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Questions tagged [mathematical-statistics]

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

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Perfusion Analysis Counts as Survival Analysis?

In perfusion analysis, the patient is injected with some dose of medicine. A machine detects, over time, the dose of medicine in the patient's body. In other words, the data for each patient is time ...
温泽海's user avatar
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Combine back- and forecast errors for cross-validation

Suppose I have a procedure to predict the timeseries value $Y_{t+k}$, where $t$ is the current period and $k \geq 1, 2, \dots$. Now, I want to estimate the procedure's out-of-sample performance. The ...
bodhi's user avatar
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Consistency of a test - convergence of quantile

I have given statistical model $((0,1)^n, \mathcal{B}(0,1)^n,\mathcal{P}_n)$, where $\mathcal{P}_n=\{ P_{\theta}^{\otimes n} \ |\ \theta \in (0, \infty) \}$ and each $P_{\theta}$ has density function $...
tychonovs-scholar's user avatar
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17 views

Nonhomogenous Geometric Distribution Approach

I am trying to solve this problem by considering a geometric distribution with unequal probabilities. First, I am using the Irwin-Hall Distribution to deduce that for n independent uniform random ...
Hamzah10's user avatar
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How to estimate the CI of RERI [closed]

This thread is involved in statistical tech to measure interactions. The RERI is short for "relative excess risk due to interaction". I know it's quite a bit difficult to understand but, to ...
Tom Hsiung's user avatar
1 vote
1 answer
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+50

How to disentangle effects of components in system?

I have a system where one property (delay) depends on its components, a valve $V$ and the piping $P$ in a hydraulic system: $d = f(V, P)$ I have reference measurements with one type of valve and ...
Ken Grimes's user avatar
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1 answer
45 views

Power of One Sided t-test

Let $X_1, \ldots, X_n$ be a sample from $N(\mu, \sigma^2)$ for unknown $\mu \in \mathbb{R}$ and unknown $\sigma > 0$. Fix $\mu_0 \in \mathbb{R}$. The one-sided hypothesis is $H_0: \mu \leqslant \...
温泽海's user avatar
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Proof of Strong consistency of Beta posterior distribution

Suppose that we have random variable $X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0})$ with $p_{0}$ true unknown probability in $[0,1]$. Now, I want to implement Bayesian machinery to ...
Fiodor1234's user avatar
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An example problem of converting a maximum likelihood problem into a restricted maximum likelihood problem

I have a question about this derivation. What is an example value of the actual matrix $A'$ such that $A'X=0$, $A'A=I$, and $\frac{1}{n}\Sigma((A'Y_{i}-mean(A'Y))^{2}=\frac{1}{(n-1)}\Sigma((Y_{i}-...
A Friendly Fish's user avatar
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Why do we assume samples have the same variance when deriving standard error? [closed]

In all derivations I've seen of the standard error formula $\sigma/\sqrt{n}$, it is assumed all the samples in the sampling distribution have the same variance ($\sigma^2$). Why is it assumed they all ...
statataka's user avatar
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How to calculate $P(f_1(X) = \text{max}(f_1(X), \dots, f_K(X))$ when $X$ is multivariate Normal?

Let's say I have a multivariate distribution $\mathbf{X} \sim \text{MVN}(\mathbf{\mu}, \mathbf{\Sigma})$ and a set of $K$ scalar functions of $\mathbf{X}$, $f_1(\mathbf{X}), \dots, f_K(\mathbf{X})$. ...
Noah's user avatar
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Calculate the propagated uncertainty by measurement error in linear fit coefficients

I am currently taking a physic experiment class and am required to analyse the uncertainty of data caused by measurement error. They have provided me the standard way to propagate the error, namely by ...
Monolitho's user avatar
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How can I find the margin of error of the extrema point?

