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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Generalization of degrees of freedom for t distribution for coefficients after multiple imputation

Donald Rubin has shown that regression coefficient estimates have fatter tails after multiple imputation and has provided a formula for the degrees of freedom to use as a t-distribution approximation ...
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What is the original source for Jeffreys 7 degrees of freedom suggestion?

In a comment on this question Nick Cox mentioned that Jeffreys suggested that errors in data approximate t distributions with 7 degrees of freedom. Does anyone have a source/citation for this?
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Calculating the t statistic

I recently came across a blog post (https://cssanalytics.wordpress.com/2014/02/12/probabilistic-momentum-spreadsheet/) that discusses calculating a t-stat as part of their model. However, the formula ...
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How do I check if a random variable follows the T Distribution?

Let us say I have a data for a random variable X -- Is there a convenient test that can tell me how well it fits the T distribution? Code in R would be helpful.
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1answer
41 views

Help Beginner Q: Explanation on pooled variance and when it is used

If I am conducting a difference in means hypothesis test, when do we use the pooled variance and why? Lets say the population variance was unknown for two samples, the sample sizes for the two means ...
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1answer
29 views

Inference about two samples

I have two different large samples, one is normally distributed (0,1) and the other one follows a t-distribution (df = 4). Can I compare whether the means or the st. deviations of both samples are the ...
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1answer
21 views

Formula for confidence interval level doesn't give correct result

I am kind of new in calculation of confidence interval, so I hope it will not be a too basic question. I have a normal distribution (with slightly longer left tail) as you can see in the picture. The ...
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2answers
66 views

Student t-distribution parameter/s and MLE

So I always thought of the Student t-distribution as having only 1 parameter, v, the degrees of freedom (as described by wikipedia). When I searched however on how to find the MLE of v I keep coming ...
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Non-central t-distributions with different degrees of freedom

I have two non-central t distributions with CDF given by $t_{n,a}$ and $t_{m,a}$ with $n>m$ degrees of freedom respectively and the same non-centrality parameter $a$. The question is: for which ...
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quantile of standardized t distribution

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is ...
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26 views

parameter estimation of skewed student-t distribution in R

I am trying to fit skewed student-t distribution to fitted residual returns obtained from ARMA(1,1)-GARCH(1,1) The histogram of my data looks like I am using fGarch's sstdFit to estimate the ...
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22 views

How to rescale data for multivariate t-pdf?

I try to evaluate the pdf of a multivariate t-distribution in Matlab. Unfortunately the function is only defined for a correlation matrix and not for a covariance matrix. I guess i can rescale the ...
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1answer
21 views

Trying do determine how to rank water flow measurements?

I'm doing a study that involves waterflow measurements as it relates to erosion prioritizing. I'd like to rank between (0-1) a population of 160 values with a minimum of 2.1 and a maximum of 5.9 The ...
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Ranking data sets by means etc. (unequal sample size and variance)

I want to rank several lists of data by their tendency to be larger or their capacity to be larger when extrapolated. Is there a way to do this effectively?
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37 views

How do you determine when to use certain test statistics?

I've been trying to formulate this question and have struggled, so if it seems ill-worded, I apologize: A statistics book I have has a table in the back that says: For an inference test regarding a ...
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27 views

fitting t distribution with lighter tails

I am trying to fit a t distribution in R using the fitdistr function in the MASS package as follows ...
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31 views

Derivation of GARCH student t log-likelihood

I got the general PDF of the student t distribution, that is: $\frac{\Gamma[\frac{(\nu+1)}{2}]}{\Gamma(\frac{\nu}{2})}\,\frac{1}{\sqrt{\pi\,\nu}}\,\bigg[1 + \frac{x^2}{\nu}\bigg]^{-(\nu+1)/2}$ ...
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39 views

Expectation of t distribution

Find $E(T^{2r})$ of Student's t-distribution. My teacher has told me the answer.... it's something like $$ \frac{n^r \Gamma(r+(1/2)) \Gamma(n/2-r)}{\Gamma(1/2) \Gamma(n/2)} $$ From this I calculated ...
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48 views

Degrees of freedom in t dist in ML estimation using ugarchfit in R

I'm quite new to R, and is currently working on my master's thesis and have a question in that relation. I have been using the ugarchspec and ugarchit functions (rugarch package) to estimate my ARMA(...
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How to simulate from a t copula?

This is a question related to: How to simulate from a Gaussian copula? Suppose that I have two univariate marginal distributions, say $F$ and $G$, which I can simulate from. Now, construct their ...
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35 views

Maximum-likelihood estimation of student t distribution [duplicate]

I am a little confused about the Maximum Likelihood estimation of the student t distribution. Although it is said here that there is no closed form parameter estimation, I found in the MatLab ...
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13 views

Robust confidence interval for observations high/low variance

How to build consistent and robust confidence interval for lower (close to zero, =invariant) and higher variance observations? Example using R: The example includes confidence interval based on ...
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11 views

Standard deviation of sample mean differences used as the basis for calculating standard error of the two samples

I came across this problem in the OpenIntro Statatistics (it is very free) textbook: Now, in calculating the t-statistic for the difference of means, the same book shows that the Standard Error in ...
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Projection of a spherical t-distribution

I'm working with an $n$-dimensional spherical t-distribution for which the density is defined as $f(x) = \frac{\Gamma(\frac{n+\nu}{2})}{(\pi \sigma^2\nu)^{\frac n2}\Gamma(\frac \nu2)} \left( 1 + \...
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1answer
158 views

Does the standard error of the mean approach 0 as the number of samples increases?

The standard error of the mean (SEM) is often given as $s / \sqrt n$, where $s$ is the standard deviation and $n$ the number of samples. Does this mean that if we were to calculate the SEM with more ...
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1answer
63 views

Why does Naive Bayes use gaussian pf rather than Student's t?

Every source in the literature I could find about naive Bayes mentions using a gaussian's probability density function, using the mean and variance estimated from the data itself. This strikes me as ...
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143 views

Estimating DCC GARCH using a multivariate t-distribution in Matlab

What I want to do is estimate the DCC equation using a multivariate t-distribution, preferably in Matlab. There is a package/toolbox of Sheppard and Shephard. But that one only allows you to use a MVN....
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63 views

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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89 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 - \mu)^2}{\nu\...
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1answer
87 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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68 views

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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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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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
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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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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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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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139 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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153 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
43 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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187 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
50 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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519 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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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
70 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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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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26 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? I'...
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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
43 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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67 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 ...