Chi-bar-square tail probability for boundary likelihood-ratio tests

Upper-tail probability of the mixture $$ \bar{\chi}^{2} = \sum_{i=0}^{q} w_i\,\chi^{2}_{s+i}, \qquad w_i=\binom{q}{i}2^{-q}, $$ with $q$ constraints active on the boundary of the parameter space and $s$ interior constraints. This is Case 9 of (Self and Liang 1987) , which generalises the $\tfrac{1}{2}\chi^{2}_{0}+\tfrac{1}{2}\chi^{2}_{1}$ result of (Chernoff 1954) . For $q=0$ the mixture collapses to the ordinary $\chi^{2}_{s}$ of (Wilks 1938) .

Usage

chi_bar_square_pvalue(
  statistic,
  n_boundary = 1L,
  df_interior = 0L,
  weights = NULL
)

Arguments

statistic: Numeric vector of observed likelihood-ratio statistics $D=2(\hat{\ell}_1-\hat{\ell}_0)$.
n_boundary: Integer $q$: number of constraints active on the boundary (e.g. null ratios set to zero).
df_interior: Integer $s$: number of interior (two-sided) constraints tested simultaneously.
weights: Optional numeric vector of length n_boundary + 1 giving the mixing weights $w_0,\ldots,w_q$. Defaults to the binomial weights.

Value

Numeric vector of upper-tail probabilities.

Details

The binomial weights hold when the block of the Fisher information corresponding to the active constraints is diagonal after orthogonalisation against the nuisance block. Under correlated constraints the mixing weights depend on the geometry of the approximating tangent cone and the binomial choice is only an approximation; Self and Liang also exhibit a nuisance-parameter configuration whose limit is not a chi-square mixture. Supply weights explicitly, or calibrate by simulation with bootstrap_decovart(), whenever $q>1$ and the constrained cell types share signal.

Atom at zero

$\chi^{2}_{0}$ is a point mass at the origin, so its strict upper tail $\Pr(\chi^{2}_{0}>D)$ is zero for every $D\ge 0$. The component therefore only removes weight: with one active constraint and no interior constraint the p-value is $\tfrac{1}{2}\Pr(\chi^{2}_{1}>D)$, capped at $0.5$ when $D=0$. Note stats::pchisq() switches to the closed tail at exactly q = 0 for df = 0; the strict convention is used here so the p-value stays continuous in statistic.

References

Self SG, Liang K (1987). “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions.” Journal of the American Statistical Association, 82(398), 605–610. ISSN 0162-1459. doi:10.1080/01621459.1987.10478472 .

Chernoff H (1954). “On the Distribution of the Likelihood Ratio.” The Annals of Mathematical Statistics, 25(3), 573–578. ISSN 0003-4851. doi:10.1214/aoms/1177728725 .

Wilks SS (1938). “The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses.” The Annals of Mathematical Statistics, 9(1), 60–62. ISSN 0003-4851. doi:10.1214/aoms/1177732360 .

Examples

# One active boundary constraint: 50:50 mixture of chi2_0 and chi2_1.
chi_bar_square_pvalue(2.71, n_boundary = 1L)
#> [1] 0.0498605
# Interior (Wilks) calibration.
chi_bar_square_pvalue(3.84, n_boundary = 0L, df_interior = 1L)
#> [1] 0.05004352