/**
 * Noncentral chi-squared CDF via the Poisson-mixture representation:
 *
 *   F(x; k, λ) = Σ_{j=0}^∞ Pois(j; λ/2) · chiSquaredCDF(x, k + 2j)
 *
 * Truncated once the cumulative Poisson mass covers `1 − 1e-12` of the total
 * (past the Poisson mode, so truncation never fires during the rising phase).
 *
 * @example
 * noncentralChi2CDF(10, 3, 2); // ~0.89856 (scipy ncx2.cdf)
 */
export declare function noncentralChi2CDF(x: number, df: number, nc: number): number;
/**
 * Noncentral F CDF via the Poisson-mixture representation over the numerator
 * degrees of freedom:
 *
 *   F(x; d1, d2, λ) = Σ_{j=0}^∞ Pois(j; λ/2) · fCDF(x·d1/(d1+2j), d1+2j, d2)
 *
 * Truncated the same way as {@link noncentralChi2CDF}.
 *
 * @example
 * noncentralFCDF(2, 3, 10, 4); // ~0.46636 (scipy ncf.cdf)
 */
export declare function noncentralFCDF(x: number, dfn: number, dfd: number, nc: number): number;
/**
 * Noncentral Student-t CDF via the mixture representation
 * `T = (Z + δ) / sqrt(V/ν)`, `Z ~ N(0,1)`, `V ~ χ²_ν` independent:
 *
 *   F(t; ν, δ) = E_V[ Φ(t·sqrt(V/ν) − δ) ] = ∫₀^∞ Φ(t·sqrt(v/ν) − δ) · χ²_ν(v) dv
 *
 * Evaluated by composite Simpson's rule over `v` (central-χ² density; `δ`
 * only enters the normal-CDF term). The upper integration bound is set
 * generously past the χ²_ν tail so the truncation error is negligible
 * relative to the ~1e-4 Simpson discretization error at the panel count used.
 *
 * @example
 * noncentralTCDF(1.5, 10, 2); // ~0.30479 (scipy nct.cdf)
 */
export declare function noncentralTCDF(t: number, df: number, nc: number): number;
/** Options shared by the circular-statistics functions. */
export interface CircularOptions {
    /** Upper bound of the angular range. Default `2π`. */
    high?: number;
    /** Lower bound of the angular range. Default `0`. */
    low?: number;
}
/**
 * Circular (angular) mean, mapped back into `[low, high)`.
 *
 * `mean = atan2(Σsinθ, Σcosθ)`, where `θ` is `angles` rescaled into
 * `[0, 2π)` when a non-default `[low, high)` range is given (matching
 * `scipy.stats.circmean`).
 *
 * @example
 * circmean([0.1, 0.2, 6.2]); // ~0.07236 (scipy circmean; wraps near 0)
 */
export declare function circmean(angles: readonly number[], opts?: CircularOptions): number;
/**
 * Circular variance `1 − R`, where `R = |Σcosθ + iΣsinθ| / n` is the mean
 * resultant length. `R ∈ [0, 1]`, so `circvar ∈ [0, 1]`.
 */
export declare function circvar(angles: readonly number[], opts?: CircularOptions): number;
/**
 * Circular standard deviation `sqrt(−2·ln(R))` (matches `scipy.stats.circstd`
 * with the default `normalize=False` low/high dispersion measure).
 */
export declare function circstd(angles: readonly number[], opts?: CircularOptions): number;
/**
 * Von Mises probability density function (the circular analogue of the
 * normal distribution).
 *
 *   f(θ; μ, κ) = exp(κ·cos(θ − μ)) / (2π·I₀(κ))
 *
 * `I₀` is the modified Bessel function of the first kind, order 0
 * (`besselIScalar` — the shared special-function scalar backing the public
 * `besselI`).
 *
 * @example
 * vonMisesPDF(0, 0, 2); // ~0.51589 (scipy vonmises.pdf(0, 2))
 */
export declare function vonMisesPDF(theta: number, mu: number, kappa: number): number;
/** Options for {@link mcnemar}. */
export interface McNemarOptions {
    /** Apply the continuity correction (`|b−c| − 1`). Default `true`. */
    correction?: boolean;
}
/** Result of {@link mcnemar}. */
export interface McNemarResult {
    chi2: number;
    pValue: number;
}
/**
 * McNemar's test for paired nominal data on a 2x2 table
 * `[[a, b], [c, d]]` (only the discordant pairs `b`, `c` matter):
 *
 *   chi2 = (|b − c| − correction)² / (b + c)
 *
 * with the continuity correction (`1`) applied by default, matching
 * `statsmodels.stats.contingency_tables.mcnemar`. `pValue = 1 −
 * chiSquaredCDF(chi2, 1)`.
 *
 * @example
 * mcnemar([[10, 5], [3, 12]], { correction: false }); // { chi2: 0.5, pValue: ... }
 */
export declare function mcnemar(table: readonly (readonly number[])[], opts?: McNemarOptions): McNemarResult;
/** Result of {@link cochranQ}. */
export interface CochranQResult {
    Q: number;
    pValue: number;
    dof: number;
}
/**
 * Cochran's Q test — the extension of McNemar's test to `k > 2` matched
 * binary treatments. `data` has one row per subject, one column per
 * treatment (0/1 entries).
 *
 *   Q = (k−1)·(k·ΣCⱼ² − N²) / (k·N − ΣRᵢ²)
 *
 * where `Cⱼ` are column sums, `Rᵢ` are row sums, and `N = ΣRᵢ`. `dof = k −
 * 1`; `pValue = 1 − chiSquaredCDF(Q, k−1)`. Matches
 * `statsmodels.stats.contingency_tables.cochrans_q`.
 *
 * @example
 * cochranQ([[1, 1, 0], [1, 0, 0], [1, 1, 1], [0, 1, 0], [1, 1, 0]]);
 */
export declare function cochranQ(data: readonly (readonly number[])[]): CochranQResult;
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