Questions tagged [mathematical-statistics]

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

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I have troubles with getting a neural network to learn a function [duplicate]

I am currently trying to make an artificial neural network learn a function that takes a 1 dimensional input and the output is also of dimension 1. However, I have some troubles making the network ...
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Different results with different link functions

I have used the R package eventglm to construct pseudo observations, and want to estimate the relative risk and the risk difference for exposure with age adjustment....
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MLE for Two component mixture model

Chapter 8 section 8.5.1 of the Elements of Statistical Learning book describes a simple mixture model for density estimation and the associated EM algorithm for carrying out maximum likelihood ...
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Effects of ignoring random effects to type 1 error

I was been told that failing to account for repeated measures can cause more type 1 errors, since sample sizes are over-counted. However, I simulated some data under mix effect model while ignore the ...
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Normality assumption of structural shocks in Vector Autoregression model

I’m currently working on forecasting yield curve evolution over time using a state space model where the transition dynamics are described using a vector autoregressive model of order 1. This model ...
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What is the variance of the error term for the regression of the sample mean of $x$ on the sample mean of $y$ that is heteroskedastic?

I am having difficulty finding an appropriate expression for the variance of $\bar{e}_{k}$ in $$ \bar{y}_{k} = \tilde{\beta} \bar{x}_{k} + \bar{e}_{k} $$ where $i = 1, \dots, n $ individuals from a ...
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Can distribution of i p-values from regression of Y over Xi provide evidence for/against Xi interactions?

Suppose we have $i$ explanatory variables, and for each of these, we construct a linear model to regress a single dependent variable $Y$ over each explanatory $X_i$. This produces a distribution of $i$...
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How to choose between one-way, two-anova and multliple regression?

I am working on the data set that consists of Patients' id (after stroke), Time (they can walk after going through the program), the Program they follow and the Visit number when the Time was measured....
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What is the impact of duplicate data on the variance of regression coefficient? [duplicate]

What is the impact of duplicate data on the variance of the regression coefficient?. Does increasing the size of data always certainly decrease the variance of the model coefficients? Suppose I have ...
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What is meaning of such notations in general?

I read a notation on a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models" by Tengyuan Liang, Subhabrata Sen, and Pragya ...
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Find asymptotic variance of the moment estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Derivation of F distribution from quotient distribution

I'm looking for the derivation of the F distribution. There is a fairly understandable (for me) derivation presented on this site: http://www.milefoot.com/math/stat/pdfc-fdist.htm. On this site, the ...
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In elastic net regularisation, will dividing the OLS term the number of observations cause misleading results when cross-validating?

Two formulations of the elastic net regression function Consider sklearn's implementation of elastic net regularisation (Wikipedia link). From the docs, it works by ...
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Issue with bounded in probability

I have tried to prove the following problem that I read in the lecture and it seems not transparent to me. Suppose that $Y_{i}$ be independent random variables (with $i=1,2,3, \dotsc$). Each has the ...
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How to determine how much variation was due to the differences between a group?

I am working on the data set that consists of Patients (after stroke), Time (then can walk after a going through the program), and Program they follow. ...
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Find fisher information matrix for optimization estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$ and we have ...
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How to have a constant error[b-a] for a range of values [1d array] in decending order that are not linear but are semi-linear?

I am trying to generate n numbers in decending order (for the purpose of decreasing order of weights) such that their sum should be 100 and more importantly the difference between anytwo numbers ...
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Error Terms in the ANOVA Summary Table

my lecturers require us to write the error term for the effect/source in the ANOVA summary table. Such as below: I don't understand how to find which S(error) term goes to which source. And I do know ...
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Can somebody help me understand the sentences in more readable expresions?

I was reading a paper about "bayesSimIG" and I have problem in understand the following paragraph.I have read it many times and did a lot of research for it and have understood what each ...
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Proof of statistical assumptions behind the methods or tests

When reading statistics literature, they provide the assumptions of when specific tests/methodologies can be used. Take factor analysis as an example; you may find this inside a paper: "Generally ...
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Is there a boundary between statistical and machine learning methods?

I want to better understand what exactly is considered "machine learning" or not. I realize there might not be a categorical answer for this, but that's what I want to understand. ...
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football probability

I have the following problem: Suppose that the probability of a football team (T) to win the match with 2-1 is p1 and, given that T scores a goal, the probability that the best player of the team ...
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Detailed comparison of two methods for obtaining the ridge regression solution

I have come across two different ways of obtaining the ridge regression solution, which are as follows: Method1:-(obtained from here) $RSS(\beta) = (Y-X\beta)^T\cdot(Y-X\beta)+\lambda\beta^T\Omega\...
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Bounds on the conditional variance of a truncated binomial

I have a binomial variable $R$ drawn from $binom(N, p)$, and I'm interested in the variance of $R$, given $R \ge Q$. The pmf of this variable $R^*$ is $$ \phi_{R^*}(l) = \frac{\phi(l, N, p)}{P} $$ ...
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How can I calculate Posterior Distribution, analytically with given information?