I create a 4th or 5th degree fit curve to find the extrema point of distribution. However, how will I calculate the margin of error of the extrema on x values? Is there any statistical method or ...
Erkan Güler's user avatar
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Is this the correct way to go about determining correlation between upstream and downstream temperature?

I am new to stats and r in general, so please bear with me. I have been trying to find an answer to this question for about the past week and have spent a ton of hours researching but am still ...
Matt Schaaf's user avatar
2 votes
1 answer
53 views

Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n

For random variables $X_1, \cdots, X_n$, we denote the order statistics by \begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ &...
Ishigami's user avatar
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In a bivariate linear regression why is $Y = \alpha X + \beta + U$ where $\alpha$ and $\beta$ are real constants & $U$ is an r.v. an assumption?

Suppose that I want to conduct a bivariate regression between random variables $Y$ and $X$. In the textbooks that I'm reading from, primarily Introductory Econometrics and Estimation and Inference in ...
Musicfacter's user avatar
1 vote
1 answer
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Rao-Blackwell Theorem

I'm having problems on understanding the Rao-Blackwell theorem. In particular I don't understand why the resulting estimator is the one with minimum variance between ALL unbiased estimators of the ...
Onofrio Olivieri's user avatar
3 votes
1 answer
43 views

Suppose $(X,Y)$ have copula $C(u,v)$, does $(aX,aY)$ have the same copula for $a>0$?

Suppose $(X,Y)$ have copula $c(u,v)$ in the sense of $Pr(X\leq x,Y\leq y)=Pr(F_X(X)\leq F_X(x),F_Y(Y)\leq F_Y(y))=Pr(U\leq u, V\leq v)=c(u,v)$, where $u\equiv F_X(x)$ and $v\equiv F_Y(y)$ and $c(u,v)$ ...
ExcitedSnail's user avatar
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2 votes
1 answer
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Let $X(t)$ be a Gaussian process. Does $\mathbb{E}[X(t)^2 X(s)^2] = \mathbb{E}[X(t)^2 ] \mathbb{E}[X(s)^2 ] + 2 (\mathbb{E}[X(t) X(s)])^2 $?

As the title says, can I apply Isserlis' theorem to $\mathbb{E}[X(t)X(t)X(s)X(s)]$?
hipHopMetropolisHastings's user avatar
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Manual Calculation using STL Decomposition

Does anyone know how to manually perform calculations using STL Decomposition? I have this data: Date Count 2017-01-31 68 2017-02-28 59 2017-03-31 75 2017-04-30 71 2017-05-31 70 2017-06-30 68 ...
Devri Zefanya's user avatar
6 votes
2 answers
480 views

Examples of distribution for which first-order condition is not enough for MLE

As stated by the title, I am looking for an example (if any exists) of a distribution for which annulling the gradient of the (log-)likelihood function w.r.t. the parameters is not enough to ensure we ...
MysteryGuy's user avatar
2 votes
2 answers
141 views

Mathematical Prediction of Linear Mixed Models Random Intercept

Given data $\{(x_{i,j}, y_{i,j})\} \subset \mathbb{R}^2$, with $i = 1, \ldots, k$ classes and $j = 1, \ldots, n_i$. The linear mixed model is: \begin{equation*} y_{i,j} = a + b x_{i,j} + u_i + \...
温泽海's user avatar
  • 447
3 votes
2 answers
197 views

sample size in chi-squared test

The chi-square test of independence is a type of non-parametric test, but in cases of small sample sizes, the Fisher's exact test should be used instead. My understanding of non-parametric methods is ...
Ivan's user avatar
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7 votes
3 answers
463 views

Derivative of the Score Function in Fisher Information

I'm studying Fisher Information and am trying to develop an intuitive understanding. Keep in mind I only have bachelor level mathematics background so I would appreciate an answer that is more ...
Ryan's user avatar
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1 vote
0 answers
33 views

Is it possible to compare the output probabilities of two machine learning models? [closed]

