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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When does the sum of two $t$-distributed random variables follow a $t$ distribution?

In the scope of a project, I need to find the sum of two independent $t$-distributions. I know that in the general case, the sum of two $t$-distributed random variables is not $t$-distributed. However,...
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Deriving the CDF of student $t$-distribution

I am trying to derive the cumulative density function(cdf) of $t$-distribution, define as in https://en.wikipedia.org/wiki/Student%27s_t-distribution#Cumulative_distribution_function I derive it by ...
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Understanding the relationship between a 'sampling distribution of a statistic' and a 'population distribution' in the context of a 'hypothesis test'

I would like to confirm that I am understanding the relationship between a sampling distribution of a statistic (an example of a 'statistic' would be a sample mean $\bar{x}$) and a population ...
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2 votes
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T-Statistics vs. Z-Statistics [duplicate]

If Z-statistics, when population variance is unknown, is a close approssimation of the T-statistics, which is better suited for the case, why do we overcomplicate things by using Z if sample size is ...
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7 votes
3 answers
457 views

Variance estimate for Student's t-distribution with heavy tails

The variance of a Student's t random variable when the degrees of freedom $\nu$ is greater than $2$ is $\nu/(\nu-2)$. In R, when I try to estimate the variance using the usual estimator $$1/(n-1)\sum_{...
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Calculating the variance for a complex transformed non central t-distribution

I am currently stumped and wanted to know if somebody could advise me on how to calculate the variance if, assuming $T_{\delta,\nu}$ is a noncentral t-distribution with degrees of freedom $\nu$ and ...
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How do I make different distributions objectively comparable?

I have some model, $f(x)$, that takes in some input and produces some output. Now my input are random variables distributed according to some distribution. The output are also random variables with ...
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Degrees of Freedom for Linear Combination of Variables

I have three mean estimates, all from variables assumed to follow a Normal distribution. The following is a mock example. x1 is estimated at 4, with a standard error of 5 and 6 degrees of freedom x2 ...
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2 votes
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Calculating confidence interval for binomial distribution [duplicate]

Suppose we have a sample $X_1, X_2, \ldots, X_n \stackrel{\text{iid}}{\sim} Binomial(\theta)$, where $n$ is known to be large. I would like to calculate the 95% confidence interval for $\theta$, and I ...
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Hypothesis Testing of t-distribution

An auto manufacturing company wants to estimate the variance of kilometers per litre for its one of the auto model. A random sample of 25 cars of this model showed that the variance of kilometres per ...
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Independence of Random Variables - from t distribution and its definition [duplicate]

I have a somewhat strange question about the independence of random variables. It comes from the definition of t-distribution. In this definition, we need two independent random variables and we can ...
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2 votes
1 answer
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Student's t-distribution and ML in R [duplicate]

I am a newbie to R and am trying to learn it. I have some financial data which is normal distributed. Now, the Student's t-distribution is given by $$g_v(z):= \...
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3 votes
1 answer
184 views

How can I sample from a shifted and scaled Student-t distribution with a specified mean and sd in R?

I'm currently building some Bayesian models with the brms package and the default intercept prior is student_t(3, 0, 6.3) and so ...
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Which is the noncentral multivariate t distribuiton

In various documentations, I found out two definitions of the multivariate noncentral student $t$ distribution. The most commonly density (PDF) found is (e.g., wikipedia): $$\mathcal{T}(x;\mu, \sigma^...
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1 vote
1 answer
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The distribution of the difference between two correlated non-central t distribution

Suppose a binormal population $\{X, Y\}$ with means $\mathbf{\mu} = \{\mu_1,\mu_2\} \ne \{0,0\}$ and covariance $\Sigma= \sigma^2\begin{bmatrix}1 & \rho\\ \rho &1 \end{bmatrix}$. Let $S^2$ be ...
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How to get rid of a nuisance parameter when the two parameters are multiplied

I have the distribution of a sample statistics which is given by a noncentral t distribution. This distribution is dependent on the population correlation ($\rho$) which is unknown. However I have the ...
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1 vote
1 answer
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How to prove equality of standard errors for two-sample t-test and linear regression?

Using classical linear model assumptions, we know that $$\frac{\hat \beta_j - \beta_j}{se(\hat \beta_j)} \sim t_{n-k-1}$$ meaning that the ratio of regression coefficients to their standard error ...
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1 answer
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Average of t-distributed random variables

I have 10 t-distributed random variables that I'm averaging over. They are unlikely to be independent but for simplicity let's just assume that they are. Each random variable is parameterised by mean $...
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Why doesn't the t distribution depend on variance of the original normal distribution?

