Cumulative distribution function (cdf)

Cumulative distribution function (cdf)#

Calculating the cdf from the pdf#

ctx.cdf_from_pdf(x, cf)#

where ctx is dec, mpm, or gmp.

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\):

\[\text{cdf}_X(x) = \int_{0}^{x} \text{pdf}_X(x) \mathrm{d}t .\]

Using this method can be a viable option when the pdf, but not the cdf, is available in closed form.

Calculating the cdf from the pmf#

ctx.cdf_from_pmf(x, cf)#

where ctx is dec, mpm, or gmp.

If \(X\) is a purely discrete random variable, then it attains values \(x_{1},x_{2},\ldots\) with probability \(p_{i}=p(x_{i})\), and the CDF of \(X\) will be discontinuous at the points \(x_{i}\):

\[F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i}).\]

Calculating the cdf from the characteristic function#

ctx.cdf_from_cf_continuous(x, cf)#

where ctx is dec, mpm, or gmp.

Calculates the cdf from the characteristic function using the procedure of Gil-Pelaez.

Assuming that the characteristic function is absolutely integrable over \((-\infty, \infty)\), Gil-Pelaez derived the following inversion formula which requires integration of a real-valued function only. In particular,

\[\text{cdf}_X(x) = \frac{1}{2} - \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{e^{-itx} C_X(t) - e^{itx} C_X(t)}{it} \mathrm{d} t = \frac{1}{2} - \frac{1}{\pi} \int_{0}^{\infty} \Im \left ( \frac{ e^{-itx} C_X(t)}{t} \right ) \mathrm{d}t.\]

where \(\Im (z)\) denotes the imaginary part of \(z\). We also have

\[\int_{0}^{\infty} \Im \left ( \frac{ e^{-itx} C_X(t)}{t} \right ) \mathrm{d}t = \int_{0}^{\infty} \Im \left ( \frac{C_X(t)}{t} \right ) \cos(t x) \mathrm{d}t - \int_{0}^{\infty} \Re \left ( \frac{C_X(t)}{t} \right ) \sin(t x) \mathrm{d}t.\]

Using the right-hand side of this equation allows for efficient use of the quadrature formula of Fillon, with the half-period \(\omega = \pi/x\).

The python code is currently in Charfun.py

The following code provides a test-suite for the numerical inversion of the characteristic function:

class tests_charfunc(rv_cont):

    def __init__(self, rv2, x = 5, a = 0, b = 2):

        cdf_value = rv2.cdf(x)
        print ("rv2.cdf(x): ", cdf_value)

        rv2.set_x(x)
        plot(rv2.gil_pelaez_imag, [a, b], points=200)
        print

        rv2.set_x(x)
        plot(rv2.gil_pelaez_cos, [a, b], points=200)
        print

        rv2.set_x(x)
        plot(rv2.gil_pelaez_sin, [a, b], points=200)
        print

        rv2.set_x(x)
        I0 = quad(rv2.gil_pelaez_imag, [0, +inf])
        print("Integral: ", I0)
        result0 = 0.5 - I0/pi
        print("result0:", result0 )
        print("diff0:", result0 - cdf_value)


        rv2.set_x(x)
        I1 =quadosc(rv2.gil_pelaez_cos, [0, inf], period=1*pi/x) # half period
        print("I1:", I1 )

        rv2.set_x(x)
        I2 =quadosc(rv2.gil_pelaez_sin, [0, inf], period=1*pi/x) # half period
        print("I2:", I2 )

        I3 = I1 + I2
        print("I3:", I3 )
        print("Int diff:", I3 - I0)
        result3 = 0.5 - I3/pi
        print("result3:", result3 )
        print("diff3:", result3 - cdf_value)

Example: non-central chi-squared distribution

The following code shows the difference between using the generic integration (error: diff0: -6.1218939724378892896e-6) and quadrature for oscillatory functions (error: diff3: 8.4703294725430033907e-22).

mp.dps = 20
print()
print ("Hello mpDistributions local ! ")
print()


a = 0.0
b = 2

n = mpf("5")
x = mpf("10")
rv2 = mpr().chisquare(n)

tests_charfunc(rv2, x, a, b)

This produces the following output (plots are to be added):

Hello mpDistributions local !

rv2.cdf(x):  0.92476475385348782128

Integral:  -1.334418597712864291
result0: 0.92475863195951538339
diff0: -6.1218939724378892896e-6
I1: 0.11817924829245123086
I2: -1.4526170785024453884
I3: -1.3344378302099941575
Int diff: -0.000019232497129866511475
result3: 0.92476475385348782128
diff3: 8.4703294725430033907e-22

Calculating the cdf and sf from the characteristic function (lattice distribution)#

ctx.cdf_from_cf_lattice(x, cf)#

where ctx is dec, mpm, or gmp.

The corresponding inversion formula for discrete distributions on the nonnegative integers is

\[\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=0}^x e^{-itz} \right) \mathrm{d}t.\]
\[\text{sf}(x) = 1-\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=x+1}^{N(N+1)/2} e^{-itz} \right) \mathrm{d}t.\]

Calculating the cdf from the factorial moments (lattice distributions)#

The following relationships hold between the probabilities and the factorial moments in the case of a discrete distribution:

\[\text{Pr}[X=x] = \sum_{j \ge x} (-1)^{x+j} \binom{j}{x} \frac{\mu'_{[j]}}{j!} = \sum_{r \ge 0} (-1)^{r} \frac{\mu'_{[x+r]}}{x!r!}\]

and

\[\sum_{i \ge x} \text{Pr}[X=i] = \sum_{j \ge x} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!}\]

See also Johnson(2005), page 59.

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