Let's suppose I have two classification machine learning models: $\text{Model}_1$ and $\text{Model}_2$. Each of them are not necessarily the same algorithm, and have not been trained necessarily with ...
Poisson Parade's user avatar
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31 views

Notation to report the measurement of a parameter

The estimation of a parameter ($p$) is customary reported in Physics and other fields with, $ p = \hat{p} \pm \Delta p$, where $\hat{p}$ is an estimator, and $[\hat{p} - \Delta p, \hat{p} + \Delta p]$ ...
Diego Ravignani's user avatar
1 vote
0 answers
72 views

Unbiased estimator of covariance^2

Assuming that the sample covariance $c_{ij}$ is an unbiased estimator of the true covariance $p_{ij}$, how do we find an unbiased estimator $\Theta$ which follows $\mathbb{E}(\Theta)=p_{ij}^2$? I made ...
Paulo Ranazzi's user avatar
7 votes
3 answers
154 views

What is meant by the probability of a sample having a value of $x$ is $ng(x)$?

Reading from Wikipedia: The probability of one sample having a value of $x$ is $n g(x)$. Assuming that the notation is consistent throughout the page, I would take $g$ to either be the probability ...
Galen's user avatar
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2 votes
1 answer
82 views

Is $f(a) = EX^{1+a}EX^{-(1+a)}$ non-decreasing?

$X$ is a non-negative random variable and $a$ is a non-negative real number. Define $$f(a)= EX^{1+a}EX^{-(1+a)}.$$ Is $f(a)$ non-decreasing with $a$? Original problem: when I read a paper, I encounter ...
Voyager's user avatar
  • 305
2 votes
1 answer
61 views

Right continuity of cdf

Before asking, I want to let you know that I realize already there are different proofs for the right continuity of the cdf, however I would like to know if my proof of this is correct, as I assume it ...
curious's user avatar
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0 answers
10 views

The generalized likelihood ratio test of H0: µ<=µ0 v.s. H1: µ>µ0 with unknown σ [duplicate]

Assume:random samples in N(µ,σ²) when null hypothesis(H0) is true, why MLE is min{µ0,X̄} ? How can get MLE in restricted parameter space(µ<=µ0) ? please help me
강래현's user avatar
1 vote
0 answers
14 views

Calculate the contributions of percentage change of a ratio compossed by a/b

I would like to know how to calculate the change contribution to a ratio a/b where b is equal to a plus other variables: ...
Ayoze Alfageme's user avatar
11 votes
3 answers
167 views

How can restricted randomization to achieve covariate balance lead to imbalance in unobserved variables?

In literature, designing an experiment is considered a trade-off between covariate balance and robustness. For example Harshaw et al. (2024) writes In an effort to make the estimators more precise, ...
retodomax's user avatar
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5 votes
2 answers
132 views

Do tail bounds on probability translate into bounds on expectations?

Suppose I have a bound of the form: $$P(X \geq t) \leq \exp(-t^2).$$ Can I say anything about the expectation of $X$, $E[X]$? In particular, can I get a bound on $E[X]$? Here's the specific case that ...
snickerdoodles777's user avatar
0 votes
1 answer
38 views

Variance around true value, not mean

The water temperature is 19 C, which is true value. I have measured the temperature of the water and I received such results: 18.7, 18.8, 19.1, 18.8. Can I calculate variance here around true value (...
Max Heron's user avatar
0 votes
0 answers
12 views

Mapping two Dirichlet Distributions into a comparative Dirichlet

Assume I observe some draws from 2 choice options, and want to infer the probabilities of various outcomes, e.g. non-negative integers up to a limit L. I could simply use 2 Dirichlet distributions to ...
Max Montana's user avatar
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0 answers
8 views

Help with PSD graph interpretation and Time Series Analysis

I am pretty new to time series analysis and I am currently studying it. My latest work requires me to work with a time series that was generate by VASP AIMD. I tried power spectrum density analysis ...
David's user avatar
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0 answers
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Help with completing a derivation of usefulness of cross-validation

This question is raised as a result of my attempt to answer this other question of mine. Let's refer to all our prior knowledge, both explicit and implicit, as $X_\text{true}$. Almost always, we are ...
Feri's user avatar
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1 vote
0 answers
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Sufficient statistic as iso-surfaces in the distribution density. Is it possible to generalise to multiple parameters?

