# Tagged Questions

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 n increases the t-value increases in a hypothesis test, but the t-table is just the opposite. Why?

The formula for $t$ in a hypothesis test is given by: $$t=\frac{\bar{X}-\mu}{\hat \sigma/\sqrt{n}}.$$ When $n$ increases, the $t$-value increases according to the above formula. But why does ...
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### Student's t-distribution

I have two functions that provide an implementation of the t-distribution. A webpage with a Javascript algorithm. The algorithm takes two input x-value and ...
169 views

### Grouping similar subjects based on slopes

I have 500 subjects for which I am studying their response to temperature increase using a linear regression on each degree of temperature (from 10°C to 28°C). Thus, it was possible for me to compute ...
210 views

### The ratio of the dependent normal rv and chi rv is?

(I hope this question may be too easy for some experts here). It is well-known that if $X$ is normal rv and $Y$ is a chi-square rv, then $Z=X/\sqrt{Y/n}$ follows student's $t$ distribution, where $n$ ...
111 views

### The mgf and cf of Student's t distribution

A student's t distributed rv $X$ has characteristic function but no moment generating function. I wonder if cf(X)=$E[e^{itX}]$, why we cannot take $t=-iu$ to get the mgf $E[e^{uX}]$? (This question ...
90 views

### Generate data from a t-distribution with specified mean and standard deviation

How does one randomly sample from a T-distribution in R. From what I've found, the function rt in R doesn't let you specify the mean and standard deviation. For a ...
57 views

### Ratio of lengths of two confidence intervalls

I have two random variables: (1) With standard normal distribution. Confidence interval $I_1$, which is centered and has probability of $(1-\alpha)$ (2) With T-distribution. Confidence intervall ...
54 views

### How to prove that a t-distribution can be written as a ratio distribution?

If $X \sim N(0,1)$ and $Y \sim \chi^2(n),$ then it's "known" that $Z = X/\sqrt{Y/n}$ is $t$ distributed. Is there anywhere a proof for this? That in the end one can see the $t$ distribution?
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### Can the t-distribution be defined as the distribution on the true mean of a sampled normal?

Wikipedia says now, here in the introduction: http://en.wikipedia.org/wiki/Student%27s_t-distribution "... then the t-distribution (for n-1) can be defined as the distribution of the location of the ...
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### Generic test for unimodality given sample

Are there any generic tests to validate if a given sample follows a unimodal distribution, like a Gaussian, Cauchy, Student's t or a chi-square?
231 views

### Generating random numbers from a t-distribution

How can I generate random numbers that follow a student-t distribution? From several sources I understand that this can be done using a random sample of size $n$ drawn from a normally distributed ...
77 views

### Does inference from a heteroskedasticity-consistent covariance matrix follow the t-distribution or the normal?

Background I'm currently looking into how reliable confidence intervals are on bounded scores (EQ-5D) and I compare regular asymptotic, robust and botstrap-based confidence intervals. The robust ...
114 views

### Statistical notation in plain English

I have a very limited understanding of statistics, and am having trouble translating an answer to this question from quant.SE, "Separating the wheat from the chaff", the answer being: Larry Harris ...
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### Can we never have a normal distribution?

Since the population variance can never be estimated, will we always have a t-distribution instead of normal?
129 views

### quantile threshold of t distribution and normal distribution

I want to calculate the quantiles of a t distribution and a normal distribution fitted to my data (can be found here: http://uploadeasy.net/upload/nk0f.rar). The t distribution is leptokurtic, ...
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### T values of ARIMA parameter estimates

The SAS proc ARIMA documentation states that "The t values reported in the table of parameter estimates are approximations whose accuracy depends on the validity of the model, the nature of the model, ...
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### quantile transformation with t distribution

I'm wondering about the monotonic quantile transformation in case of a t distribution. First, for explanation, lets consider a normal distribution: \begin{align*} f(l | \mu , \sigma ...
139 views

### Correct formula for standardized Student's t-distribution

I am wondering about the correct formula of the standardized Student's t distribution. In the rugarch package on page 15 it is given as: Substituting $\frac{(\nu-2)}{\nu}$ into 49 we obtain the ...
355 views

### Estimating parameters of Student's t-distribution

What are the maximum-likelihood estimators for the parameters of Student's t-distribution? Do they exist in closed form? A quick Google search didn't give me any results. Today I am interested in the ...
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### Normality of a lognormal variable having a spike in 0?

