Quantile regression with a Dirichlet-process scale mixture

Kottas & Gelfand's point is that fixing the error distribution to a single asymmetric Laplace is stronger than the quantile restriction requires; the tau-quantile restriction is preserved by any scale mixture, so a Dirichlet process is placed on the mixing distribution of the scale. The location parameter is unaffected, which is why the regression coefficient can be read off the check-loss fit, \(_hat = argmin sum rho_tau(y_i - x_i primeb)\) with rho_tau(u) = u (tau - 1{u < 0}), and only the error model changes. The fit uses the shared fixed-iteration IRLS helper, which takes exactly the same path in both language arms

Usage

Npbqr(y, X, tau = 0.5, alpha = 1, niter = 40L, eps = 0.001)

Arguments

y

Response, length n.

X

Design WITHOUT an intercept column; one is added.

tau

Quantile in (0, 1).

alpha

DP concentration for the scale mixture, positive.

niter

IRLS iterations.

eps

Residual floor in the IRLS weight.

Value

List with beta, estimate, sigma, check_loss, e_k, e_k_digamma, tau, alpha, n, p.

Details

The scale part is closed form: the ALD maximum likelihood scale is sigma_hat = (1/n) sum rho_tau(r_i), and the DP mixing distribution induces E[K_n] = sum alpha / (alpha + i - 1) distinct scales a priori.

References

Kottas, A. & Gelfand, A. E. (2001). Bayesian semiparametric median regression modeling. Journal of the American Statistical Association, 96(456), 1458-1468. doi:10.1198/016214501753382363

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Npbqr(V, V)
#> $beta
#> [1] 2.41019e-14 1.00000e+00
#> 
#> $estimate
#> [1] 2.41019e-14
#> 
#> $sigma
#> [1] 5.342948e-15
#> 
#> $check_loss
#> [1] 4.274359e-14
#> 
#> $e_k
#> [1] 2.717857
#> 
#> $e_k_digamma
#> [1] 2.717857
#> 
#> $tau
#> [1] 0.5
#> 
#> $alpha
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $p
#> [1] 2
#> 
#> $method
#> [1] "Quantile regression with a DP scale mixture (Kottas-Gelfand 2001)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"