The image below shows that the posterior distribution is as follows with given information: I wonder how the posterior has been calculated, analytically.
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Conditioning of join gaussian over a line

I need to compute the conditional probability of bivariate normal distribution over a line. Let's suppose that X and Y both are normal distributions and that they are independent. Let's suppose that ...
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Gauss-Markov with $p>n$

Let $p$ be the number of parameters in a linear regression model, let $n$ be the number of observations, and let $p>n$. $$\mathbb E[Y\vert X] = \beta_0 +\beta_1X_1 +...+\beta_pX_p$$ Does the Gauss-...
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Derivation of residual bivariate expression to find the value for a multiple linear regression coefficient

Could someone help me with this derivation? How do I get to this expression of $k$? $y_i=\beta_0+kT_i+\beta_1X_{1i}+...+\beta_kX_{ki}+u_i$ $k = \frac{Cov(Y_i, \tilde{T_i})}{Var(\tilde{T_i})}$ where $\...
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Mathematic modelling vs. machine learning

As a biologist with some background in mathematics I am really interested in mathematical modelling of biological processes, like tumor development or therapy responses. However, looking at recent ...
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Determining Percentile Ranking of Test Score Given Individual Score, Population Mean, and Standard Deviation

I have recently received an exam grade of 160/200. In addition to this, I have been given the information that the mean score amongst all students was 114.25 and the standard deviation was 45.62. ...
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Why do discrete choice models (such as MNL) not require test set?

The central challenge of Machine learning models is perform well on unseen data. The data is randomly split into train and test set. The test set acts as surrogate for unseen data and is used to ...
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sufficient, minimal, complete

Are all complete statistics functions of each other? For example if I have T and S complete statistics Can you always write T in terms of S and S in terms of T?
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variance of the difference of two independent variables is the sum of variances

It is said that variance of the difference of two independent variables is the sum of variances (topic - Confidence intervals. Two means. Independent samples) I tried to experiment with small dataset ...
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Does the likelihood ratio test violate the likelihood principle?

I've been going over Berger's famous example of negative binomial vs binomial sampling leading to two different p-values conditional on the same observed data. To summarize, suppose we observe 9 tails ...
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Is it possible to prove that this function of the truncated binomial is decreasing?

Suppose a random variable is binomially distributed with $G$ trials and success probability $p$. Consider the expected proportion of successes, given that the proportion of successes is at least $k$: $...
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Are there possibilities to determine 95% confidence interval for right skewed data?

The dataset that I'm using for my thesis is right skewed. It is about lead times (days) I tried log10 transforming it in SPSS but it still does not meet the requirement p>0,05 (Shapiro-wilk). So ...
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What is the conditional expectation of two random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
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Comparing current and new version of Autoencoder

I am building an LSTM Autoencoder (unsupervised model) to detect anomalies in a time series dataset. The input is telemetry data from routers and I want to detect anomalies in the throughout of ...
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3 votes
1 answer
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two sample t-test for non-normal population

I assume this question has been beaten to death and thus I am just looking for a reference which goes through the details. Assuming all populations we deal with have finite means and variances (even ...
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What is my best option to evaluate a predictive (or proxy) variable?

I have a list of thunder flashes, and I'm trying to find if a meteorological variable (CAPE) is a good predictable variable (or proxy) for theses flashes. In my thoughts, I want to evaluate this proxy ...
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How to generate a general trend from hundreds of time-series data

I have hundreds of time-series data from individual experiments[2] with a similar trend that reading rises as time goes by (evaluated by my naked eyes). So what statistical tools should I use to ...
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Visualization of $L(X)$

Let $L^2_+$ the set of all $2$-dimensional nonnegative random vectors $X = (X_1, X_2)^⊤$ with finite and positive marginal expectations, and let $Ψ^{(2)}$ the class of all measurable functions from $\...
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How to find expected value from cumulative distribution function?

Hello everyone, I'm currently doing research based on the model in Competitive fit-revelation sampling and mixed pricing strategy (Wu &Deng, 2021). And I don't understand how they can conclude the ...
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Identification of a pattern based on given features in time-series data

I am working with electricity time-series data collected at 15 minutes intervals. I am looking for a procedure/theory to find the pattern/sequence in the time-series data based on given features. As I ...
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Suppose $X$ and $Y$ have joint pdf $f(x,y) = 0.5,$ for $0 \le x \le 1;$ $0\le y\le2.$ Evaluate $P (Y >X).$

Kind of lost where to start with this. Could be easy but I don't know where to start simply by knowing $f(x, y) = 0.5.$ How can I find the probability $P(Y > X).$
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What is the best statistical test for correlation between 2 ordinal variables? [duplicate]

"1st variable is ranking 2nd is Likert scale I am to correlate the motivation scale (Likert) to job preference (ranking)"
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sufficient statistics for bernoulli distribution

Let Y1, . . . , Yn be a random sample of size n where each Yi ~ Bernoulli(p), and let Y = $\sum$ Yi for i = 1, . . . , n. The estimator is W= (Y+1)/(n+2) Is the estimator a sufficient statistics for ...
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2 votes
1 answer
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Model performance when ground truth is not available

I am building an LSTM Autoencoder (unsupervised model) to detect anomalies in a time series dataset. The input is telemetry data from routers and I want to detect anomalies in the throughout of router....
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2 votes
1 answer
55 views

Proof that a necessary condition for characteristic roots to lie inside unit circle is $\sum\limits_{i=1}^{n} a_{1} < 1$

I have been trying to show that given $$P_{n}(\alpha) = \alpha^{n} - a_{1}\alpha_{n-1} - a_{2}\alpha^{n-2}... - a_{n} = 0,$$ the $\alpha$'s that solve this equation (real-valued or complex) lie in the ...
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1 vote
1 answer
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How to test for an "interaction" between two interactions

I plan to apply a LMM with the following structure: Y ~ A1 + A2 + A3 + A1:A2 + A1:A3 + (1|subject). I expect the two interactions to display opposing patterns (...
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