I have two very right-skewed datasets which I must study for difference in means. Given the skewness, I transformed using log 10 scale after adding 1 to be able to take the log. In other words: ...
504 views

### Confidence intervals for mean, when variance is unknown

When calculating the confidence interval for the population mean and the variance is unknown, I take it that you have to use the t-distribution. However, do you use the one-tailed or two-tailed test ...
62 views

### What is the distribution of the ratio of two t-distributed random variables?

x is t-distributed; y is t-distributed. How is x/y distributed? Does it have a closed-form formula?
242 views

### Formula to calculate a t-distribution

I am writing an application that will be dealing with <30 observations in a normal distribution. My understanding is that this point I would need to use t-distribution. The thing is, this is easy ...
34 views

### Degrees of Freedom for Multivariate t distribution

I have a data.frame of retruns - ...
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### Code for fitting multivariate t distribution (ECME)

I am looking to fit some data with central multivariate t distributions. I understand an ECME algorithm is the most promising way to do this. I wondered if anyone knew of any existing code for ...
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### Entropy of noncentral multivariate t distribution

I am looking at calculating the entropy of a multivariate noncentral t-distribution. I found in the central case (mean is zero) the entropy is the same as multivariate normal with a correction for ...
569 views

### Is Hansen's skewed-$t$ distribution the same as the skewed-$t$ distribution which is a special case of GH Distribution?

I recently studied two asymmetric t distribution both with a name of skewed-$t$. I am confused with their differences or are they actually the same? The first one is introduced by Hansen (1994) with ...
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### Testing for mutual exclusivity

So first, sorry if I use non-standard vocabulary, since I am not an expert. Please feel free to correct. I would like to know if a set of genes in a patient cohort are mutated by a mutually exclusive ...
49 views

### Test with Student's t

What kind of null hypothesis can be tested using a Student's t? It's very likely to be one of the questions of the statistics exam. In order to test a null hypothesis using the Student's t, does it ...
228 views

### t-distribution confidence intervals for non-Gaussian data but large n

I have a question concerning a claim I read in statistics books concerning the applicability of the t-distribution to compute confidence intervals for large $n$ if the data is not normally distributed ...
296 views

### With a small sample from a normal distribution, do you simulate using a t distribution?

I want to simulate temperature data for some "what-if" calculations. The problem is, I only have a time series of 10 actual temperature data values. I want to use temperature as an input to the ...
92 views

### Finding t distribution values for uniform and exponential functions

I know that a basic one-sample $t$ statistic can be calculated by $$t = \frac{x - \mu}{s / \sqrt{n}}$$ I also know that the following 4 functions $x\sim\mathcal{U}(0,1)$, $x\sim\mathcal{Exp}(2)$ ...
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### How to estimate the probability that the mean of an unknown distribution is over a threshold given small sample size

I am trying to quantify my concerns regarding a proposed incinerator in our community. The company is basing its potential to emit dioxins (a class of chlorinated organic compounds with a reference ...
314 views

### When is the distribution of $(\overline{x}-\mu)/{\rm SE}(\overline{x})$ normal and when is it $t$?

When estimating the mean of a population, two distributions seem to be useful, the normal distribution and the t-distribution. Is it correct to say that $t=(\overline{x}-\mu)/{\rm SE}(\overline{x})$ ...
59 views

### “t-Chart” value

I wonder what is wrong with my own answer for the problem which I have shared below, could you please help me some about that: " Suppose that we have 12 observation units $( n=12 )$ in a ...
112 views

### Is there standard nomenclature for Confidence Distributions?

I am wondering if there are standard naming conventions in general for confidence distributions or for the particular cases of the non-centrality parameters of (non-central) $t$- and ...
3k views

### Calculating P-value for Spearman's rank correlation coefficient example on Wikipedia

The Wikipedia article Spearman's rank correlation coefficient contains an example for calculating ρ. At the end of the sections is the statement "...with a P-value = 0.6864058 (using the t ...
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### t-test and t-distribution etymology

Okay, I'm aware of the fact that Gosset came up with t distribution, but what is the etymology of "t". How did "t" end up in t-test and t distribution?
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### What is the distribution of this nearly-Hotelling statistic?

Let $X$ be an $n \times l$ matrix, and $F$ an $n \times p$ matrix, with the rows of $X$ and $F$ drawn i.i.d. from multivariate Gaussians. (The independence applies to rows: the $X$ and $F$ may be ...
1k views

### How exactly does Wilks' Lambda distribution generalize the Hotelling distribution?

According to Wikipedia, Wilks' Lambda distribution generalizes Hotelling's distribution. I am having some problems seeing how this works. I can see how Hotelling's distribution generalizes Student's ...
We know that density for a student-t distribution is given as \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} ...
What is the median of the non-central t distribution with non-centrality parameter $\delta \ne 0$? This may be a hopeless question because the CDF appears to be expressed as an infinite sum, and I ...