R Under development (unstable) (2023-12-07 r85661 ucrt) -- "Unsuffered Consequences"
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> library(bbmle)
Loading required package: stats4
> x <- 0:10
> y <- c(26, 17, 13, 12, 20, 5, 9, 8, 5, 4, 8)
> d <- data.frame(x,y)
> LL <- function(ymax=15, xhalf=6)
+     -sum(stats::dpois(y, lambda=ymax/(1+x/xhalf), log=TRUE))
> options(digits=3)
> mfit0 <- mle2(y~dpois(lambda=exp(interc)),
+               start=list(interc=log(mean(y))),data=d)
> mfit1 <- mle2(y~dpois(lambda=exp(loglambda)),
+               start=list(loglambda=log(mean(y))),data=d)
> 
> coef(mfit0)
interc 
  2.45 
> residuals(mfit0)
 [1]  4.254  1.605  0.428  0.134  2.488 -1.926 -0.749 -1.043 -1.926 -2.221
[11] -1.043
> AIC(mfit0)
[1] 87.5
> BIC(mfit0)
[1] 87.9
> vcov(mfit0)
        interc
interc 0.00787
> ## fitted(mfit0)  ## fails, looks for default value
> predict(mfit0)  ## FIXME: doesn't expand properly (need implicit lambda~1 formula??)
[1] 11.5
> set.seed(1001)
> simulate(mfit0)
 [1] 18 10 10 15  7  9 10 14 10 13 15
> anova(mfit0,mfit1)
Likelihood Ratio Tests
Model 1: mfit0, y~dpois(lambda=exp(interc))
Model 2: mfit1, y~dpois(lambda=exp(loglambda))
  Tot Df Deviance Chisq Df Pr(>Chisq)
1      1     85.5                    
2      1     85.5     0  0          1
> summary(mfit0)
Maximum likelihood estimation

Call:
mle2(minuslogl = y ~ dpois(lambda = exp(interc)), start = list(interc = log(mean(y))), 
    data = d)

Coefficients:
       Estimate Std. Error z value  Pr(z)    
interc   2.4463     0.0887    27.6 <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

-2 log L: 85.5 
> summary(mfit1)
Maximum likelihood estimation

Call:
mle2(minuslogl = y ~ dpois(lambda = exp(loglambda)), start = list(loglambda = log(mean(y))), 
    data = d)

Coefficients:
          Estimate Std. Error z value  Pr(z)    
loglambda   2.4463     0.0887    27.6 <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

-2 log L: 85.5 
> 
> proc.time()
   user  system elapsed 
   1.37    0.21    1.53