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Questions tagged [mathematical-statistics]

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

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How to measure the consistency of improvement on different conditions?

I want to measure whether the speed improvement of method 1 over method 2 is consistent on different conditions. Below are two examples of the speedup values of method 1 over method 2 on 5 conditions. ...
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1answer
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Intuition behind product distribution pdf

Say we have two distributions $X$ and $Y$. I know that the pdf of the distribution $Z = X + Y$ is given by: $f_Z(z) = \int_{-\infty}^{\infty}f_X(x)f_Y(z-x)dx$ The intuition is that you sum up the ...
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2answers
177 views

Finding Pareto distribution's Kurtosis

I have no clue how to solve this question: Questions: Given that each of a,b,c,d and e is a digit from {0,1,2,3,...,9} and f is an alphabet from {A,B,C,D,...,Z} X has a Pareto distribution with ...
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Bonferroni confidence region for shifted Laplace parameters

Consider the shifted Laplace distribution with the density: $$f(y)=\frac{\theta}{2}e^{-\theta|y-\mu|}\quad, \quad y\in \mathbb R$$ Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
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Finding the limiting expected length of a confidence interval

I have a query regarding the limiting length of a confidence interval, related to this question. Suppose I have a sample $X_1,X_2,\ldots,X_n$ from the distribution $$f_{\theta}(x)=\frac{1}{\...
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1answer
37 views
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How to calculate conditional hazard at time zero

I am looking for a statistical method to calculate the conditional hazard before the diagnoses or study has started; in other words, time is not a variable. For example, here is a hypothetical ...
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197 views

If $X_t^2$ is stationary, is $X_t$ necessarily stationary?

I came across a proof for one of the properties of the ARCH model which says that if $\mathbb{E}(X_t^2) < \infty$, then $\{X_t\}$ is stationary iff $\sum_{i=1}^pb_i < 1$ where the ARCH model is: ...
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Consistency of unbiased estimator of error term variance in Multiple regression

Let $Y=X\beta+\epsilon$. We know that $\frac{e'e}{n-k}$ is an unbiased estimator of $Var(\epsilon)$, where $e$ is the vector of residuals, and $\epsilon$ is multivariate normal distributed in this ...
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1answer
201 views

What's the definition of multivariate mode?

In the case of grouped data where a frequency curve have been constructed to fit the data, the mode will be the value (or values) of $x$ corresponding to the maximum point (or points) on the curve. ...
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How I can sketch the proof of consistency of only one beta in multiple regression?

Now assume you additionally obtained data on average parental incomes (PI) and the ethnic composition (EC) of the pupils in school. You regress the score on STR PI EC and a constant. State the ...
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1answer
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How to show sample correlation is sample covariance for standardized values?

Given a matrix $X$ and the resulting sample correlation matrix $R$, consider the standardized observations: $$\frac{(x_{jk} - \bar x)} {\sqrt{S_{kk}}} \quad k=1,2,...,p \quad j=1,2,...,n$$ Show that ...
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Ripley K function value in a specific radius and dataset using R's Kest function

I'm having general trouble with calculating Ripley's K function values. The following is a simple spatial point pattern, where both X and Y range from 0 to 200: Here's its corresponding Ripley K ...
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15 views

Relative and Absolute Precision in Sample Size calculation [on hold]

Can someone give me a tip on how to calculate precision? There isn't a lot of wisdom here so I apologize in advance for the wording... Its an energy savings application where based on a months data (...
0
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1answer
18 views

Generate random synthetic dataset with python [on hold]

Is there any implementation of the synthetic datasets from mlbench in python? I am looking for these implementations in particular. I've found sklearn.datasets for synthetic examples, but they do not ...
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0answers
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how to prove B1 as a consistent estimator in panel data

$Yi=a+ B_1*X_i+ B_2*Z_i+\epsilon_i$, and suppose that $Zi$ is unobservable and not correlated with $X_i$. Is the OLS estimator of $B_1$ consistent by regressing $Y_i$ on a constant a and $X_i$? I ...
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1answer
156 views

