# Questions tagged [mathematical-statistics]

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

4,378 questions
0answers
11 views

### 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 ...
0answers
10 views

### 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 ...
1answer
16 views

### 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 ...
3answers
2k views

### 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-...
0answers
8 views

### 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 ...
0answers
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 ...
1answer
23 views

### 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 ...
0answers
26 views

### 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 ...
0answers
21 views

### 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 ...
1answer
18 views

### 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 ...
0answers
174 views

### 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 ...
2answers
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). ...
1answer
28 views

### 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 ...
1answer
89 views

### 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 ...
1answer
178 views

### 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 ...
1answer
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 ...
1answer
40 views

### 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?
0answers
33 views

### 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, ...
0answers
13 views

### 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?
1answer
38 views

### 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 ...
1answer
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, ...
1answer
58 views

### 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 ...
0answers
31 views

### 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 ...
0answers
15 views

### 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.
0answers
15 views

### 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 ...
0answers
11 views

### 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?
0answers
59 views

### 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: ...
0answers
10 views

### 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 ...
1answer
273 views

### 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...
1answer
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 ...
0answers
7 views

### 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 ...
1answer
169 views

### 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 ...
0answers
22 views

### 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 ...
0answers
26 views

### 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 ...
4answers
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 ...
1answer
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 ...
0answers
34 views

### 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 ...
0answers
9 views

### 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 ...
2answers
28 views

### 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 ...
2answers
53 views

### 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 ...
1answer
33 views

### 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: ...
1answer
41 views

### 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 ...
0answers
113 views

### 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 ...
0answers
18 views

### 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 ...
0answers
32 views

### Prove the relevance of the Pearson correlation distribution for hypothesis testing?

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 $X$ ...
0answers
57 views