How to estimate mean and standard deviation of a normal distribution from noisy data?

Question

I have $n$ observations, $x_i$ following a normal distribution. I would like to estimate $\mu$ and $\sigma$ from my samples. Normally I would simply estimate

$\mu=(\sum x_i)/n$ and
$\sigma^2=\sum (x_i-\mu)^2/(n-1)$.

However I know that my samples $x_i$ are noisy. I know that the 95% confidence interval of

$x_i$ is $[x_i-t, x_i+t]$

with the same $t$ for all $x_i$. How do I take the noise into account when estimating $\sigma$?

If I don’t take the confidence interval into account the estimation of $\sigma$ might be too large.

Answer 1

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