Questions tagged [mathematical-statistics]

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

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93 views

regression to the mean and its formula in R

In this book, the estimate of the regression to the mean phenomenon is said to be: $Prm=100(1-r)$. Where $Prm$ is the percent of regression to the mean, and $r$ is the correlation between the two ...
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36 views

Independence of the sample mean and sample variance

For $X_i \stackrel{indep}{\sim} N(\mu_i, 1)$, I have to show that the sample mean $\bar{X}=n^{-1}\sum_i X_i$ and $S_{XX}=\sum_i (X_i-\bar{X})^2$ are independent the theory of random projections onto ...
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16 views

2 x 2 x 2 design

I'm doing a 2 x 2 x 2 design. To be specific, it's a 2 (BETWEEN-subjects variable) X 2 (WITHIN-subjects) X 2 (WITHIN-subjects) factorial design. And three Dependent Variables I know MANOVA is the one ...
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18 views

how to estimate the precipitation value from other relative data

I have climate data but the precipitation values are missing, I would like to know if there is any formula used to estimate the precipitation value from the other recorded climate data: temperature, ...
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2answers
136 views

Distribution of the inner product between a noise-free and a noisy signal

I am working on a problem where we have a noisy measured signal, which is stored as an $N$-dimensional vector $\mathbf{Y},$ and a set of $n_s$ simulated noise-free signals $\{\mathbf{X}_i\}_{i=1}^{n_s}...
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What is the origin of the name “conjugate prior”?

I know what a conjugate prior is. But I'm confused by the name itself. Why is it called "conjugate"? A complex conjugate $z^\ast$ has a reciprocal relationship with $z$, i.e., ${z^\ast}^\ast ...
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14 views

How to analyze the data set

I have a data set with all the zip codes in a city Texas, and the number of stores each zip code has. So on excel, it looks like following: Column A (list of all the zip codes) 22160 etc; Column B (...
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13 views

Question on method for analysis

First of all, I have begun researching as a masters student and I am forcing myself on more statistical validation and significance testing - this is not my strong suit. As such, I am applying ...
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24 views

Why not use % change in regression instead of log diff?

Given that log difference = percent change why not make the series stationary by just using percent change as the regresor. a. It's not an approximation and we can interpret an coefficient as a ...
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19 views

Equation for weighted mean of percentages

I am trying to get the equation for a weighted average of percentages to be normalized between 0 and 1. Am I doing it correctly? Thank you so much!
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22 views

Linear regression with noisy observations

Given the model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$ and that one doesn't observe $x_i$ but instead $w_i = x_i + u_i$ where $$ \begin{bmatrix} Y_i \\ X_i \end{bmatrix}=\mathcal{N} \begin{pmatrix} ...
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35 views

prove change in total probability of success in binomial distribution

A binomial distribution of $n$ samples and probability of success $p$ is defined as $ P(k) = \binom{n}{k} \cdot p^kq^{n-k} $. For a given value of $r$ where $r \in \mathbb{N} \quad \text{and } r \gt ...
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42 views

Is there a word in statistics for “mean divided by absolute difference”?

For two values, let's say a and b, I calculate a mean of two numbers divided by absolute difference between them. i.e. ...
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1answer
39 views

Select a sample size form the infinite population

I have a list of every store chain in the US with at least 100 locations in the US. a) What number of those do I have to pick in order to have a representative sample of the number of locations across ...
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21 views

maximum Corelation coefficient - how the numarator and denominator becomes equal?

Pearsons corelation coefficient is defined as follows $$ r_{x,y} =\frac {\sum (x_i -\bar X)(y_i - \bar Y)}{\sqrt{\sum(x_i-\bar X)^2} \sqrt{\sum(y_i-\bar Y)^2}} $$. Now, maximum magnitude of a ...
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27 views

Finding variance and mean of an expression?

