# Weighted normal errors regression with censoring

I have some data which I would model via standard multiple regression except:

1. There is censoring (left-censored, fixed but varying censoring points which are known)
2. The errors are assumed independent normal but of non-constant variance. Weights are available.

If it was constant variance, I would use the Tobit model and survreg() function in R. Does anyone know of the/an approach when the variance is not constant (but weights for variances are available)?

• Does the solution need to be in R? – Dimitriy V. Masterov Feb 20 '14 at 1:38
• See the R survival package survreg function. – Frank Harrell May 1 '17 at 12:14

There must be some weight arguments to the survreg function? Anyhow, this can be solved by setting up a likelihood function from first principles.
You have a normal model (with independent observations) and known weight, the optimal weights are the inverse variances, so write the weight as $w_i$ taken to be the inverse of known variances. Then we can write the density as $$f(x:\mu) = \frac{\sqrt{w_i}}{\sqrt{2\pi}} e^{-\frac12 w_i (x_i-\mu)^2}$$ Assume the censoring points are at $t_i$, the first $r$ obs are fully observed and observations $r+1 \dotsc n$ censored. Then the likelihood becomes $$L(\mu) = \prod_1^r f(x_i; \mu) \prod_{r+1}^n \Phi(\sqrt{w_i}(t_i-\mu))$$ where $\Phi$ denotes the standard normal cdf. The loglikelihood becomes $$l(\mu) = -\frac12\sum_1^r w_i (x_i-\mu)^2 + \sum_{r+1}^n \log \Phi(\sqrt{w_i}(t_i-\mu))$$ where we have left out some terms not influencing the shape of the loglikelihood function. Now this function can be sent to a numerical optimization routine to find the maximum likelihood estimator of $\mu$.