However, to use all the data available to us in the inference we need to derive the joint likelihood of all the cumulative records:
\[f_{Y_{1:n}}(y_1, \ldots, y_n | \theta)\]where \(f_{Y_{1:n}}(\cdot | \theta)\) denotes the joint density of \(Y_1, \ldots, Y_n\), given parameters \(\theta\). Since \[\max \left\{X_1, \ldots, X_j, X_{j+1}\right\} = \max \left\{X_j, X_{j+1}\right\}\]for any \(j\), we conclude that each of the \(Y_i\) are independent of all of the previous except for \(Y_{i-1}\). The likelihood function must then factorize like\[f_{Y_{1:n}}(y_1,\ldots,y_n|\theta)=f_{Y_1}(y_1|\theta)\prod_{j=1}^{n-1}f_{Y_{j+1}|Y_j=y_j}(y_{j+1}|\theta).\]
Now we are ready to derive the likelihood function .
Proposition 1. 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 \(Y_j := \max_{i\le j} X_i\). We assume that \(f_X\in C^1\). Then the joint likelihood for the sequence \(Y_1, \dots, Y_N\) is given by