I want to compute the expectation and variance of the following expression. $ Z|H_{i}=(|h_{i}|^{2}|s[n]|^{2}+|w[n]|^{2}+2\mathcal{R}(h_{i}s[n]w^{*}[n]))$ where,$s[n]$ is PSK signal and $h$ is rayleigh ...
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13 views

Help establishing the distribution of this problem

Given a population of size N we consider n individuals by means of simple sampling without replacement. Consider D a subpopulation of size $N_D$ from where $n_D$ is the size of the sample that are ...
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62 views

Acceptance sampling

I have this question. "Units of different lengths are produced. If a unit has length greater than 10 metres it is defective. Lengths are assumed to be normally distributed, $ N(\mu,0.01)$ - that ...
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1answer
35 views

Interpreting HRs from stratified cox survival analysis in R

In the R code below, binary_variable_2 has two levels (e.g. "a" and "b"). The code outputs only one HR for ...
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1answer
29 views

Restricted standard deviation of survival time

I'm working through a paper on restricted mean survival time and I cannot figure out how this paper jumps from one step to the next. The restricted mean survival time $\mu$ is the mean survival time ...
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21 views

Are segments painted randomly respective to previously painted segments?

I have a sequence of $N$ binary values (+ and -). At $t=0$, all values are minuses (-) ...
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0answers
7 views

“Dominance” condition for consistency of MLE

On the Wikipedia page for maximum likelihood estimation, there are a set of four conditions listed for the consistency of the maximum likelihood estimator. I had a question about the last one, called ...
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1answer
26 views

Finding C for a PMF of a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = c/k!$ for $k=2,3,4,...$ I used exp series $\sum_{n=1}^{\infty} \frac{x^k}{k!} = e^x$ to get $ c\sum_{n=1}^{\infty} \frac{1}{k!}$ then $ce=1$ ...
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2answers
36 views

Looking to build the Mathematical proof that $ Var(\hat{y}) = \sigma^2\textbf{H} $

I'm looking to build the proof that $$ Var(\hat{y}) = \sigma^2\textbf{H} $$ So I was going down this road and seem to have gotten a bit confused: $$ Var(\hat{y}) = \sigma^2\textbf{X} (\textbf{X}'\...
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1answer
35 views

Likelihood function when there is no common dominating measure?

When we have a statistical model $\{ P_\theta, \theta\in\Theta\}$ on some common probability space, usually we define "the" likelihood function $L(\theta)$ via a Radon-Nikodym derivative ...
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1answer
21 views

annualized probability of default for loan including time component

i am struggling with this. say i am given an annual probabilty of default for a company going insolvent as 0.02. so 2%. say this client then takes out a 100k , 150 day loan on jan 1st 2018, what is ...
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19 views

Can use Factorization criterion for proving given estimator is not Sufficient?

I was learning about the Sufficiency of Estimators and Factorization criteria. Now, I noticed that whenever we prove a given estimator is not sufficient, we use a counterexample with concrete values. ...
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42 views

Expectation of Gaussian random vector over a subspace

Suppose $\boldsymbol{x} \in \mathbb{R}^d$ is a Gaussian random vector, distributed as $\boldsymbol{x} \sim \mathcal{N}(\boldsymbol{\mu}, \sigma^2 \boldsymbol{I}_d)$, where $\boldsymbol{I}_d$ is an ...
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28 views

What are the interpretations of these different types of stochastic convergences?

Thanks for passing by! I am studying an introduction to stocastic processes and I found 4 types of stocastic convergences with their definitions as follows: Let $(X_n)_{n\geq0}$ be a sequence of $r.v.$...
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19 views

Mathematics foundation under choice of varying size input problem

I am reading this paper Taxi Destination Prediction, in which trying to predict the destination of the taxi. The dataset is composed of many complete taxi trajectories. In the dataset, the input is ...
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1answer
181 views

Sources for learning (not just running) statistics/math through Python

I am interested in examples of sources (Python code, Python packages, books, book chapters, articles, links etc) for learning statistical and mathematical concepts through Python (it could also be ...
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63 views

proof of Cramer-Rao lower bound

I am trying to understand the proof for this theorem from the book Casella and Berger (2nd ed.) page 336. $W(X)$ if any estimator for samples $X_1,\ldots,X_n$ on distribution $f(X|\theta)$. I notice ...
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1answer
14 views

Does wealth index created in stata (command: pca and predict) considers all the components whoese eigen value more than 1 or just first component?

I am trying to calculate the wealth index of a rural community of Nepal. For this, I used 10 household assets variables after conducting a descriptive analysis. I used the principal component analysis....
2
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1answer
67 views

What's the definition of $O_{p}(\cdot)$ in the vector case?

