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Compare rho and w1 Distributions

Source: R/09_weights_rho.R
compare_rho_w1.Rd

Compares the marginal distributions of rho and w1 under the same hyperprior alpha ~ Gamma(a, b).

Usage

compare_rho_w1(a, b, M = .QUAD_NODES_DEFAULT)

Arguments

a

Numeric; shape parameter of the Gamma prior on alpha (a > 0).

b

Numeric; rate parameter of the Gamma prior on alpha (b > 0).

M

Integer; number of quadrature nodes. Default is 80.

Value

A list containing:

mean_rho

E(rho)

mean_w1

E(w1)

mean_equal

Logical; whether means are equal

var_rho

Var(rho)

var_w1

Var(w1)

var_ratio

Var(rho) / Var(w1)

Details

While E(rho) = E(w1), the variances differ because:

  • Var(rho | alpha) = 2*alpha / ((1+alpha)^2*(2+alpha)*(3+alpha))

  • Var(w1 | alpha) = alpha / ((1+alpha)^2*(2+alpha))

Generally, Var(rho) < Var(w1) because rho averages over all squared weights.

Examples

compare_rho_w1(a = 2, b = 1)
#> $mean_rho
#> [1] 0.4036526
#> 
#> $mean_w1
#> [1] 0.4036526
#> 
#> $mean_equal
#> [1] TRUE
#> 
#> $var_rho
#> [1] 0.05744825
#> 
#> $var_w1
#> [1] 0.08968429
#> 
#> $var_ratio
#> [1] 0.6405609
#> 
compare_rho_w1(a = 1.6, b = 1.22)
#> $mean_rho
#> [1] 0.508368
#> 
#> $mean_w1
#> [1] 0.508368
#> 
#> $mean_equal
#> [1] TRUE
#> 
#> $var_rho
#> [1] 0.07096137
#> 
#> $var_w1
#> [1] 0.1052062
#> 
#> $var_ratio
#> [1] 0.6744981
#>