# Truncated Weibull Distribution Family Function Maximum likelihood estimation of the 2-parameter Weibull distribution with lower truncation. No observations should be censored. ```r truncweibull(lower.limit = 1e-5, lAlpha = "loglink", lBetaa = "loglink", iAlpha = NULL, iBetaa = NULL, nrfs = 1, probs.y = c(0.2, 0.5, 0.8), imethod = 1, zero = "Betaa") ``` ## Arguments - `lower.limit`: Positive lower truncation limits. Recycled to the same dimension as the response, going across rows first. The default, being close to 0, should mean effectively the same results as `weibullR` if there are no response values that are smaller. - `lAlpha, lBetaa`: Parameter link functions applied to the (positive) parameters `Alpha` (called $\alpha$ below) and (positive) `Betaa` (called $\beta$ below). See `Links` for more choices. - `iAlpha, iBetaa`: See `CommonVGAMffArguments`. - `imethod, nrfs, zero, probs.y`: Details at `weibullR` and `CommonVGAMffArguments`. ## Details MLE of the two parameters of the Weibull distribution are computed, subject to lower truncation. That is, all response values are greater than `lower.limit`, element-wise. For a particular observation this is any known positive value. This function is currently based directly on Wingo (1989) and his parameterization is used (it differs from `weibullR`.) In particular, $\beta = a$ and $\alpha = (1/b)^a$ where $a$ and $b$ are as in `weibullR` and `dweibull`. Upon fitting the `extra` slot has a component called `lower.limit` which is of the same dimension as the response. The fitted values are the mean, which are computed using `pgamma.deriv` and `pgamma.deriv.unscaled`. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`. ## References Wingo, D. R. (1989). The left-truncated Weibull distribution: theory and computation. **Statistical Papers**, 30 (1), 39--48. ## Author(s) T. W. Yee ## Note More improvements need to be made, e.g., initial values are currently based on no truncation. This `VGAM` family function handles multiple responses. ## Warning This function may be converted to the same parameterization as `weibullR` at any time. Yet to do: one element of the EIM may be wrong (due to two interpretations of a formula; but it seems to work). Convergence is slower than usual and this may imply something is wrong; use argument `maxit`. In fact, it's probably because `pgamma.deriv.unscaled` is inaccurate at `q = 1` and `q = 2`. Also, convergence should be monitored, especially if the truncation means that a large proportion of the data is lost compared to an ordinary Weibull distribution. ## See Also `weibullR`, `dweibull`, `pgamma.deriv`, `pgamma.deriv.unscaled`. ## Examples ```r ## Not run: nn <- 5000; prop.lost <- 0.40 # Proportion lost to truncation wdata <- data.frame(x2 = runif(nn)) # Complete Weibull data wdata <- transform(wdata, Betaa = exp(1)) # > 2 okay (satisfies regularity conds) wdata <- transform(wdata, Alpha = exp(0.5 - 1 * x2)) wdata <- transform(wdata, Shape = Betaa, # aaa = Betaa, # bbb = 1 / Alpha^(1 / Betaa), Scale = 1 / Alpha^(1 / Betaa)) wdata <- transform(wdata, y2 = rweibull(nn, Shape, scale = Scale)) summary(wdata) # Proportion lost: lower.limit2 <- with(wdata, quantile(y2, prob = prop.lost)) # Smaller due to truncation: wdata <- subset(wdata, y2 > lower.limit2) fit1 <- vglm(y2 ~ x2, maxit = 100, trace = TRUE, truncweibull(lower.limit = lower.limit2), wdata) coef(fit1, matrix = TRUE) summary(fit1) vcov(fit1) head(fit1@extra$lower.limit) ## End(Not run) ```

truncweibull function