\[\mathcal{L}_{Y_{1:n}}(y_1, \dots, y_N) = \prod_{i\in R} f_X(y_i) \prod_{i\not\in R} F_X(y_i)
\label{eqn:cumJointLikelihood}\]
where \(R \subseteq \{1, \ldots, n \}\) is the set of indices where the record was broken and a new maximum was established.
Proof: See appendix. \(\square\)
Proposition 2. Let \(X_1, \dots, X_n\) be a collection of i.i.d. continuous random variables with common PDF \(f_X\) and CDF \(F_X\), and define \(Y_j := \min_{i\le j} X_i\). We assume that \(f_X\in C^1\). Then the joint likelihood for the sequence \(Y_1, \dots, Y_N\) is given by
\[\mathcal{L}_{Y_{1:n}}(y_1, \dots, y_N) = \prod_{i\in R} f_X(y_i) \prod_{i\not\in R} (1-F_X(y_i))
\label{eqn:cumJointLikelihood}\]
where \(R \subseteq \{1, \ldots, n \}\) is the set of indices where the record was broken and a new minimum was established.
Proof: Analogous to proposition 1. □