Modification of Sigmoid function

I need to model my data into a function like shown in the following picture. But how I can do this mathematically? Is there any similar function to model data that should increase on smaller values ...
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1answer
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Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$

I am unable to compute conditional probability(x|x>y) in the above question. Also, I am unable to determine the region of integration for calculation of the above expectation.
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Create an A/B Sample Size Calculator using Evan Miller's Post

To learn more about A/B Testing sample sizes selection I am attempting to use Evan Miller's popular blog-post to recreate a sample size calculator (https://www.evanmiller.org/sequential-ab-testing....
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Joint probability distribution of some Gumbel differences

I would like your help to double check my derivations below involving the joint probability distribution of some Gumbel differences. Consider $K$ i.i.d. random variables $\epsilon_1,...,\epsilon_K$, ...
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0answers
18 views

Kolmogorov Smirnov test and dependence on number of parameters

Somewhere I heard that Kolmogorov Smirnov test depends on number of parameters, is true? If yes, so I have problem because I have used ks.test(...) in R for normal distribution with 2 parameters and ...
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0answers
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What are some introductions to classical statistics that emphasize unifying principles? [duplicate]

I'd like to know an introduction to classical statistics, that: Emphasizes connections and unifying principles (I checked this question and the links posted therein, but didn't find an introduction ...
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1answer
23 views

Finding the uniformly most powerful test for hypothesis

Let $\mathbf{X}=(X_1,...,X_n)^T$ is a simple sample where $X$ belongs to exponential distribution family $\mathcal{P}=\{ f(x;\mu,\sigma \}, -\infty<\mu<\infty, 0<\sigma<\infty.$ Density is ...
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1answer
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Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2) $. My aim is to calculate the ...
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1answer
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Bootstrap for goodness of fit

Suppose I want to test whether a die is fair. I toss 60 times with outcome $X = (40,20,0,0,0,0)$ where $i$th coordinate denotes times of $i$th face occurs.Consider Hypothest test($\alpha = 0.05$): $$...
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Wassertein “least squares” and symmetries

So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$ These have symmetry (exchanging $\mu_1^j$ ...
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1answer
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Statistic to measure no.of cancelled members

These percentages indicate how many male vs female members cancelled a subscription we offer. Male Female 22% 13% Looking at the ...
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0answers
37 views

Proof of Berkson's Paradox

I'd be very thankful if someone could help me with the proof of Berkson's Paradox. I found this quite helpful thread which I understand How to prove Berkson's Fallacy?. But I'm actually trying to ...
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1answer
197 views

Topologies for which the ensemble of probability distributions is complete

I have been struggling quite a bit with reconciling my intuitive understanding of probability distributions with the weird properties that almost all topologies on probability distributions possess. ...
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Fill missing values of precipitations [on hold]

I share you my web app to fill missing values of precipitation. web app to fill missing vales
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0answers
19 views

Addition of Gaussian with Gaussian [duplicate]

Hi I have following equations, where $T_1$,$T_2$ and $T_3$ are distributed as Gaussian.I want to know their mean and Variance. finally, I want to add them. How can I do that? All variables are ...
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1answer
184 views

Testing correlation and the t-statistic used in Simple Linear Regression

Given $H_0$ : $\rho=0$ and $H_A$ : $\rho\neq0$, we use the test statistic $t_{n-2}$ , which is $\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$. I have to show that $\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$ equals $\frac{\...
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Find the mean, variance, and standard deviation for the following linear combinations of X and Y [closed]

Consider the following two independent random variables, X and Y Random variable X has a mean of 18 and a standard deviation 9. Random variable Y has a mean of 24 and a standard deviation 4.
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What is between regression and ordinal classification (or called ordinal regression)?

