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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Multivariate t-distribution with different degrees of freedom per dimension

One way to define the multivariate t-distribution is: $ p(x | \mu, \Sigma, n) = c \left(1 + \frac{1}{n} \left(x -\mu\right)^T\Sigma^{-1}\left(x-\mu \right)\right)^\frac{-(n+K)}{2}$ with $K$ the number ...
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44 views

Fit generalised extreme event distribution in MATLAB/R

I have a location-scale t-student distribution $X$ $$ f(x,\mu,\sigma,\nu) = \frac{1}{\sigma} \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu} \Gamma (\frac{\nu}{2})} \left( 1 + \frac{(x - ...
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45 views

Is the Student-t distribution a Lévy stable distribution?

Let $X$ have a Student-t distribution, so that \begin{align*} f_X(x|\nu ,\mu ,\beta) = \frac{\Gamma (\frac{\nu+1}{2})}{\Gamma (\frac{\nu}{2}) \sqrt{\pi \nu} \beta} \left(1+\frac{1}{\nu}\left(\frac{x - ...
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Equality vs. Equality in Distribution ($t$-distribution for example)

A technical question that came up to mind as I was reading up on linear models today. Consider the $t$-distribution with $\nu$ degrees of freedom ($t_\nu$) for example. Let's say $T \sim t_{\nu}$; ...
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20 views

Informativeness of t-score?

My organization recently did a morale survey. Results were given as t-scores. I had never seen that before, so I embarked on a web journey. I found a very similar survey. My question isn't about the ...
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2answers
48 views

Uses and estimation of Student-t distribution

My question relates to a confusion between how the Student-t distribution is often documented versus how it is used. In the documentation the Student-t is used (from Wikipedia): when estimating ...
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1answer
39 views

What would be the method of moments (MOM) estimator of $\nu$ for t-distribution?

I recognize that method of moments is not the best way to estimate $\nu$ for the t-distribution, but I am just wondering how this would be calculated since $E[X^n] = 0$ if $n$ is odd. Here we're ...
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Importance Sampling of a Student-t with mean 0, $\sigma^2$ and degrees of freedom unknown

I want to calculate the posterior parameters of a Student-t by importance sampling. In order to tune the importance density I sample from the prior first. My data is demeaned, the degrees of freedom ...
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22 views

Sample from sum of squared independent $t_\nu$ distributed variables

I would like to specify radius r in a case where $P(Z_1^2 + Z_2^2>r)=a$, $a$ is known, where $Z_1,Z_2$ are i.i.d $t_\nu$-distributed. Since, this can be really challenging to do analytically, I ...
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15 views

Product of two multivariate t densities

Does the product of two multivariate t densities result in the density of a known distribution, that is easy to simulate from?
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125 views

Visualization of normality condition for t-test in R

I am working through the concept of the need for normality in the underlying population when performing a t-test. This is nicely expounded by @Glen_b here. The gist of the explanation, I think, is ...
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1answer
128 views

Issues with Importance sampling for flat prior

I am trying to draw Bayesian inference via importance sampling for a parameter $\xi$ attached with an (unbounded) flat prior. This seems problematic as this is clearly not a probability measure but ...
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1answer
42 views

Student's distribution tails

I do not understand why in some figures they put $t_{1-\alpha/2}$. Why is the subscript $1-\alpha/2$ used? I know that $t_{\alpha/2}$ and $-t_{\alpha/2}$ are used to construct confidence intervals. ...
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1answer
181 views

How to show mean and standard deviation of Normal distribution?

