$M_{Z^{2}}(t)=\int_{\mathbb{R}}\exp(tz^{2})\frac{1}{\sqrt{2\pi}}\exp(\frac{-z^{2}}{2}) dz$$ That is, we use ...
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Bonferroni confidence region for shifted Laplace parameters

Consider the shifted Laplace distribution with the density: $$f(y)=\frac{\theta}{2}e^{-\theta|y-\mu|}\quad, \quad y\in \mathbb R$$ Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
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How to estimate the MLE for a probit panel data model in R? [closed]

Can you recommend a good package in R to calculate the MLE for a simple panel data probit model? I would like to use GHK simulator to do it. But What I found is not for panel data in R.
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Statistical Reasoning of Noise Images on Random Pixel Generator

http://www.pixelmonkeys.org/#theory It is always explained that even billions of images are generated per second, it is almost impossible to see a natural image ( whether it is clear or distorted as ...