For continuous distributions, there is a geometric intuition behind sufficient statistics that regards a multivariate probability density as several iso-surfaces. This works at least for cases where a ...
Sextus Empiricus's user avatar
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0 answers
28 views

Conjugate prior for a beta distribution [duplicate]

What is the conjugate prior of the beta distribution? All I can find is the wikipedia page on conjugate prior. Is this correct? And does anyone know where it came from? Like a specific paper? Thanks
Thomas's user avatar
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0 votes
0 answers
24 views

How to show that RMSE is more sensitive to outliers than the MAE?

I am reading this book where it states that for $\ell_p$ norms: The higher the norm index, the more it focuses on large values and neglects small ones. This is why the RMSE is more sensitive to ...
ado sar's user avatar
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0 answers
29 views

What is the meaning of term "whiten" data in relation to Mahalanobis Distance?

I'm writing my thesis and I have trouble in understanding the paper: https://people.bu.edu/bkulis/pubs/ftml_metric_learning.pdf My major is not mathematics, but I can understand the basic so I hope ...
jupyter's user avatar
  • 101
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0 answers
8 views

Does there exist a matrix Bernstein type inequality for weakly dependent random variables?

When reviewing the literature, I managed to find the following papers, which deduce Bernstein type inequalities for weakly dependent random variables (where weak dependence is defined in some special ...
Stephen Jiang's user avatar
0 votes
1 answer
63 views

Maximum Likelihood of Standard Deviation

I'm trying to get a better understanding about the distribution and uncertainty of the sample standard deviation. Since I'm not a mathematician, I try to compare the math literature with some ...
Mexx's user avatar
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0 votes
0 answers
22 views

What transformation to adjust empirical data variance 'smoothly'?

Let's assume we have empirical data for an outdated process that generated it. We have reason to believe the process of today is similar (e.g. same distribution) but with a different variance. General ...
yeahman269's user avatar
1 vote
0 answers
50 views

What is the motivation for the use of $n-m$ in the method of moments in Wedderburn 1974?

In an answer to this question Is there a Relationship Between Variance and Chi-Square? I wrote So to find the dispersion parameter, one has to use a different trick. In Wedderburn 1974 this is done ...
Sextus Empiricus's user avatar
2 votes
1 answer
42 views

Proof Markov's Inequality

I have a question regarding the proof of Markov's Inequality, attached as a picture to the post, which is quite basic: It is comphrensible that $\int_{a}^{\infty} xf(x) \, dx \geq \int_{a}^{\infty} af(...
george1994's user avatar
3 votes
1 answer
44 views

Rough answer for the maximum of absolute value of $n$ standard gaussians (Computer Age Statistical Inference Problem 1.3)

I am working through "Computer Age Statistical Inference" as a self-study and am stuck on the follow exercise (1.3): The details of equation 1.6 are unimportant for the exercise, so far as ...
naveace's user avatar
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0 votes
0 answers
12 views

is there a closed form of the CF (characteristic function) of a bivariate von Mises distribution?

is there a closed form of the CF (characteristic function) of a bivariate von Mises distribution? And if I have two parameters that follow von Mises distribution, but my two parameters will be mixed ...
cassidi's user avatar
1 vote
1 answer
28 views

How the randomisation probability is updated for each new patient entering adaptive clinical trial?

Disclaimer: You may need to read the entire paper to answer this question :) I am reading this paper to learn about adaptive clinical trials. Here in page 1723 under "Statistical methods" ...
Chirag Patil's user avatar

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