TDIST

Institute of Measurement Science SAS - Distribution of a linear combination of independent symmetric random variables, e.g. Student's t

TDIST Computes the distribution (PDF, CDF or QF - quantiles) of a linear combination of independent SYMMETRIC (zero-mean) random variables (RVs), say Y = sum_i \lambda_i X_i, with X_i having specific distributions:
- STUDENT's t distribution with 0 < df < Inf,
- NORMAL distribution N(0,1),
- RECTANGULAR (uniform) distribution R(-1,1),
- symmetric TRIANGULAR distribution T(-1,1),
- symmetric ARCSINE distribution (U-distribution) U(-1,1),
TDIST is based on numerical inversion of the characteristic function (Gil-Pelaez method). The required integration is performed by the 14-points Gaussian quadrature over N sub-intervals of [0, Tmax].

SYNTAX:
[yfun,xfun,results] = TDIST(X,df,lambda,funtype,options)
INPUT:
X - vector of appropriate (funtype) x values if X = [] then xfun is generated automatically
df - vector of degrees of freedom of independent STUDENT's t RVs, t_df, for 0 < df < Inf. Further,
set df = Inf for the NORMAL RVs, N(0,1),
set df = -1 for the RECTANGULAR RVs, R(-1,1),
set df = -2 for the symmetric TRIANGULAR RVs, T(-1,1),
set df = -3 for the symmetric ARCSINE RVs, U(-1,1),
set df < -10 for the symmetric mixture of CHI2 and -CHI2 RVs with nu = abs(df+10) degrees of freedom.
lambda - vector of coefficients of the linear combination
funtype - default value is 1 (calculates the CDF)
The following funtypes are legible:
0: TDIST calculates CDF and PDF, yfun = [CDF,PDF].
1: TDIST calculates the cumulative distribution function, CDF at X, yfun = CDF.
2: TDIST calculates the probability density function, PDF, at X, yfun = PDF.
3: TDIST calculates the quantile function, QF at X, yfun = QF.
4: TDIST calculates the characteristic function, CHF at X, yfun = CHF.
EXAMPLE 1: (CDF of a linear combination of RVs defined by df)
df = [Inf 1 -1 -2 -3];
lambda = [1 1 5 1 10];
funtype = 1;
[cdf,x] = tdist([],df,lambda,funtype);
EXAMPLE 2: (PDF of a linear combination of RVs defined by df)
df = [Inf 1 -1 -2 -3];
lambda = [1 1 5 1 10];
funtype = 2;
[pdf,x] = tdist([],df,lambda,funtype);
EXAMPLE 3: (QF of a linear combination of RVs defined by df)
options.isPlot = false;
df = [Inf 1 -1 -2 -3];
lambda = [1 1 5 1 10];
funtype = 3;
prob = [0.9 0.95 0.99]';
qf = tdist(prob,df,lambda,funtype,options);
disp([prob qf]);

EXAMPLE 4: (CHF of a linear combination of RVs defined by df)
options.isPlot = true;
df = [Inf 1 -1 -2 -3];
lambda = [1 1 5 1 10];
funtype = 4;
t = linspace(0,pi);
chf = tdist(t,df,lambda,funtype,options);

EXAMPLE 5 (Create PDF as a CHEBFUN function and use it to compute CDF)
df_true = [1 2 3 10];
df = -10 - df_true; % symmetric chi2-mixture distributions
lambda = [1 1 1 1];
funtype = 2;
N = 2^10;
xmax = 130;
x = -xmax * cos((0:N)*pi/N);
f = tdist(x,df,lambda,funtype);
pdf = chebfun(f,[-xmax,xmax]);
integrate = sum(pdf);
cdf = cumsum(pdf);
xnew = linspace(-50,50);
figure
plot(xnew,cdf(xnew))

EXAMPLE 6: (Create a CHEBFUN for the QF)
options.isChebfun = true;
options.n = 2^8;
options.N = 2^9;
df = [Inf 1 -1 -2 -3];
lambda = [1 1 5 1 10];
funtype = 3;
[qf,prob,results] = tdist([],df,lambda,funtype,options);
QF = results.chebfun;
disp(QF([0.9 0.95 0.99]'))

The algorithm requires evaluation of the BesselK and BesselJ functions.

The algorithm requires evaluation of the BesselK and BesselJ functions.

REFERENCES:
[1] GIL-PELAEZ, J. Note on the inversion theorem. Biometrika 38 (1951), 481–482.

[2] WITKOVSKY , V. On the exact computation of the density and of the quantiles of linear combinations of t and F random variables. Journal of Statistical Planning and Inference 94 (2001), 1–13.

[3] WITKOVSKY , V. Matlab algorithm TDIST: The distribution of a linear combination of Student’s t random variables. In COMPSTAT 2004 Symposium (2004), J. Antoch, Ed., Physica-Verlag/Springer 2004, Heidelberg, Germany,1995–2002.

[4] DRISCOLL, T. A., HALE, N., TREFETHEN, L. N. Chebfun Guide. Pafnuty Publications, Oxford, 2014.

Viktor Witkovsky ([e-mail])
Ver.: 01-Dec-2014 01:30:48