Student t Distribution help(TDist)

TDist                  package:stats                   R Documentation

The Student t Distribution

Description:

     Density, distribution function, quantile function and random
     generation for the t distribution with ‘df’ degrees of freedom
     (and optional non-centrality parameter ‘ncp’).

Usage:

     dt(x, df, ncp, log = FALSE)
     pt(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
     qt(p, df, ncp, lower.tail = TRUE, log.p = FALSE)
     rt(n, df, ncp)
    
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.

      df: degrees of freedom (> 0, maybe non-integer).  ‘df = Inf’ is
          allowed.

     ncp: non-centrality parameter \delta; currently except for ‘rt()’,
          only for ‘abs(ncp) <= 37.62’.  If omitted, use the central t
          distribution.

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:

     The t distribution with ‘df’ = n degrees of freedom has density

     f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)
    
     for all real x.  It has mean 0 (for n > 1) and variance n/(n-2)
     (for n > 2).

     The general _non-central_ t with parameters (df, Del) ‘= (df,
     ncp)’ is defined as the distribution of T(df, Del) := (U + Del) /
     sqrt(V/df) where U and V are independent random variables, U \~
     N(0,1) and V \~ Chi^2(df) (see Chisquare).

     The most used applications are power calculations for t-tests:



Let T= (mX - m0) / (S/sqrt(n)) where mX is the ‘mean’ and S the
     sample standard deviation (‘sd’) of X_1, X_2, ...., X_n which are
     i.i.d. N(mu, sigma^2) Then T is distributed as non-central t with
     ‘df’{} = n-1 degrees of freedom and *n*on-*c*entrality *p*arameter
     ‘ncp’ = (mu - m0) * sqrt(n)/sigma.

Value:

     ‘dt’ gives the density, ‘pt’ gives the distribution function, ‘qt’
     gives the quantile function, and ‘rt’ generates random deviates.

     Invalid arguments will result in return value ‘NaN’, with a
     warning.

Note:

     Setting ‘ncp = 0’ is _not_ equivalent to omitting ‘ncp’.  R uses
     the non-centrality functionality whenever ‘ncp’ is specified which
     provides continuous behavior at ncp = 0.

Source:

     The central ‘dt’ is computed via an accurate formula provided by
     Catherine Loader (see the reference in ‘dbinom’).

     For the non-central case of ‘dt’, contributed by Claus Ekstroem
     based on the relationship (for x != 0) to the cumulative
     distribution.

     For the central case of ‘pt’, a normal approximation in the tails,
     otherwise via ‘pbeta’.

     For the non-central case of ‘pt’ based on a C translation of

     Lenth, R. V. (1989). _Algorithm AS 243_ - Cumulative distribution
     function of the non-central t distribution, _Applied Statistics_
     *38*, 185-189.

     For central ‘qt’, a C translation of

     Hill, G. W. (1970) Algorithm 396: Student's t-quantiles.
     _Communications of the ACM_, *13(10)*, 619-620.

     altered to take account of

     Hill, G. W. (1981) Remark on Algorithm 396, _ACM Transactions on
     Mathematical Software_, *7*, 250-1.




The non-central case is done by inversion.

References:

     Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S
     Language_.  Wadsworth & Brooks/Cole. (Except non-central
     versions.)

     Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) _Continuous
     Univariate Distributions_, volume 2, chapters 28 and 31.  Wiley,
     New York.

See Also:

     ‘df’ for the F distribution.

Examples:

     require(graphics)
    
     1 - pt(1:5, df = 1)
     qt(.975, df = c(1:10,20,50,100,1000))
    
     tt <- seq(0,10, len=21)
     ncp <- seq(0,6, len=31)
     ptn <- outer(tt,ncp, function(t,d) pt(t, df = 3, ncp=d))
     t.tit <- "Non-central t - Probabilities"
     image(tt,ncp,ptn, zlim=c(0,1), main = t.tit)
     persp(tt,ncp,ptn, zlim=0:1, r=2, phi=20, theta=200, main=t.tit,
           xlab = "t", ylab = "non-centrality parameter",
           zlab = "Pr(T <= t)")
    
     plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32),
          main="Non-central t - Density", yaxs="i")