I just find it counter intuitive that, for an arbitrary normal distribution with 0 mean and unknown variance, the t statistic of $\frac{\bar{X}}{\hat{se}} $ is completely independent of the variance ...
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4 answers
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What is the benefit of regression with student-t residuals over OLS regression?

Sometimes I see advice to fit regressions with student-t residuals rather than using OLS (which is equivalent to assuming normally distributed residuals) if the distribution of the residuals is heavy-...
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integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$ Here first part is ...
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2 votes
1 answer
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why do some use ν=n instead of ν=n-1 in the t distribution

I have always accepted that ν=n-1, but then I come across multiple authors showing a derivation of the t distribution by marginalizing out the variance from the normal-inverse-gamma distribution, or ...
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QQ-Plot for data on skewed t distribution in R

I have been trying to plot my financial data against a skewed t distribution in a QQ plot in R for some time now, but I have only managed to do it against a symmetric t distribution so far. I use the ...
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3 votes
1 answer
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Distribution of $\frac{Z_1^2 + Z_2^2}{Z_1+Z_2}$ where $Z_1, Z_2$ are standard normals

Let $Z_1, Z_2 \sim \mathcal{N}(0,1)$ be i.i.d random variables. I wish to find the distribution of \begin{align} \frac{Z_1^2 + Z_2^2}{Z_1+Z_2} \,. \end{align} It is well known that $W = Z_1^2 + Z_2^2 ...
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What is a test statistic? [duplicate]

Quick question: Are test statistics like the "F" and "t-statistic" the number of standard deviations on an F and t-distribution, respectively? For example, if I got a "t" ...
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Can we do a t-test to a one-sample problem when variance is known?

The reason why we need to use t-test when $\sigma$ is unknown is that t-statistic $\dfrac{\overline{X}-\mu_0}{S_X/\sqrt{n}}$ itself does not have a $\sigma$ in it and it follows a $t_{n-1}$ ...
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2 votes
1 answer
112 views

Why use a t-distribution instead of a normal distribution for small sample tests? [duplicate]

It is commonly said that sample sizes <~30, you are unable to use a z-test for a difference of means due to the CLT. Instead it is said that your sample follows a t-distribution, and that you ...
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0 votes
0 answers
32 views

Computing Expected Values for Skewed generalized $t$ distribution

If $X$ has a Skewed generalized t distribution , whose PDF is given by $$f\left(x;\mu,\sigma,\lambda,p,q\right)=\frac{p}{2\nu\sigma q^{1/p}B\left(1/p,q\right)\left(\frac{\vert x-\mu+m\vert}{q\left(\...
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3 votes
3 answers
90 views

How do I determine the distribution of an additional sample $x_{n+1}$ given sample mean, variance $\bar x, s^2$ of a normally distributed $x$?

Suppose I draw $n$ samples $x_1...x_n$ of a random variable $x$, which is normally distributed with an unknown mean $\mu$ and variance $\sigma^2$. From those samples, I compute a sample mean $\bar x$ ...
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1 answer
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Frequentist Justification for using Student's t-Distribution to Construct Confidence Intervals

In my most recent statistics class, the general1 frequentist approach to constructing confidence intervals2 for an estimator (of any kind) was described as follows: Specify the estimator and the data ...
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Why is in order to normalise t-distribution the following function is used $t = z*s$, shouldn't it be $t=z*\frac{\sigma}{s}$?

I am trying to answer the following question: Suppose there is a new genetically modified species of pine, with an expected fully grown height of 14 metres and a standard deviation of one metre. The ...
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2 votes
2 answers
71 views

Why I get high observation when I generate data from t-distribution in R

I want to generate 200 samples from t-distribution with the degree of freedom=1 and sample size is 10 and in R I use this code ...
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1 vote
0 answers
35 views

How many degrees of freedom do the estimators of a panel regression have?

I have multiple observations for multiple people stored in a panel. Here's a sample of my data (where id is a unique person identifier and ...
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4 votes
1 answer
136 views

Finding mean and variance of t-distribution to solve for constant c

Suppose that 5 Random Variable X1, X2,... X5 are independent and each has standard normal distribution. A constant C such that the random variable $$C\frac{X_{1} + X_{2}}{(X_{3}^2 + X_{4}^2 + X_{5}^2)...
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1 answer
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What statistical parameters change when I exchange the null and alternate hypothesis?

