Questions tagged [mathematical-statistics]

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

4,378 questions
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Test differences between rates

I ran a digital advertising test with 4 different groups of demographic. 4 different segments visited the website > goal : purchase Segment A - 1000 visitors - 10 purchases > 1,0% purchase rate ...
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Multiple state transition probabilities in Markov model

I have a problem where, in a multi-state Markov model with monthly timesteps, I have a state X where, in an average of 1 year, individuals exit from the state in this proportion: 56% of individuals ...
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How to modify a linear regression model when the variance is not constant?

I have a very basic question. I have a model that has residuals that don't follow a constant variance pattern, but rather it has a clustered pattern (see the figure attached). Any idea of how to deal ...
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Why does independence imply zero correlation?

First of all, I'm not asking this: Why does zero correlation not imply independence? This is addressed (rather nicely) here: https://math.stackexchange.com/questions/444408/why-does-zero-correlation-...
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How to use parcel route data to calculate a threshold [on hold]

I have a parcel delivery business and I want to calculate a threshold for each route. I need this to compensate more when a driver delivers above a threshold. This will encourage drivers to help their ...
182 views

Bias Variance decomposition derivation question/proof (from Wikipedia)

I have a question on this derivation of the bias-variance decomposition. At some point they have this part of the expression --> $\mathbf{E}[2y\hat{f}]$ and they say that $\epsilon$ and $\hat{f}$ are ...
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Mean and variance of a variable (inside a function) without known its distribution, but known mean & variance of the function

Let $$Z_k = A\, e^{i(2\pi B+\phi_k)}, \qquad k =1,2,3\dots$$ be a complex function with dependent on $\phi_k$ and others are real constants. Assume that the mean $\mathbb{E}[Z_k] = \mu_{z_k}$ and ...
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Getting sets of random correlated variables

For the training of a machine learning model I need to add additional features, and these features are correlated. I need to run the model N times adding these features with random values, and for ...
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Which statistical methods to be used?

I am analyzing data in order to show the reasonings of interindividual variabilities of the results of a laboratory data that was measured repeatedly over 100 hours in a single group. But the timings ...
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class grade and weigh [on hold]

A statistics class has the following activities and weights for determining a grade in the course: test 1 worth 15% of the grade, test 2 worth 15% of the grade, test 3 worth 15% of the grade, homework ...
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How do I get the p value of AD test using the results of scipy.stats.anderson()

I am conducting an Anderson Darling test for normality using scipy.stats.anderson() command in python. I am getting test statistic, critical values at various significance levels as the output. The ...
287 views

Prediction interval for lasso regression with time series data

I am currently working with time series data. My objective is to predict the a certain value at time t given some other variables that we will know the same day ( but prior to our objective variable). ...
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Home-brewing GARCH implementation

Motivation I want to wrap up my own GARCH implementation to make sure I have understood the underlying model/assumption. to leverage forecast::auto.arima to ...
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Central Limit Theorem only needs sample size, N?

I think explaining the central limit theorem needs two elements: the sample size and the number of samples drawn. But nobody seems to talk about the number of samples drawn when they are making some ...
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Estimating conditional probability with many samples

I am confused about the estimation of conditional probabilities. Suppose I want to predict a binary outcome variable $Y = 0,1$ given $n$ categorical features $X = (X_1, \ldots, X_n)$, i.e. to ...
372 views

Is Complete Statistic Uncorrelated with Ancillary Statistic

By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be ...
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Proof $E[Z'TZ]^2=\operatorname{tr}^2(T)+\operatorname{tr}(T^2)$ [duplicate]

How to prove second moment of a quadratic form where $Z$ has normal distribution with mean zero and covariance matrix identical?
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Very Crisp and Concise Statistics Book [closed]

I had been struggling with various Statistics books and online MOOCs but I keep losing interest because of the following reasons: Formulas are unnecessarily complicated. e.g. extra super/subscripts, ...
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Fixed effects one way Anova model [closed]

Can somebody explain answer of this with explanation? I guess answer should be (a) but not sure about it?
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Variance comparison of $XY$ and $\min(X,Y)$

There are two independent random variables and each is Gaussian with $\mathrm{E}(X)=mx, \mathrm{E}(Y)=my$, variance of X, i.e. $\mathrm{Var}(X)=vx, \mathrm{Var}(Y)=vy$, respectively. I would like to ...
202 views

How to read checkresiduals graphics in R?

