Estimate Distribution Parameters Using Method of Moments

Estimates parameters for various distribution families from the Generalized Kumaraswamy family using the method of moments. The implementation is optimized for numerical stability and computational efficiency through Nelder-Mead optimization and adaptive numerical integration.

Usage

gkwgetstartvalues(x, family = "gkw", n_starts = 5L)

Arguments

x

Numeric vector of observations. All values must be in the open interval (0,1). Values outside this range will be automatically truncated to avoid numerical issues.

family

Character string specifying the distribution family. Valid options are: "gkw" (Generalized Kumaraswamy - 5 parameters), "bkw" (Beta-Kumaraswamy - 4 parameters), "kkw" (Kumaraswamy-Kumaraswamy - 4 parameters), "ekw" (Exponentiated Kumaraswamy - 3 parameters), "mc" (McDonald - 3 parameters), "kw" (Kumaraswamy - 2 parameters), "beta" (Beta - 2 parameters). The string is case-insensitive. Default is "gkw".

n_starts

Integer specifying the number of different initial parameter values to try during optimization. More starting points increase the probability of finding the global optimum at the cost of longer computation time. Default is 5.

Value

Named numeric vector containing the estimated parameters for the specified distribution family. Parameter names correspond to the distribution specification. If estimation fails, returns a vector of NA values with appropriate parameter names.

Details

The function uses the method of moments to estimate distribution parameters by minimizing the weighted sum of squared relative errors between theoretical and sample moments (orders 1 through 5). The optimization employs the Nelder-Mead simplex algorithm, which is derivative-free and particularly robust for this problem.

Key implementation features: logarithmic calculations for numerical stability, adaptive numerical integration using Simpson's rule with fallback to trapezoidal rule, multiple random starting points to avoid local minima, decreasing weights for higher-order moments (1.0, 0.8, 0.6, 0.4, 0.2), and automatic parameter constraint enforcement.

Parameter Constraints: All parameters are constrained to positive values. Additionally, family-specific constraints are enforced: alpha and beta in (0.1, 50.0), gamma in (0.1, 10.0) for GKw-related families or (0.1, 50.0) for Beta, delta in (0.01, 10.0), and lambda in (0.1, 20.0).

The function will issue warnings for empty input vectors, sample sizes less than 10 (unreliable estimation), or failure to find valid parameter estimates (returns defaults).

References

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

Examples

# \donttest{
# Generate sample data from Beta distribution
set.seed(123)
x <- rbeta(100, shape1 = 2, shape2 = 3)

# Estimate Beta parameters
params_beta <- gkwgetstartvalues(x, family = "beta")
print(params_beta)
#>    gamma    delta 
#> 2.421561 2.455624 

# Estimate Kumaraswamy parameters
params_kw <- gkwgetstartvalues(x, family = "kw")
print(params_kw)
#>    alpha     beta 
#> 1.972954 1.895691 

# Estimate GKw parameters with more starting points
params_gkw <- gkwgetstartvalues(x, family = "gkw", n_starts = 10)
print(params_gkw)
#>     alpha      beta     gamma     delta    lambda 
#> 1.2590661 3.1762475 1.9558583 0.0100000 0.9977211 
# }