There are many articles explaining the difference between regression and ordinal classification, most of them mentioned that regression is for continuous response while ordinal classification is for ...
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1answer
68 views

distribution for scaled Maximum of n independent Weibulls for $n \to \infty$

Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. $F(X_1\leq x) = 1-e^{-(\lambda x)^k}$ define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$ according to ...
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1answer
192 views

Generating a positive semi-definite covariance matrix (using KL transform)

I have a set of input data X consisting of S&P 500 returns, that provides me with a covariance matrix C that is non positive semi-definite. The reason for the non-semi definite nature of the ...
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1answer
192 views

Difference between probability density functions and sampling distributions

I was wondering what is/are the fundamental difference(s) between a probability density function for a mean and sampling distribution of a mean? Can we say that ...
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2answers
324 views

Can a statistic depend on a parameter?

Can a statistic depend on a parameter? By definition, a statistic $T(\mathbf{X})$ is a function that depends on the r.v. taken from a population. In Berger's 'Statistical Inference', in the paragraph ...
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1answer
34 views

Question about statistical test for classification problem in a very unique case!

I'm working on a non-standard dataset with deep learning, and I want to prove that my method is better. We have a dataset composed from dataset A and dataset B where each one is composed of ...
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0answers
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Distribution of dot products of two random independent unit vectors in $D$ dimensions

Duplicate of the stats stack exchange question here; however, I need some help with some of the steps in the accepted answer. A uniform distribution on the unit sphere $\mathbb{S}^{D-1}$ is ...
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1answer
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What is the difference between the Binomial Distribution and the Negative Binomial distribution?

I am trying to understand the negative binomial distribution(also called gamma-Poisson distribution. What is the difference of it with the Poisson distribution anyway?), but it looks kind of similar ...
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2answers
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Intuitive interpretation on an expectation equation

It's provably correct that for a nonnegative random variable denoted as $Z$. The expectation of $Z$ can be written as follows: $$\mathbb{E}[Z] = \int_{x=0}^{\infty}\Pr[Z\geq x]dx.$$ Well, it can be ...
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1answer
218 views

Expected squared distance from origin of training points vs. test points

This is from Exercise 2.4 (Page 39) of Elements of Statistical Learning: The edge effect problem discussed on page 23 is not peculiar to uniform sampling from bounded domains. Consider inputs drawn ...
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Did Deborah Mayo refute Birnbaum's proof of the likelihood principle?

This is somewhat related to my previous question here: An example where the likelihood principle *really* matters? Apparently, Deborah Mayo published a paper in Statistical Science refuting Birnbaum'...
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Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3) [duplicate]

Let us consider the random vector $X=[X_1,X_2,X_3]$, which follows a multivariate normal distribution. That is, for each entry $X_i$: $X_i \sim N(\mu_i, \Sigma_i)$. What I am trying to compute is the ...
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Help Identifying the Name of a Theorem

Recall the following theorem: Let $X$ be a random variable with probability density function $f_x(x)$ and let $g$ be monotone over the support of $f_x$. The random variable $Y=g(X)$ has density $$...
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How to do random simulations on R [closed]

Draw 1000 Uniform random variable. You know the true mean. You can use the standard deviation of your data or you can calculate the true standard deviation and use that. How often does your data fall ...
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which statistical test to use (5 binary IVs and 1 ratio DV)

I'm conducting a study where I have data that include 5 Binary independent variables (0 or 1) and a ratio dependent variable (performance 0.0 to 1.0). I want to check the effect of the IVs on the DV ...
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1answer
713 views

sample autocovariance function vs autocovariance function

What is the difference between both functions? I have been reading, but I cannot see the difference. I suppose that the sample autocovariance function is the estimation of the autocovariance function. ...
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0answers
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Updating the variance of a Normal Distribution Using Bayes [closed]

I have a prior belief for the mean $\mu_p$ and std $\sigma_p$ of a normally distributed variable $X$. (no dataset). So I have the mean of these parameters, but NOT their variance (I'll just presume ...
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Does a “Normal Distribution” need to have mean=median=mode?

I've been in a debate with my graduate-level statistics professor about "normal distributions". I contend that to truly get a normal distribution one must have mean=median=mode, all the data must be ...