For a scalar random sequence $X_n$, we write $X_n=O_p(a_n)$ if for every $\epsilon$ there exists $M_{\epsilon}$ such that $\limsup_{n}\Pr(|X_n/a_n|>M_{\epsilon})<\epsilon$. What's the extension ...
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1answer
24 views

proof of the the prior predicative distribution

According to wikipedia, the posterior predicative distribution is $p(\tilde x | X) = \int p(x | \theta) p(\theta | X) d\theta$ How does one reach at the given equality? I have $p(\tilde x | X) = \int ...
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2answers
84 views

Finding the covariance of two random sums

I am trying to derive the covariance of two sample sums. Some notation and details: $x_i$ and $y_i$ are numeric values of two characteristics corresponding to member i of a finite population of N ...
2
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0answers
23 views

Random coefficients and fixed effects

In a nonlinear model, for each time $t\in\{1,..,T\}$ and product $ j\in \{1,..,J_t\} $ individual $i \in \{1,...,I_{t}\} $ chooses an amount $y_{i,j,t}$ modeled as $ q_{i,j,t} = \frac{1}{\tau_j k_{j,t}...
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1answer
18 views

Finding the mode of multiple sample modes

I'm running an experiment where I'm collecting samples of different size (numeric data only) and computing the mean, median and mode of each sample. I'm interested in finding out the mode across all ...
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2answers
107 views

Predict daily count of delivered packages

I have a historical information for delivered packages (send time and receive time). Also I have an information for packages in transit (send time only). For example, all of the packages in transit ...
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4 views

Test time series performance across subjects against chance

I'm wondering if anyone had any advice on testing where (if at all) the average of several time series deviates against chance. That is, I have an output from an SVM, computed at several time points, ...
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18 views

Is it possible to produce two [random] graphs that always pass each other

My question is whether it is possible to create two graphs that go up one point or fall one point at a time [e.g. every minute] in a random walk, and they will pass over each other for sure all the ...
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0answers
7 views

Suitable statistical test for multiple categories data?

I am compiling my research result which ended up like a table below: ...
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0answers
25 views

Advantages and disadvantages of windsorizing, truncation and logistic transformation of explanatory variables in a logistic regression contect

I am using a logistic regression model to develop a ranking function; e.g. customer A has a higher chance of not paying the invoice than customer B. My data contains many explanatory variables with ...
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0answers
15 views

Sample size to test a three-way interaction

I'm want to run a study with 2 categorical IVs and 1 continuous IV. The DV is continuous. It's my understanding I will need to run a multiple regression to test the effects. How do I determine the ...
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14 views

Condition numbers, invertibility and multicollinearity

The following is an excerpt from Greene's Regression Analysis (Seventh Edition): a) What does it mean to be "difficult" to invert a matrix accurately? Shouldn't all matrixes be either ...
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1answer
32 views

Doubt in independence of 2 random variables

If 2 random variables are independent, then $f(x,y) = f(x)f(y)$. Is converse true? $F(x,y)=F(x)F(y)$. Is converse true? $E(x,y)=E(x)E(y)$. Is converse true? where $F$ is cdf and $f$ is pdf I recently ...
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1answer
72 views

How do I find the MLE of the APE distribution in R?

The random variable $Y$ is said to have a two-parameter APE distribution, denoted by $\text{APE}(\alpha, \lambda)$, with the shape parameter $\alpha>0$ and scale parameter $\lambda>0$ if the ...
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9 views

How to convert Incidence rate ratio into probability?

I understand that converting odds ratio into probability one can divide the odds by the odds plus one. Is it the same formula to convert Incidence rate ratio into probability? I'm trying to interpret ...
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36 views

What is the best method to compare two populations within a bigger group?

I have a data frame with two columns: rows correspond to different people, one column is the country of the respondent. This is a categorical variable, the other column is the yes/no answer to the ...
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1answer
37 views

Multilayer Perceptron | Likelihood of parameters

I am referring to a simple multiplayer perceptron (e.g. only one hidden layer and one output layer). Notation: Assume we have $K$ dimensional output. The number of samples we have is $N$. And we ...