Cross Validated Asked by Paul M on December 29, 2020

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

- There is censoring (left-censored, fixed but varying censoring points which are known)
- 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

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