The null hypothesis of the score tests is that in every vector
(columns of the matrix X), the probability of record at
time \(t\) is \(1/t\) as in the classical record model, and
the alternative depends on the alternative and probabilities
arguments. The probability at time \(t\) is any value, but equal in the
\(M\) series if probabilities = "equal" or different in the
\(M\) series if probabilities = "different". The alternative
hypothesis is more specific in the first case than in the second one.
Furthermore, the "two.sided" alternative is tested with
the usual Lagrange multiplier statistic, while the one-sided
alternatives use specific statistics based on scores.
(See Cebrián, Castillo-Mateo and Asín (2022) for details on these tests.)
If alternative = "two.sided" & probabilities = "equal", under the
null, the Lagrange multiplier statistic has an asymptotic \(\chi^2\)
distribution with \(T-1\) degrees of freedom. It has been seen that for
the approximation to be adequate \(M\) should be greater than \(T\).
Otherwise, a simulate.p.value can be computed.
If alternative = "two.sided" & probabilities = "different", the
asymptotic behaviour of the Lagrange multiplier statistic is not
fulfilled, but the Monte Carlo approach to simulate the p-value is
applied.
If alternative is one-sided and probabilities = "equal",
the statistic of the test is
$$\mathcal{T} = \sum_{t=2}^T \frac{(t S_t-M)^2}{M(t-1)} I_{\{S_t > M/t\}}.$$
The p-value of this test is estimated with Monte Carlo simulations,
since the compute the exact distribution of \(\mathcal{T}\) is very
expensive.
If alternative is one-sided and probabilities = "different",
the statistic of the test is
$$\mathcal{S} = \frac{\sum_{t=2}^T t (t S_t - M) / (t - 1)}{\sqrt{M \sum_{t=2}^T t^2 / (t - 1)}},$$
which is asymptotically standard normal distributed in \(M\). It is
equivalent to the statistic of the weighted number of records
N.test with weights \(\omega_t = t^2 / (t-1)\)
\((t=2,\ldots,T)\).