Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
5,583
questions
1
vote
1answer
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 ...
2
votes
0answers
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 ...
0
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0answers
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 ...
0
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0answers
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, ...
5
votes
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}...
13
votes
2answers
430 views
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 ...
-1
votes
0answers
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 (...
1
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0answers
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 ...
1
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0answers
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 ...
1
vote
0answers
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!
0
votes
0answers
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
$$
\begin{bmatrix}
Y_i \\
X_i
\end{bmatrix}=\mathcal{N}
\begin{pmatrix}
...
2
votes
1answer
35 views
prove change in total probability of success in binomial distribution
A binomial distribution of $n$ samples and probability of success $p$ is defined as
$ P(k) = \binom{n}{k} \cdot p^kq^{n-k} $.
For a given value of $r$ where $r \in \mathbb{N} \quad \text{and } r \gt ...
2
votes
0answers
42 views
Is there a word in statistics for “mean divided by absolute difference”?
For two values, let's say a and b, I calculate a mean of two numbers divided by absolute difference between them. i.e.
...
1
vote
1answer
39 views
Select a sample size form the infinite population
I have a list of every store chain in the US with at least 100 locations in the US.
a) What number of those do I have to pick in order to have a representative sample of the number of locations across ...
0
votes
0answers
21 views
maximum Corelation coefficient - how the numarator and denominator becomes equal?
Pearsons corelation coefficient is defined as follows
$$ 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 ...
0
votes
0answers
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 ...
0
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0answers
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 ...
0
votes
0answers
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 ...
3
votes
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 ...
1
vote
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 ...
1
vote
0answers
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 (-)
...
1
vote
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 ...
-1
votes
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$ ...
1
vote
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}'\...
2
votes
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 ...
1
vote
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 ...
0
votes
0answers
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. ...
0
votes
0answers
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 ...
2
votes
0answers
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 $r.v.$...
0
votes
0answers
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 ...
2
votes
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 ...
2
votes
0answers
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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
votes
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}...
1
vote
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 ...
2
votes
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 ...
1
vote
0answers
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, ...
0
votes
0answers
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 ...
0
votes
0answers
7 views
Suitable statistical test for multiple categories data?
I am compiling my research result which ended up like a table below:
...
0
votes
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 ...
0
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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 ...
0
votes
0answers
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
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 ...
