signal_strength: Estimate intercept and signal strength

In zq00/glmhd: Estimates the inflation and variance of the MLE from a high-dimensional GLM

Estimate intercept and signal strength

Description

signal_strength estimates (\beta_0, \gamma) using the estimated \kappa_s and observed proportion of successes.

Usage

signal_strength(
  rho_prime = rho_prime_logistic,
  kappa_hat,
  intercept = FALSE,
  p0 = NA,
  verbose = FALSE,
  tol = 1e-04
)

Arguments

rho_prime	A function that computes the success probability \rho'(t) = \mathrm{P}(Y=1 | X^\top \beta = t), here \beta is the coefficient. The default is logistic model.
kappa_hat	Numeric. Estimated dimension where the data becomes linearly separable
intercept	Logical TRUE if the model contains an intercept.
p0	Numeric. Proportion of outcomes Y=1.
verbose	Should progress be printed? If TRUE, prints progress at each step.
tol	Numeric. Tolerance to be used in fsolve function.

Details

Assume that Y depends on X as

\mathrm{P}(Y=1\,|\,X) = \rho'(X^\top \beta + \beta_0),

and let the signal strength be \gamma = \mathrm{Var}(X^\top \beta)^{1/2}. The pair (\beta_0, \gamma) satisfies

• They are on the phase transition curve \kappa(\beta_0, \gamma).

  \hat{\kappa} \approx \kappa(\beta_0, \gamma)

• The observed proportion of Y=1 should be close to the expected proportion.

  p_0 \approx \mathrm{P}(Y = 1\,|\, \beta_0, \gamma) = \mathrm{E}{\mathrm{Ber}(\rho'(\beta_0 + \gamma Z)}

  where Z is a standard normal variable.

We solve the above system of two equations to obtain an estimate of (\beta_0, \gamma)

Value

If the model does not contain an intercept, returns estimated gamma_hat. Otherwise, returns a list with two components

gamma_hat

Estimated signal strength.

b_hat

Estimated intercept.

Examples

## Not run: 
# no signal case
# should return 0, returns 0.0127
signal <- signal_strength(kappa_hat = 0.5, intercept = FALSE)
signal$gamma_hat

## End(Not run)

zq00/glmhd documentation built on April 7, 2023, 7:45 a.m.