I have the following data from a weight loss survey from a diet program. I calculated the following things from the data and formulated my null hypothesis and alternate hypothesis as follows. My ...
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0 votes
1 answer
37 views

Sum of two different freedom degree multivariate t dsitributions

there are two multivariate t distributions whose freedom degree $\nu$ are different to each other. $$ \mathbf{x} \sim \mathcal{T}(\nu_x, \mathbf{0}, \Sigma_x) $$ $$ \mathbf{y} \sim \mathcal{T}(\nu_y, \...
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5 votes
1 answer
174 views

Independent copula vs Student-$t$ copula with zero correlation matrix?

Suppose I have the random variables $X_1, \dots, X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution). Will there be any difference between the assumption $X_1, ...
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13 votes
2 answers
990 views

Not getting 95% coverage for 95% t-distribution CI

I'm simulating a bunch of 95% confidence intervals on samples taken from a normal distribution. Since the data is normal, then, I think, my 95% confidence should translate into a 95% coverage ...
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5 votes
1 answer
112 views

Does Student's T require normally-distributed data?

Student's T distribution can be derived from the fact that: $$ \frac{\frac{\bar X - \mu}{\sigma/\sqrt{n}}}{\sqrt{S^2/\sigma^2}} $$ Is an approximately normally distributed variable in the numerator (...
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1 vote
0 answers
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How to get the expected value and variance of $\log{||x||^2}$ where x $\sim$ Student t-distribution

In the following paper A Novel Parameter Estimation Algorithm for the Multivariate t-Distribution and Its Application to Computer Vision, under section 3.2, the authors state some results for the ...
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2 votes
0 answers
207 views

What are the sampling distributions underlying a product of gaussian variables (moderated effect)?

I'm trying for a while now to understand the differential power of main effects (two random variables) and moderated effects (the product of these two random variables). At this point, I'm mostly ...
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7 votes
1 answer
221 views

What is the pdf of a normal distribution divided by the square root of a log-normal over n?

We know that if $Z \sim \mathcal{N}(0,1)$, $W \sim \chi^2(n)$ and are independently distributed, then the variable $Y = \frac{Z}{\sqrt{W/n}}$ follows a $t$-distribution with degrees of freedom $n$. ...
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4 votes
1 answer
177 views

Student's t as a power law distribution

I'm currently reading about power laws and I have came across an answer stating: The density function of a Student's t-distribution with $n$ degrees of freedom is: $$f(x) \sim (1 + x^2 / n)^{-(n+1)/2}...
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1 vote
0 answers
35 views

T distribution for other functions than the mean [closed]

If there are $X_1,\cdots,X_n \sim N(\mu,\sigma^2)$ independent random variables, then their mean $n \cdot \overline X = X_1 + \cdots + X_n $ has the property that it is independent of $(n-1)S = \sum_{...
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0 votes
0 answers
29 views

Making a t-distribution with sample size less than 30 [duplicate]

Our teacher said that we can use the graph of a standard normal distribution whenever the sample size is greater than or equal to 30 because of CLT. How can we make a graph of a $t$-distribution if ...
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Marginal and joint distributions of normal and student's

Is there an equivalence between two normal (student's) distributions. in the sense that, if my two variables (X,Y) are normal (student's) then their joint distribution is also normal (student's). I ...
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4 votes
3 answers
99 views

Excess kurtosis in the error term

I am modelling the residual errors terms in my time-series model using Student's $t$-distribution. This distribution allows for more kurtosis (‘heavy tailedness’) than the Gaussian distribution. ...
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0 votes
1 answer
85 views

Effect on GARCH innovations after scaling by a constant

I wish to fit the innovations resulting from a GARCH (1,1) process to either a student-t or an NIG distribution. For stability, I had to scale my data before applying GARCH. How will this affect the ...
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2 votes
1 answer
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Paradoxical result when computing a confidence interval when population standard deviation is not known

I have been trying to construct a $1-\alpha$ confidence interval for the mean. The distribution from which a sample is drawn is exponential distribution with density: $$p(x) = \frac{1}{\beta}e^{-x/\...
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How to prove that the t distribution doesn't belong to the exponential family?

Or in other words, is there anyway prove that the t distribution doesn't belong to the exponential family without going through all that calculation? Since the density has the gamma function in it ...
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