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Questions tagged [t-distribution]

t is the distribution of the t-statistic that results from a t-test. Use this tag only for questions about the distribution; use [t-test] for questions about the test.

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Count probability of average value of random variable with normal distribution without variance value known [on hold]

Please verify my calculation for example placed below. Random variable $X$ has distribution $N(-10; \sigma)$. $S^{2}_{10} = 25$. Count $P(\overline{X}_{10} < 85)$. To resolve the example I am ...
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How to “honestly” calculate likelihood of 2D normal with small sample size? [on hold]

I estimate covariances from data and want to calculate likelihood. For 1D case I know - if the sample size is $<40$, I use Student's t-distribution to calculate likelihood of the data since my ...
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22 views

How should I calculate mean CI - from raw data or mean values?

I have experimenatal data from medical industry. In the experiment 4 observations were weighted, each one 2 times for more precise measurement. As a final weight, a mean from each pair is reported. ...
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10 views

Gradient signifiance test with tolerance and measurement uncertainties

I'm trying to test whether a likelihood has converged to walk around a single number or if it is increasing. In my (EM) algorithm, the noiseless likelihood cannot decrease, only ever increase. In my ...
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1answer
79 views

One sided test $H_0:\mu=0$ or $H_0:\mu\leq 0$? [duplicate]

I want to test $$H_0: \mu \leq 0 \,\,\,\,\,\, vs \,\,\,\,\,\, H_1: \mu > 0.$$ I am using a t test, so the statistic $T$ has $\nu$ degrees of freedom which depend on the sample size. What is ...
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1answer
23 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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20 views

Distribution of coefficient on the error correction term in ECM and VECM

According to statistic academic literature, the cointegration test on coefficient $\alpha$ of the error term included in ECM or VECM does not follow a standard distribution. My question is: If so, ...
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19 views

Degrees of freedom when manually doing (Welch) T-test?

I have two samples A and B and want to test if the (Pearson) auto-correlation of A is greater than that of B. So far I've computed the two autocorrelations, $r_a$ and $r_b$ and found their standard ...
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1answer
55 views

Confidence intervals over mean difference with unknown but equal variance

The problem is the following: I have 2 Stores, called Store1 and Store2 I take a sample of the number of item sold in each store over a certain period of time From the first store the sample has ...
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1answer
45 views

Do we assume a t distribution for the estimate of the difference of normal distributions?

I am studying hypothesis testing. When performing a one sample t test, we assume a t distribution for the sample mean estimates of the true mean. When conducting a two independent sample test, it ...
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1answer
28 views

95% limits of a normally distributed parameter

How do I find the 95% limits of the population distribution of a normally distributed parameter? I've taken the mean and SD from 10 different readings of the parameter. Will the 95% limits be mean +/-...
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1answer
106 views

Find the pdf of $\frac{Z_1}{\sqrt{(Z_1^2+Z_2^2)/2}}$ where $Z_1,Z_2$ are i.i.d standard normal

Given $Z_1, Z_2$ are i.i.d standard normal random variables. Let $$V:=\frac{Z_1}{\sqrt{(Z_1^2+Z_2^2)/2}}$$ Derive the pdf of $V$. The numerator and denominator of $V$ are dependent, so the square ...
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What is the relationship of student t-distribution and gaussian function [duplicate]

The shape of student t-distribution looks pretty much like a Gaussian function, what is their relationship? Can t-distribution be represented by a closed form formula?
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1answer
67 views

multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
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1answer
68 views

What kind of distribution is this?

I'm sorry is this is too obvious, but I'm having a hard time trying to find a distribution for my data. It is clearly not a normal distribution. It does not seem to be skewed, but seems to have fat ...
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1answer
31 views

Converting a prior probability of the null into t-test parameters

Suppose I am in the rare situation where I know for certain the prior probability of the null hypothesis. And yes, I mean the marginal probability of the null (not the probability of the null, ...
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Value at Risk of portfolio with student t distributed assets

I am trying to compute the VaR of a portfolio of cryptos, and trying to do it by 2 methods (VaR is defined as the minimum potential loss that a portofolio may suffer in the x% worst cases over a given ...
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2answers
189 views

How do the t-distribution and standard normal distribution differ, and why is t-distribution used more?

