Let \(X_1, X_2, \dots , X_n\) be the lengths of \(n\) restriction fragments. Suppose that a biotechnique can measure fragment lengths accurately up to a given length c. That is, if \(X_i < c\), then the technique gives correct value \(X_i\) Show that MLE of \(\lambda = \frac{n-T_n}{S_n+cT_n}\) where \(S_n= \sum_{j=1}^n X_jI(X_j < c)\) and \(T_n = \sum_{j=1}^n I(X_j \geq c)\)

## Solution¶

This is another of my favorite problems that requires clever use of exponential variables. I also answered a similar question on Mathematics stackexchange

Few observations:

\(P(X_i < x) = 1 - \lambda e^{-x}\) if \( 0 \leq x < c\)

\(P(X_i = c) = P(X_i \geq c) = e^{-\lambda c}\) (This one looks counter intuitive at first look, but that is what the \(X_i=c\) if \(X_i \geq c\) returns)

Now,

\(

\(\log L= (n-T_n) \log \lambda -\lambda(S_n+cT_n)\)

\(\frac{\partial \log L}{\partial \lambda} = \frac{n-T_n}{\lambda}-(S_n+cT_n)\)

which gives \(\hat \lambda = \frac{n-T_n}{S_n+cT_n}\)