# tt: The Truncated Student-t Distribution In crch: Censored Regression with Conditional Heteroscedasticity

 tt R Documentation

## The Truncated Student-t Distribution

### Description

Density, distribution function, quantile function, and random generation for the left and/or right truncated student-t distribution with `df` degrees of freedom.

### Usage

```dtt(x, location = 0, scale = 1, df, left = -Inf, right = Inf, log = FALSE)

ptt(q, location = 0, scale = 1, df, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

qtt(p, location = 0, scale = 1, df, left = -Inf, right = Inf,
lower.tail = TRUE, log.p = FALSE)

rtt(n, location = 0, scale = 1, df, left = -Inf, right = Inf)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `location` location parameter. `scale` scale parameter. `df` degrees of freedom (> 0, maybe non-integer). `df = Inf` is allowed. `left` left censoring point. `right` right censoring point. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].

### Details

If `location` or `scale` are not specified they assume the default values of `0` and `1`, respectively. `left` and `right` have the defaults `-Inf` and `Inf` respectively.

The truncated student-t distribution has density

f(x) = 1/σ τ((x - μ)/σ) / (T((right - μ)/σ) - T((left - μ)/σ))

for left ≤ x ≤ right, and 0 otherwise.

where T and τ are the cumulative distribution function and probability density function of the student-t distribution with `df` degrees of freedom respectively, μ is the location of the distribution, and σ the scale.

### Value

`dtt` gives the density, `ptt` gives the distribution function, `qtt` gives the quantile function, and `rtt` generates random deviates.

`dt`

crch documentation built on Sept. 10, 2022, 1:06 a.m.