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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7 views

How do I go about choosing the right degrees of freedom for a simulation study?

How does one go about choosing the right degrees of freedom when sampling from a T-Distribution? I understand n-1 degrees of freedom is the theoretical rule of thumb, but my sample size is quite large ...
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Why use the student's t-test rather than z-score?

Suppose we are given IID r.v's $X_1, \ldots, X_n$ that are not necessarily normally distributed. Mean $\mu$ and standard deviation $\sigma$ are unknown and we want to construct a confidence interval ...
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31 views

Verifying Sampling Distribution of a Statistic

In the given question, I can easily show that option A and C are true but i am not sure about option B. I know that |$X_2$ +$X_3$| can be written as $($$(X_2 + X_3)^2$$)^{1/2}$ and then it can be ...
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Limit of $t$-distribution as $n$ goes to infinity

I found in my intro to stats textbook that $t$-distribution approaches the standard normal as $n$ goes to infinity. The textbook gives the density for $t$-distribution as follows, $$f(t)=\frac{\Gamma\...
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Moments of truncated Student's $t$-distribution

I performed random sampling on a Student's $t$-distribution. I used SciPy to calibrate my parameters and then truncated my allowable values to the maximum and minimum observation in the data for ...
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1answer
76 views

Is the t distribution a member of the exponential family?

From what I understand, the exponential family is defined as $$f(y;\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)}+c(y,\phi)\right) $$ I've read (but not seen shown anywhere), that the ...
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Newton-Raphson method to solve for dof when performing MLE of a multivariate Student-t distribution using EM

I am reading the derivation of EM algorithm to estimate the maximum likelihood of a multivariate Student-t distribution $\mathcal{T}(\mathbf{x} \vert \pmb{\mu}, \pmb{\Sigma}, \nu)$ in Kevin Murphy's ...
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How do I read a t-table correctly?

Motivating Example: It is of interest to the medical profession to identify whether a given change in a measured quantity is real or due to noise, for purposes of a diagnosis. Precision from ISCD (...
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43 views

t-student or f-distribution?

I am reading 'Applied multivariate statistical analysis' by Richard Johnson and I do not understand that first to explain this test, talks about the t-student distribution. And then out of nowhere he ...
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41 views

What is Half-Inverse-Student-t distribution?

In the paper of Vanhatalo et al 2018, section 5.3 page 15, they use Half-Inverse-Student-t distribution as a prior: I have never heard of such distribution and I have problems finding it. What is it?
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Prior distribution of variance for normal distribution

In the wikipedia of "Student's t-distribution", "Bayesian inference" part, it is claimed that the prior distribution of the variance is taken to be $p(\sigma^{2} \mid I) \propto 1/\sigma^{2}$ but I ...
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Show that t distribution variance doesn't exist when df = 2

When DF = 1, t distribution is just Cauchy distribution whose mean does not exist. $E(T) = E(\frac{Z_1}{\sqrt{Z_2^2}}) = E(\frac{Z_1}{Z_2}) = E(Z_1)E(\frac{1}{Z_2})$ where the second expectation ...
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In-sample likelihood cannot be lower for student-t model compared to Gaussian model

I am looking at the solution of a question that asks us to briefly comment on how we can compare two linear models with different distributed $\epsilon_i$. For the first linear model $\epsilon_i$ is t-...
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Are confidence intervals only for a specific population parameter, like the mean? What about the whole population range, e.g. for outlier detection?

As someone with quite a weak understanding of statistics, I'd like to know: Is it possible to calculate confidence intervals for a population range in general from a sample? What I have learned so ...
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Comparing WAIC from normal linear model and a student T with z scored data?

I am running multiple models and looking at Widely Available Information Criteria for them. Is it unreasonable to run two linear models, the only difference between likelihood distributions (one ...
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1answer
48 views

What is the integral of the t-distribution function?