If a sample is normal with observations independent and identically distributed: $\mu|\sigma^2 \propto N(\beta \,,\,\sigma^2/\, n_0)$ How can I show that $\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim ...
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1answer
47 views

Students t-test

In comparing B6 levels in serum vs plasma, 20 volunteers were chosen. The objective is to determine if B6 concentrations differ when taking from serum or plasma. Each volunteer gave two samples, one ...
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2answers
283 views

t-distribution having heavier tail than normal distribution

In my lecture note, it says, "t-distribution looks like normal, though with slightly heavier tails." I understand why it would look normal (because of Central Limit Theorem). But I am having hard ...
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42 views

Sampling distribution question

I've been struggling with this question for 3 days and I cannot find the way to do this. I know the answer, but I can't solve it clearly. When I try to solve it, I get stuck here: How can I ...
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1answer
34 views

Simple Student Copula simulation

I want to simulate a t copula with given correlation parameter $\Sigma$ and $k$ degrees of freedom. I can't find any literature about practical simulation, so I am trying new approaches. (I also have ...
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18 views

Building confidence interval with t-distribution

I am wondering wether i have to use normal or t distribution in the following situation I have a population of 1000 individuals. I'm interested in $P$ the sum of all $x_i$ where $x_i$ is the value of ...
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1answer
25 views

Comparing a distribution law with its density function in R [closed]

I want to generate a sample of a student t distribution with a certain length and a certain degree of freedom and compare it in the same graphic with its density funcion. How could i aproach that? ...
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53 views

Student's t-distribution value calculation

I need to calculate the t-distribution values. While referring to literature I found that I need degrees of freedom for this. But while referring to my reference paper I find that author has used some ...
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1answer
41 views

Fitting upper and lower student-t separately

I'm doing some research on fitting marginal distributions to data, in particular distributions with a focus on heavy tails. In one report, albeit several years old, the authors suggested fitting a ...
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46 views

improving accuracy for sample skewness and kurtosis under Student's t distribution with low degrees of freedom

There is no skewness in Student's t distribution. Given the degrees of freedom, the kurtosis estimator is given by $\frac{6}{\nu - 4}$. However, when sampling form a t-distribution with low degrees of ...
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2answers
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What is the equivalent of Student's t-distribution on an interval?

I have been using Student's t-distribution as a proxy for the Gaussian distribution estimated from a limited number of points in order to calculate alpha-intervals (for instance 95% confidence ...
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1answer
65 views

Is the $t$ test a special case of the generalized linear model?

The GLM consists of methods which model responses following a distribution from the exponential family. These distributions have a probability function which can be expressed as: $$ f(x; \theta) = ...
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1answer
32 views

$t$-distribution and finding probability of sample mean

There is a question from a statistics book: If four observations are taken from a normal distribution what is the probability that the difference between the observed mean $\bar x$ and the true mean ...
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38 views

Non-standardized Student's t distribution

If I compute the quantity $$ t = \frac{\mu}{\sigma}, $$ for a mean value $\mu$ and standard deviation $\sigma$ computed from some $n$-vector, $v$, is it then correct to state that $t$ follows a ...
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23 views

Externally studentized residuals

Assume the our model is simply $$Y = X\beta + \epsilon$$ with $\epsilon$~ $N_n(0,{\sigma}^2I)$. If we want to identify the existing outlier in our dataset, we use the externally studentized ...
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1answer
67 views

Sum of two elliptical distributions will it be again elliptical distribution?

What will be the sum of two elliptical distributions will it also be elliptically distributed? from this What is the distribution of the difference of two-t-distributions It says that sum of t ...
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2answers
88 views

Correct adjective for an RV with a Student's $t$-distribution

If a variable has a normal distribution, we say it is normally distributed. If it has an exponential distribution, we say it is exponentially distributed. What is the proper way to reference a ...
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1answer
49 views

Is my work correct (easy problem, confidence intervals)

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...
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3answers
131 views

Need a seminal reference

Does anyone know of the seminal reference for the multivariate $\mathcal{T}$ process? The oldest reference to it I can find is Gamerman 1997: Markov Chain Monte Carlo: Stochastic Simulation for ...
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1answer
75 views

Deegrees of freedom of Student's distribution

I'm trying to figure out the distribution of this statistic: $$S=\frac{\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}}{\sqrt{\hat{\sigma}^2/\sigma^2}}$$ Where: $\overline{X}=\frac{1}{n} \sum_{i=1}^n ...
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1answer
220 views

