Maximum Likelihood – Fitting Left Truncated Weibull Distribution

Question

I want to fit some samples to the right tail of a Weibull distribution. To fix the notation:

• the samples are $\{X_i\}_{i=1,\ldots,n}$,

• all samples are greater than a fixed threshold $L$,

• the $X_i$'s are distributed according to a Weibull distribution left truncated at $L$:
  $$
  X_i \sim F(x)=1-\mathrm e ^{-\frac{x^\alpha-L^\alpha}{\beta^\alpha}}.
  $$

For simplicity in the following I assume $L=1$ and replace $X_i$ with $Y_i=X_i/L$. I want to estimate $\alpha$ and $\beta$ with MLE. The log-likelihood function is:
$$
\ell(\alpha,\beta;Y_i) = \sum_i\left(-\frac{Y_i^\alpha-1}{\beta^\alpha}+\log\alpha-\alpha\log\beta+(\alpha-1)\log Y_i\right).
$$

By setting $\partial\ell/\partial\beta = 0$ I obtain
$$
\beta^\alpha = \frac{\sum_i (Y_i^\alpha-1)}{n}.
$$

Now to find $\alpha$ I can proceed in two different ways. I can set $\partial\ell/\partial\alpha = 0$ and then substitute $\beta$ in the resulting equation. This way I get:
$$
0= \frac{1}{\alpha} + \frac{\sum_i\log Y_i}n – \frac{\sum_i Y_i^\alpha\log Y_i}{\sum_i (Y_i^\alpha-1)}.
$$

I can also define $\tilde\ell(\alpha)$ by substituting the expression for $\beta$ in $\ell$:
$$
\tilde\ell(\alpha)=-n+n\log\alpha-n\log\left(\frac{\sum_i (Y_i^\alpha-1)}{n}\right)
+(\alpha-1)\frac{\sum_i\log Y_i}n,
$$
and setting $\partial\tilde\ell(\alpha)/\partial\alpha=0$, reaching the same result as above.

The equation for $\alpha$ can then be solved numerically.

EDIT I found a couple of mistakes in the derivations. Now everything works as expected.

Leaving the previous text below:

Now I have two problems:

Problem 1. The two equations for $\alpha$ are different and not equivalent. I suspect this is one or more calculation mistake on my
side, but I can't find it.

Problem 2. When I apply the formulas to some simulated data, the estimates for $\alpha$ are wildly wrong.

Answer 1

Without reading through your derivation, I imagine there was an issue with your implementation of the optimization.

If you wanted to implement it numerically, here's a quick example (in R).

set.seed(10101)

n = 1000
a = 3
b = 0.1

x = rweibull(n, a, b)
L = quantile(x, 0.2)

x = x[x>=L]

lik = function(pars, x, L) {
  
  n = length(x)
  
  -(n*(log(pars[1]) - log(pars[2])) + (pars[1]-1)*sum(log(x/pars[2])) - sum((x/pars[2])^pars[1]) + n*(L/pars[2])^pars[1])
  
}

o = optim(par = c(1, 2),
          fn = lik,
          method = "BFGS",
          L = L,
          x = x)

o$par