The t-distribution probability density function is given by: $f(x) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})}(1+\frac{x^2}{\nu})^{-\frac{\nu+1}{2}}$ Is there an integral ...
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150 views

Showing t-distribution from multivariate standard normals

I came across a paper that assumes the following has a t-distribution: Let $W = \frac{\mathbf{a}'\mathbf{X}}{\sqrt{\mathbf{X}'\mathbf{X}}}$ and $\mathbf{a}' \in \mathbb{R}^n$ with $\mathbf{a}'\...
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30 views

Under what conditions should I use an approximate Z-score vs a t-test? [duplicate]

I am struggling to understand the limiting assumptions of simple hypothesis testing using Z and T statistics under different scenarios. In a case where X is normally distributed, and n > 30, and $\...
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40 views

Confidence intervals when fitting non-linear curves

I have data for a device that dispenses material and I want to use an exponential decay model in python to relate the flow rate to the mass left inside the device, in particular $flow=a-b\times e^{-c\...
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165 views

Different results for simulated and closed form distribution of T-statistic under the alternative hypothesis

While running some simulations to get a better grasp of the concept of statistical power I stumbled upon an unexpected result. I was trying to simulate the sampling distribution of test statistic $T$ ...
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Confidence interval for the mean

If we want to create a confidence interval to estimate the mean of a specific population and we want the margin of error not to exceed $10$ at confidence level of $90\%$ given that some study suggests ...
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Noncentral t-distribution — relationship to shifted/scaled normal distribution

Let $x$ be 100 random samples from a $N(10,4)$ distribution. Suppose that I want to calculate the likelihood of these data, given my knowledge that $\mu=10,\sigma=4$. For the normal distribution, this ...
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Is there an algebra error in this derivation in a Coursera class of the posterior of a Normal with a Student-distributed mean?

This is a screenshot from Coursera's class "Bayesian Statistics: Techniques and Models", Week 1, "Non-conjugate models" lecture (any one can audit the class and access the materials for free): This ...
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estimate the parameters of t-distribution and fitting with MLE

Here is Fitting the t-Distribution by Maximum Likelihood t-method in book Statistics and Data Analysis for Financial Engineering with R examples page 113 and 168. ...
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how to understand Bootstrap t - method

Here is the definition of bootstrap t-method in book Statistics and Data Analysis for Financial Engineering with R examples page ...
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151 views

Fitting a multivariate T Copula to three variables in R

I have three variables with correlation coefficient of respectively 0.3; -0.2 and 0.1. I want to fit a T copula but when I use the fitCopula function it gives me a single value for rho, which leads to ...
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1answer
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GARCH (1,1) - stationarity in case of insignificant alpha?

the questions are about GARCH-t (1,1) [t-distribution]. The first question in GARCH-t (1,1) model, the alpha (ARCH) is insignificant. How to rewrite the model? The second one, in case of ...
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41 views

Normal versus student-t distribution

I estimated two GARCH models, one with the Normal distribution and one with the Student-t distribution. The conditional volatility shows less noise for the model based on the Student-t distribution. ...
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Multivariate T distribution?

Suppose $(Y_1,Y_2)^T \sim N((0,0)^T,\Sigma)$, where $$ \Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}. $$ $T_1= \frac{\bar{Y_1}}...
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Why we don’t make use of the t-distribution for constructing a confidence interval for a proportion?

To calculate the confidence-interval (CI) for mean with unknown population standard deviation (sd) we estimate the population standard deviation by employing the t-distribution. Notably, $CI=\bar{X} \...
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Tcdf and invT with incomplete beta [duplicate]

I am working in python. I have a function for incomplete beta. Iow would I calculate tCDF and invT using this? the incomplete beta function that I have is this: ...
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Comparison between Bivariate Student t with dependence matrix and without dependence matrix

Let $\boldsymbol{Y} = (Y_1, Y_2)^T $ be a $n \times 2$ random vector and $\boldsymbol{X}_1, \boldsymbol{X_2}$ are $n \times p$ regressors matrices. Finaly, $\Sigma^{1/2} = \Sigma \times \Sigma$, where ...
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184 views

Calculating confidence interval for RMSE

I'm reading a book on machine learning where the author uses the Random Forest Regression model to fit a dataset. The confidence interval for the root mean squared error is then computed using the ...
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1answer
162 views

What does it mean exactly to divide a distribution by another distribution?