For statistical inference (e.g., hypothesis testing or computing confidence intervals), why do we use the t-distribution instead of the standard normal distribution? My class started with the standard ...
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Derive p value from t distribution [duplicate]

I have difficulty of understanding p-value from An Introduction to Statistical Learning by Gareth James • Daniela Witten • Trevor Hastie Robert Tibshirani (2015) : 67 Consider simple linear ...
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60 views

Intuitive explanation for t vs z in confidence intervals

Skip to the conclusion if TL;DR What is an intuitive explanation for what a t-score gives you when computing confidence intervals? Background of my understanding of confidence intervals I just ...
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1answer
323 views

Simulation of t copula in Python

I am trying to simulate a t-copula using Python, but my code yields strange results (is not well-behaving): I followed the approach suggested by Demarta & McNeil (2004) in "The t Copula and ...
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1answer
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Find $\lim_{n \downarrow 1} t_{n-1, \alpha/2} / \sqrt{n}$ and prove the limit

This question is related to another recent question on CV.SE relating to the weighting function that commonly appears in confidence intervals for a population mean:. $$w(\alpha, n) \equiv \frac{t_{n-...
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1answer
156 views

How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?

Consider a family of distributions with PDF (up to a proportionality constant) given by $$p(x)\sim \frac{1}{(1+\alpha x^2)^{1/\alpha}}.$$ How is it called? If it does not have a name, how would you ...
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298 views

Prove that $t_{n-1, \alpha/2} / \sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$

Statistical problems involving confidence intervals for a population mean can be framed in terms of the following weighting function: $$w(\alpha, n) \equiv \frac{t_{n-1,\alpha/2}}{\sqrt{n}} \quad \...
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2answers
162 views

Confidence interval for the mean - Normal distribution or Student's t-distribution?

I have taken $N$ measurements $x_i$ of the time taken by a computer program to run a particular calculation, and would like to calculate a 95% confidence interval for the mean runtime. We can ...
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Studentization ad pivotal quantity for the slope in simple linear regression

Take the usual linear model: $$ Y=\alpha+\beta X + e$$ In good hypothesis (IID sample, normal error assumption $e \sim N(0,\sigma_e^2) $ and homoschedasticity) the distribution of the sample slope $\...
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Can BIC prescribe a different model specification after changing unconditional distribution assumption?

I modelled an ARMA GARCH via minimization of BIC. The result was in accordance with the simple ARMA model i.e. both prescribed AR(1) as the optimal (in BIC terms) conditional mean model. By looking at ...
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1answer
250 views

MLE for Linear Regression, student-t distributed error

Say I have a $n$ pairs $(X_i, Y_i)$ and I have the relationship $$Y_i = \beta_0 + \beta_1X_i +\epsilon_i$$ where the error terms $\epsilon_i$ are iid from a student t-distribution with constant (known)...
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1answer
224 views

meta-analysis of linear regression coefficients

I have the summary results of a linear regression model with 26 covariates which has been fitted to two independent samples (with different sizes). Now I am planning to use the linear regression beta ...
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21 views

Student t - critical value [duplicate]

i need critical value for $ t_{249}(0,975) $. I can generate in R in easy way, but in school we will have just tables and in tables are only critical values for degrees of freedom = 100,200,500. How ...
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1answer
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The necessity of t-distributions in computing confidence intervals

If the width of a 95% CI can be computed as 1.96 * SEM, where SEM is just SD/sqrt(N), then why is the t-distribution involved in ...
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62 views

Joint distribution of sample means from multivariate normal with unknown covariance matrix

The distribution of a (standardised) sample mean from a univariate normal distribution with unknown variance is given by the Student's t-distribution (Student's t-distribution). What about a joint ...
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84 views

Interpreting the best-fit student t distribution (based on maximum likelihood estimation) for my datasets

I have fitted (through maximum likelihood estimation of parameters) a number of leptokurtic probability density functions (i.e. logistic hyperbolic-secant Laplace log-logistic student's t (with ...
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2answers
59 views

one tail hypothesis testing, Gaussian versus t-distribution

Since the t-distribution represents more uncertainty than the Gaussian distribution I am accustomed to expecting a greater p-...
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2answers
96 views

Fourth moment of unit-variance t-distribution

I am considering the unit-variance t-distribution. I have read that the fourth moment in such case is given by 3(v-2)/(v-4) where v is the degrees of freedom. Can someone explain how does this follow?...
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1answer
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Why doesn't the TOST equivalence testing procedure use non-central $t$ distribution to determine the $p$ value?

