Partially answered in comments:
Is this not the prototypical ANOVA (components of variance) problem? Do you have two replicate measurement for each x ("balanced" data) or have you replicated only one of them ("unbalanced" data)? – whuber
Since the OP didn't respond to the implicit questions by whuber, but I can show (with simulated data) how this could be done in R
when there are replicate measurements for each $X_i$:
library(lme4)
set.seed(7*11*13) # my public seed
X <- 1:10
r <- 3
mu <- rep(X,rep(3,length(X)))
obs <- rnorm(length(mu),mu,1)
ind <- as.factor(mu)
mydata <- data.frame(obs, mu, ind)
mod <- lmer(obs ~ 1 + ind + (1 | ind), data=mydata )
mod
Linear mixed model fit by REML ['lmerMod']
Formula: obs ~ 1 + ind + (1 | ind)
Data: mydata
REML criterion at convergence: 74.71
Random effects:
Groups Name Std.Dev.
ind (Intercept) 3.14
Residual 1.19
Number of obs: 30, groups: ind, 10
Fixed Effects:
(Intercept) ind2 ind3 ind4 ind5 ind6
1.6086 -0.6777 1.2451 2.5058 3.8129 4.7079
ind7 ind8 ind9 ind10
5.1083 7.2207 6.7395 7.7469
If you really know the true values for your specimens (as measured by some much more precise instrument, say) you can do:
mod2 <- lmer(obs ~ 0 + offset(mu) + (1 | ind), data=mydata )
mod2
Linear mixed model fit by maximum likelihood ['lmerMod']
Formula: obs ~ 0 + offset(mu) + (1 | ind)
Data: mydata
AIC BIC logLik deviance df.resid
96.7257 99.5281 -46.3629 92.7257 28
Random effects:
Groups Name Std.Dev.
ind (Intercept) 0.000
Residual 1.135
Number of obs: 30, groups: ind, 10
No fixed effect coefficients