In the notes I'm working through, distributions are often "divided" by other distribution, and while I sort of understand what is meant, i would rather a rigorous explanation. Let me provide an ...
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35 views

Inconsitency in comparing log-likelihoods of z- and t-values with appropriate distributions?

I am working through "Glen McPherson: Applying and interpreting statistics", where he introduces the t-distribution on p.125. As a small experiment i tried to do the following (pyhton, scipy, numpy, ...
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MQL5: Question regarding MathProbabilityDensityT()

I use the MathProbabilityDensityT() function to get the p-value from the t-statistic and the degrees of freedom such as: ...
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How does software calculate exact p-values [duplicate]

I have calculated my t-value = 2.588 and the number of degrees of freedom is df=8. Looking up the t-value in a t-distribution ...
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1answer
767 views

How to calculate probability using T distribution in R and When to use T distribution

Can someone help me to solve below question using R programming. A Government company claims that an average light bulb lasts 270 days. A researcher randomly selects 18 bulbs for testing. The ...
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2answers
35 views

Can we replace the t-Student distribution by the Normal distribution in this context?

As far as I have studied, given a normal random sample, we can build the confidence interval of the mean $\mu$ if we know the variance through the relation \begin{align*} \frac{\sqrt{n}(\overline{X}-\...
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225 views

Adjusted Pearson Goodness-of-Fit Test - Rugarch Package

I fitted a GARCH(1,1), GARCH-M and EGARCH of first order (using maximum likelihood) to my return dataset using both, Gaussian normal and Student-t distribution assumption for the error term. When ...
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1answer
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Is this a t-distribution problem?

The problem says that the probability of student failure is 0.4 A random sample of size 10 is taken from this population. I'm asked to find the probability that, at most, 30 % of students failed. I ...
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455 views

Conjugate Prior for Student T distribution

Does the Student T distribution have a conjugate prior distribution? If so, what is it and what are the parameters?
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265 views

How is Student's t-distribution related with this similarity/probability equation between data points?

In the t-SNE paper "Visualizing Data using t-SNE" and a Deep Embedded Clustering (DEC) approach "Unsupervised Deep Embedding for Clustering Analysis", they both use the Student t-distribution to ...
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How to relate beta CDF to student-t CDF? [duplicate]

We can relate the student-t and beta distributions as such: If $X$ has a Student's t-distribution with degree of freedom $\nu$ then one can obtain a Beta distribution: $$\frac{\nu}{\nu + X^2} \...
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277 views

How to approximate the student-t CDF at a point without the hypergeometric function?

Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF ...
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1answer
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What's the relationship between degrees of freedom of t distribution and tail exponent (alpha) of Pareto distribution?

I'm going to generate a set of data from a T distribution and truncate the body(so that we make it approximately Pareto distributed) of it and estimate the tail exponent(shape parameter) of the ...
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430 views

Distribution of multivariate “$Z$-score”?

Suppose $\mathbf{X}_1, \dots, \mathbf{X}_n \sim N_p(\mathbf{\mu}, \Sigma)$ where $\mu \in \mathbb{R}^p$ and $\Sigma$ is a $p \times p$ covariance matrix. Suppose $\hat{\Sigma}$ is the sample ...
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1answer
167 views

Linear regression with error dispersion dependent on the independent variable

Suppose $y=ax+z$ where $x, y, z$ are random variables with range in $\mathbf R$, $\mathbf E[x]=0$, the probability distribution $p(z|x)$ is 1) normal distribution $N(0,\sigma(x)^2)$ with mean $0$ ...
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Should I use Student's t-distribution to evaluate my measurement's confidence interval?

I'm an experimental physicist who mainly needs statistics for the calculation of uncertainties/confidence intervals. Since my results are usually normally distributed, I simply take $N$ measurements ...
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How to “honestly” calculate likelihood of 2D normal with small sample size? [closed]

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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