Student's t-Distribution CDF (TDIST)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Student's t-Distribution CDF (TDIST)", "TDIST", overlay=false, precision=6)

//@function Natural log of the Gamma function via Lanczos approximation (g=7, 9 coefficients)
//@param z Argument (must be > 0)
//@returns ln(Γ(z))
lnGamma(float z) =>
    float g = 7.0
    array<float> c = array.from(
         0.99999999999980993,
       676.5203681218851,
     -1259.1392167224028,
       771.32342877765313,
      -176.61502916214059,
        12.507343278686905,
        -0.13857109526572012,
         9.9843695780195716e-6,
         1.5056327351493116e-7)
    float zz = z < 0.5 ? 1.0 - z : z - 1.0
    float x = array.get(c, 0)
    for i = 1 to 8
        x += array.get(c, i) / (zz + float(i))
    float t = zz + g + 0.5
    float logSqrt2Pi = 0.9189385332046727
    float lnG = logSqrt2Pi + math.log(t) * (zz + 0.5) - t + math.log(x)
    z < 0.5 ? math.log(math.pi / math.sin(math.pi * z)) - lnG : lnG

//@function Regularized incomplete beta function I_x(a, b) via Lentz continued fraction
//@param x Upper integration limit in [0, 1]
//@param a First shape parameter (> 0)
//@param b Second shape parameter (> 0)
//@returns I_x(a, b) in [0, 1]
betaReg(float x, float a, float b) =>
    int MAXITER = 200
    float EPS   = 1e-10
    float TINY  = 1e-30

    float result = 0.0
    if x <= 0.0
        result := 0.0
    else if x >= 1.0
        result := 1.0
    else
        bool flipped = x > (a + 1.0) / (a + b + 2.0)
        float xx = flipped ? 1.0 - x : x
        float aa = flipped ? b : a
        float bb = flipped ? a : b

        float logPfx = aa * math.log(xx) + bb * math.log(1.0 - xx)
                     - math.log(aa)
                     - lnGamma(aa) - lnGamma(bb) + lnGamma(aa + bb)
        float pfx = math.exp(logPfx)

        float f = 1.0 + TINY
        float C = f
        float D = 0.0
        for m = 1 to MAXITER
            float m2 = 2.0 * float(m)
            float numEven = float(m) * (bb - float(m)) * xx /
                           ((aa + m2 - 1.0) * (aa + m2))
            D := 1.0 + numEven * D
            D := math.abs(D) < TINY ? TINY : D
            D := 1.0 / D
            C := 1.0 + numEven / C
            C := math.abs(C) < TINY ? TINY : C
            f *= C * D
            float numOdd = -(aa + float(m)) * (aa + bb + float(m)) * xx /
                          ((aa + m2) * (aa + m2 + 1.0))
            D := 1.0 + numOdd * D
            D := math.abs(D) < TINY ? TINY : D
            D := 1.0 / D
            C := 1.0 + numOdd / C
            C := math.abs(C) < TINY ? TINY : C
            float delta = C * D
            f *= delta
            if math.abs(delta - 1.0) < EPS
                break

        float raw = pfx * f
        result := flipped ? 1.0 - raw : raw
    result

//@function Calculates Student's t-Distribution CDF
//@param source Series to evaluate (typically close)
//@param period Lookback period for min-max normalization
//@param df Degrees of freedom (ν > 0)
//@returns CDF value P(T ≤ t) in [0, 1]
//@description The Student's t-distribution CDF is computed via the relation:
//    CDF(t; ν) = 1 − 0.5 × I(ν/(ν+t²), ν/2, 1/2)  if t ≥ 0
//    CDF(t; ν) = 0.5 × I(ν/(ν+t²), ν/2, 1/2)        if t < 0
//  where I is the regularized incomplete beta function (Lentz CF).
//  The source is min-max normalized over the lookback period, then mapped
//  to a t-statistic via linear transform: t = (x − 0.5) × tScale where
//  tScale = 6.0 maps the [0,1] range to approximately [−3, +3].
//  Reuses lnGamma (Lanczos 9-coeff) and betaReg (Lentz CF with symmetry flip)
//  from BETADIST/FDIST. Stateless pure function — no var state.
//  df=1 → Cauchy (heavy tails), df=5 → moderate tails, df→∞ → normal.
//  Trading interpretation: CDF near 1.0 = price at top of recent range
//  (assuming large df, approaches normal behavior). Heavy tails (low df) make
//  the CDF less extreme, reflecting uncertainty about outlier moves.
tdist(series float source, simple int period, simple float df) =>
    if period <= 0
        runtime.error("Period must be greater than 0")
    if df <= 0.0
        runtime.error("Degrees of freedom must be greater than 0")

    float src = nz(source)
    float hi = src
    float lo = src
    for i = 1 to period - 1
        float v = nz(source[i])
        hi := math.max(hi, v)
        lo := math.min(lo, v)
    float range = hi - lo
    float x = range == 0.0 ? 0.5 : (src - lo) / range

    float tScale = 6.0
    float t = (x - 0.5) * tScale

    float t2 = t * t
    float bx = df / (df + t2)
    float ibeta = betaReg(bx, df / 2.0, 0.5)

    t >= 0.0 ? 1.0 - 0.5 * ibeta : 0.5 * ibeta

// ---------- Main loop ----------

// Inputs
i_source = input.source(close, "Source")
i_period = input.int(50, "Period", minval=1)
i_df     = input.float(5.0, "Degrees of Freedom", minval=0.1, step=0.1)

// Calculation
tdist_value = tdist(i_source, i_period, i_df)

// Plot
plot(tdist_value, "TDIST", color=color.yellow, linewidth=2)
hline(0.5, "Midline", color=color.gray, linestyle=hline.style_dotted)
hline(0.95, "Upper", color=color.red, linestyle=hline.style_dashed)
hline(0.05, "Lower", color=color.green, linestyle=hline.style_dashed)