A plethora of high frequency data is available to study from major electronic trading systems. Here we study the market microstructure through the dynamics of a limit order book (LOB). For a LOB, it is common to model waiting times until certain events occur. We have NASDAQ OMS data, which motivates us to study the behavior of times in between events instead. Patterns observed in the data suggest an invisible switch, which controls the change in speed of limit order executions. This leads to testing the number of different speeds of executions. In particular, do the times between events follow a mixture of distributions or not? This is the Mixture Test.

It is a LRT test, however it has some idiosyncrasies. Since Hartigan (1985aa) we know that the conventional asymptotic theory has difficulties when the parameter is on the boundary of the parameter space and an issue of nonidentifiability of parameters under the null hypothesis. The former is addressed by considering the sign of the mixing probabilities. The latter is solved by deriving the limiting distribution of the LRT for a fixed but unknown value of the unidentifiable parameter. Cho and White (2007) show that the resulting LRT statistic follows a Gaussian process with a specific covariance structure.

We derive and simulate the covariance structure of a limiting distribution of LRT statistic for exponential family distributions. We propose to simulate an approximated Gaussian process using the finite number of principal components of a covariance matrix. From this we obtain the asymptotic critical values for a bounded parameter space. Details of an EM algorithm for estimating the parameters are outlined.

The methods above are applied to conduct the Mixture Test for the LOB data. The first passage time of a BM process follows the Inverse Gaussian distribution. This makes an intuitive choice to describe the time interval between two limit order executions. These results provide significant evidence in favor of a two component Inverse Gaussian mixture model. In conclusion, we develop a collection of the methods to obtain asymptotic critical values and provide a practitioner with a tool to conduct Mixture Tests.