带(x-location)/scale>0约束的MLE优化：L-BFGS-B报错求解

Let's break down why you're hitting this error and fix it step by step.

The Root Cause #

Your initial parameter c(100, 1000, 10) violates the critical constraint (x - location)/scale > 0:

• Your generated x values come from a Pearson III distribution with location=6 and positive scale=7, so all x are slightly larger than 6.
• With location=1000 and positive scale=10, x - location is negative for every data point. This makes ssgn*(x-location) negative, and dgamma() returns NaN when given negative inputs.
• Taking log(NaN) gives NaN, so your negative log-likelihood (tmp) becomes NaN, which your code converts to +Inf. L-BFGS-B can't handle non-finite function values, hence the error.

Step-by-Step Fixes #

1. Guard Against Invalid Parameters in the Log-Likelihood Function

Modify your LL3 function to check constraints before computing the density. If parameters violate the constraint, return a large finite value (instead of Inf) to guide the optimizer away from bad regions:

LL3 <- function(theta, x, display) {
  shape <- as.numeric(theta[1])
  location <- as.numeric(theta[2])
  scale <- as.numeric(theta[3])
  
  # Check shape is positive (required for gamma distribution)
  if (shape <= 1e-6) {
    return(1e10)
  }
  
  # Enforce (x - location)/scale > 0 for all x
  if (scale > 0) {
    # All x must be > location → location < min(x)
    if (location >= min(x)) {
      return(1e10)
    }
  } else if (scale < 0) {
    # All x must be < location → location > max(x)
    if (location <= max(x)) {
      return(1e10)
    }
  } else {
    # Scale can't be zero
    return(1e10)
  }
  
  # Compute negative log-likelihood only if parameters are valid
  dens_vals <- dpearson3(x, shape, location, scale, log = FALSE)
  tmp <- -sum(log(dens_vals))
  
  # Fallback for any unexpected NaNs
  tmp <- ifelse(is.na(tmp), 1e10, tmp)
  
  if(display == 1){print(c(tmp, theta))}
  return(tmp)
}

Note: I removed the redundant sum(tmp) since tmp is already a sum of log densities.

2. Tighten Parameter Bounds for L-BFGS-B

Since your data was generated with a positive scale, we can restrict the optimizer to only consider positive scales and valid locations. This reduces the chance of it wandering into invalid regions:

x_min <- min(x)
x_max <- max(x)

# Set bounds: shape > 0, location < min(x), scale > 0
lower <- c(1e-6, -Inf, 1e-6)
upper <- c(Inf, x_min - 1e-6, Inf)

3. Use a Reasonable Initial Parameter

Your original starting point was way outside the valid region. Pick initial values that satisfy the constraint:

# Start with values close to the true parameters (or at least valid)
param <- c(5, 5, 5)

4. Run the Optimizer

Now your optim call should work without errors:

control.list <- list(maxit = 100000, factr=1e-12, fnscale = 1)
fit <- optim(par = param, fn = LL3, hessian = TRUE, method = "L-BFGS-B", 
             lower = lower, upper = upper, control = control.list, x = x, display = 1)

# Check the results
fit$par

You should get estimates close to the true values shape=5, location=6, scale=7.

Why This Works #

• The pre-checks in LL3 prevent the optimizer from ever computing non-finite values.
• Tighter bounds guide L-BFGS-B to focus only on valid parameter space.
• A valid initial start ensures the optimizer begins in a region where the log-likelihood is finite.

内容的提问来源于stack exchange，提问作者Andrew