I'm looking at Lakens' (2017) primer and he tests the hypothesis that there is a difference between two groups $X_1$and $X_2$of magnitude $\Delta=E[X_1]-E[X_2] \neq0$ by subtracting $\Delta$ from the ...
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104 views

How to bootstrap Smirnov-Komogorov test to check if the data follows t distribution

I am very new to statistics, so I would greatly appreciate you help. I need to test if my data follows t distribution. I am estimating the location, scale and df parameters from the data, so I can't ...
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Mutual information for independent subsystems with non-Gaussian distributions

I seem to have found contradiction when computing mutual information for a multivariate Cauchy distribution. Below, I list a few things that I think are true. But I expect that at least one of them ...
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1answer
477 views

Generating t-distributed random numbers: lesser numbers near zero.

According to a prescription given on Wikipedia, I tried generating Student's t-distributed random numbers with three degrees of freedom. I also generated these numbers using numpy's in-built random ...
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250 views

Is T distribution symmetric ?

Is student t distribution (Multivariate) symmetric ? On Wikipedia it also says its symmetric https://en.wikipedia.org/wiki/Student%27s_t-distribution. But my adviser is saying its not symmetric all ...
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(G)LME on EEG data: distribution not fitting, assumptions of normally distributed, i.i.d. residuals not met, what to do?

this is my first post on this forum, so please let me know if you need more details. Would be really glad for some advice! I have data that I want to model using LME. I tried to find a good fit for ...
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1answer
185 views

Joint distribution of independent t-distributed random variables

The multivariate t distribution seems to be defined as a "ratio" of a vector of normal random variables and a single gamma (or chi-squared) random variable (independent from the vector of normal). ...
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1answer
112 views

Closed form for the solution of the likelihood equation: Student distribution

Let $Y$ be a random vector of $\mathbf{R}^p.$ Assume that $Y$ is $t_p(\mu,\Sigma,\nu)$ distributed. That is the density is given by $$f_p(y,\mu,\Sigma,\nu)={\displaystyle {\frac {\Gamma \left[(\nu +...
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Which statistical test to use to identify significant trends in employment and hiring data?

My company has asked me to identify statistically significant trends in male vs. hiring within each of its departments over the last 20 years. I'm unsure how exactly to test for statistical ...
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1answer
789 views

Difference between a Student-T vs Cauchy distribution

What are the real and practical differences between the student t distribution and the Cauchy distribution? The results I get when I use them as priors in Bayesian linear models appear to be ...
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1answer
105 views

Confidence intervals for expected value of almost independent variables

How may I calculate (or approximate with reasonable error) confidence intervals of expected value of non-normally distributed random variable $Y$ given its almost independent samples? I have $n = 100$...
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0answers
593 views

Log Student t-distribution calculated Mean

I'm a software developer and I have to implement several distributions in my code. I have successfully found information and implemented Student $t$-distribution, and the next one is Log $t$-...
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44 views

Covariance in bivariate, noncentral, scaled t-distribution with two different degrees of freedom

Consider a bivariate, noncentral, scaled t-distribution with two different degrees of freedom which I have created with the standard copula approach by 1) creating a zero-centered, unit-scaled ...
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In the hypothesis testing of regression slope, t-distribution is used. How can we show that beta-hat is independent of the estimated sample s.d.?

Referencing to Why is a T distribution used for hypothesis testing a linear regression coefficient?! A snapshot to the link, how can we see that $\hat{\beta}$ (depending on Y and therefore the noise) ...
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1answer
89 views

Getting logistic distribution from student's t-distribution

How can I get the probability density function (PDF) of standard logistic distribution from the PDF of symmetric student's t-distribution with location and scale parameters. In the $(\beta_1, \beta_2)$...