# Weighted normal errors regression with censoring

Cross Validated Asked by Paul M on December 29, 2020

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)?

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 $$f(x:mu) = frac{sqrt{w_i}}{sqrt{2pi}} e^{-frac12 w_i (x_i-mu)^2}$$ 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 $$l(mu) = -frac12sum_1^r w_i (x_i-mu)^2 + sum_{r+1}^n log Phi(sqrt{w_i}(t_i-mu))$$ 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 $mu$.

Answered by kjetil b halvorsen on December 29, 2020