Generate t distributed random variable in matlab

How can we generate t distributed random variables with given mean and covariance in matlab. trnd is a function in matlab but we can get random variable with different degrees of freedom but how to ...
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2answers
95 views

Anderson Darling Test on student-T distribution

I would like to do an Anderson Darling test in R using ad.test() from the package “goftest”. My null hypothesis is H0: Data follows a student-t distribution with 2 degrees of freedom. I receive the ...
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1answer
328 views

Equivalence of the dt and pt function in R

in my own curiosity I decided to question the equivalence of dt and pt in R. pt denotes the cumulative density function of the t-distribution. So this gives us $P(X \le x)$. So if I want to find the ...
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How to calculate t-score and critical domain of t-statistics test

A factory specializes in the production of metal rods. According to the standards, the expected length of the rods should be 80 cm. If we investigate a population of 10 different rod lengths, we ...
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1answer
85 views

Proof: $T \sim \chi^2(N-1)$

I'm a maths student, this is my first statics class and I'm studying Confidence Intervals. When it's time to estimate the CI for the mean of $\{X_i\}_{i=1..N}$ - gaussian random variables i.i.d. with ...
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can one approximate median of non central t- location scale distribution

Generally one can approximate the median to be the non central parameter. I want full proof of the same. can you please give me any journal paper link, in which this has been proved.
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301 views

Relationship between F and Student's t distributions

According to Wikipedia, $Y\sim F_{1,v}$ has an F-distribution if $Y=X^2$ and $X\sim t(v)$ has a Student's t distribution. The proof of this makes sense to me. But I don't understand the relationship ...
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328 views

What is exactly distributed according to t-distribution?

I try to understand the idea behind the t-distribution. Here are the steps that I have understood so far: We use a sample of N elements to estimate the population mean. In more details, we use the ...
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99 views

Generating t statistics from p values - binomial test multiple comparisons

I am using the binomial test to generate a series of p values. However, my method of multiple comparisons correction that I wish to apply to this series requires that I also have the corresponding t ...
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1answer
69 views

t-statistic before-after

I’m doing an exercise on the Udacity's Intro to Inferential Statistics course (Problem set 10b) where I need to calculate the t-statistic on a before\after treatment situation. My null hypothesis ...
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1answer
390 views

Should the standard deviation be corrected in a Student's T test?

Using the Student's T test, T-Critical is calculated via: $t = \frac{\bar{X} - \mu_{0}}{s / \sqrt{n}}$ Looking at Wikipedia article on the unbiased Estimation of the standard deviation, there ...
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88 views

Deriving a probability using a more fat tailed T distribution rather than the normal distribution in Excel

I have an estimate of the central limit and standard deviation (SD) of a physical phenomenon. The SD is specifically derived from a limited sample of 12 observations. Using the 'NORMDIST' function ...
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1answer
47 views

Transformation of any normal distribution into a standardized t-distribution

What will be the transformed Mean and transformed standard deviation if any normal distribution is transformed into a standardized t-distribution? Does t force ...
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108 views

What will be the t-value if my sample size increases to an infinity compare to z value?

I do not know where to check this. Any reference/help is much appreciated. If the sample size, n increases to an infinity then will the t-value be larger/smaller ...
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Finding z-scores from z table relating to confidence intervals

I'm having trouble finding the proper $z$ score so that I can find the $99\%$ confidence interval. $\bar{x} = 6.01231$. with an $s$ of $1.96833$ and $n$ of $26$, and I got $2.575$ for ...
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53 views

How to test a hypothesis about the mean based on an assumed normal distribution?

The entrance onto a major bridge in New York City was engineered to accommodate an average of $3800$ vehicles per hour. However, a random sample of nine observations gives an average of ...
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1answer
106 views

t Student test for two samples (and paired) in R. How to Apply for my specific data table?

I'm having a trouble trying to apply the t Student test for two samples in the data of the in the image below : The diff column is the difference of the observations. The experiment is: "A field ...