I need to check the residuals of two models in R so I can determine how bad or good are said models. First, I've started simulating an INAR(2) model and wanted to fit a more convenient model, then, ...
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How to find the relationship between two variables using regression? [closed]

I have a Raspberry Pi hooked up to a sensor, which will send a data to the device every second. As you may have guessed, what I want is to find a relation between this independent variable time and ...
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Switching $H_0$ and $H_1$ by replacing $p$ with $1 - p$

I was reading the source code of tseries::adf.test, and it writes ...
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Explain at least three limitations of using the iid normal distribution to model the simple gross returns? [closed]

I just got one limitation, simple gross return is /frac{P_{t+1}}{P_t}, so it may not symetric which means it cannot suit normal.
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Leave-One-Out Property (LOOP)

I was reading this paper, and I was looking at the leave-one-out property that they mention in Appendix B. The property is as follows: For $x\in \mathbb{R}^n$, define $x_i$ to be the $i$th element ...
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How to identify Correlated Features with given Variance [closed]

I cannot interpret how can I identify the Correlation between features with given Variances. Can someone help me out with this?
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What is the difference between Gamma GLM on log output and Gamma GLM with log link function?

Here are two models (with R code to provide some context): Model 1: Take the log of the output variable $y$, then apply a Gamma GLM using the default identity link function: ...
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Analyzing Data for multiple treatment groups

I'm currently designing a research proposal to analyse the effect of varying the amount of cash transfered in a conditional cash transfer program on raising school attendance rates. The cash ...
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ranking based on lower confidence interval

I have a database of bridge scores from a local bridge club that effectively contains for this question, three fields: name, date...
28 views

How to make a model to distinguish positions using data from sensors?

What I would like to do I need ideas about how to analyze data and make a model to distinguish positions where an object locates using data from receivers. I don't know of a similar case like my ...
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Where is the single-crossing point between a distribution and its single mean-preserving spread?

Consider two random variables, $Z_A$ and $Z_B$, with the same expected value $\mu$, having distribution functions $F_A$ and $F_B$, respectively. Let $Z_B$ be obtained by a single mean-preserving ...
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Practical usefulness of pointwise convergence without uniform convergence

Motivation In the context of post-model-selection inference, Leeb & Pötscher (2005) write: Although it has long been known that uniformity (at least locally) w.r.t. the parameters is an ...
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Gauss's Original Gaussian Distribution Proof Help

I am refering to the proof of the guassian from the famous document: https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf I have attached the pictures below of the ...
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Pupils - Correlation between exam mark and answer from 1 to 5 of a test

100 pupils have their exam mark from 1 t0 50 and separately they have reposed a questionary with 10 questions. The variable that represents the answers for the questions of the questionary are ...
398 views

Maximum likelihood function for mixed type distribution

In general we maximize a function $$L(\theta; x_1, \ldots, x_n) = \Pi _{i=1}^n f(x_i | \theta)$$ where $f$ is probability density function if the underlying distribution is continuous, and a ...
324 views

P(h|v) conditional probability proof of Restricted Boltzman Machine

I have a question about the last step of the proof of conditional probability of the restricted boltzman machine in the deeplearningbook.org Below are the derivation from the book: Shouldn't ...
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Prove that the Pitman estimator is itself complete sufficient

It's a homework question, but I just have no idea about it ... Let be random variables according to a distribution having joint density ,where is a location parameter. Assume that there exists a ...
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Deriving Multiplicative Update Rules for Regularized NMF

After reading the following CrossValidated post, I cannot derived the correct multiplicative rules for regularized NMF from this paper. They obtain the coefficients $|I_u|$ and $|U_i|$ in the ...
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Evaluate methods across datasets based on distance metrics

I want to evaluate $n$ methods, says $M_1,\cdots,M_n$ on $m$ datasets $D_1, \ldots, D_m$. For each method $M$, I have a score function $f(M,D) \in [0, \infty)$ for dataset $D$. A perfect method will ...
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normalize/standardize data

I hope someone here can point me on the right direction. I have the following dataset. I have fish adult and juvenile data that I want to compare side by side, however, the juvenile fish numbers are ...
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solve an exercise of two samples using Kolmogorov-smirnov

I'm looking for books and information like crazy and I can not find what I need. Well the example proposed is about methods that have been used in literature students and these are the data collected: ...
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Have a question about bivariate random vector [closed]

When X follows beta (2,2) and X = x, the conditional distribution of Y is the binomial distribution B (15, x). The probability mass function P (Y = y) of E (X), Var (X), E (Y), and Var (Y) obtained by ...
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What technique is recommended for analysis of repeated-measured-data by using R?

I have a repeated measured dataset, which confused me a lot. I'm a newer to statistics. The experiment design: Feeding a kind of small animal with Food type (three) *Food concentration (High Medium ...
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All different size groups to create m groups from n samples

Suppose we have $n$ samples that we want to divide in $m$ groups. How can we get the total number of groups and their sizes (if possible)? Examples: For $n = 6$ and $m = 3$ we would have $3$ groups as ...
The statistic used in testing $H_0: \rho=0$ versus $H_A: \rho \ne 0$ is: $t_r=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \sim t_{n-2}$ in which $r$ is the sample Pearson correlation of two variables, say $X$ ...