On the Tuesday the 13^th November, 2018 the WA Branch of the SSA met at Murdoch University to hear a talk by Professor Katsuto Tanaka from Gakushuin University which is a university in Tokyo held in high esteem not only because of people like Katsuto but as it is noted to be a university where the emperors children and grandchildren have studied. Statistics in Japan is well advanced as we know of the household names of statistics referred to such as Akaike, Ito, not to mention the influence of Demming in modernizing Japanese industry. It is therefore no mean feat that Katsuto has won the Japan Statistical Society Prize in 1998.  Katsuto is a visitor to Murdoch University and apart from flying commitments is here until the end of March 2019.  Katsuto has a long connection with Australia..he and his family were in Canberra while Katsuto was studying for his PhD under the supervision of Professor Ted Hannan at the Australian National University. At the end of his stay in Canberra I got to know him and we played some tennis together.  It has been a long time between drinks, so to speak, as Katsuto and I have only once met briefly in the intervening approximate 40 years, and our tennis capabilities are on the wane, yet we are now playing some social tennis while he is at Murdoch, now that exams have finished.

His presentation involved a talk around what is chapter one of his John Wiley book publication Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, 2nd edition, 2017.  The talk dealt with linear time series models on which stationarity or invertibility is not imposed,  and where nonstandard aspects of nonstationarity or noninvertibility were subsequently indicated. In particular, he demonstrated how to derive and compute asymptotic distributions of various statistics by using three approaches, which he called the eigenvalue, stochastic process, and Fredholm approaches.

He also noted the distribution of the maximum likelihood estimator when the moving average parameter is less than one in modulus and then when it is equal to one. He then discussed three approaches to inference on nonstandard statistics in the case of (c).  This unleashed a veritable ocean of mathematics, indeed in which Katsuto could clearly swim. Derivations involved weak convergence, Brownian motion, the Functional Central Limit Theorem, the Continuous Mapping Theorem, derivations of moment generating functions using an eigenvalue approach, and the stochastic approach using Girsanov’s Theorem. Then there was a discourse on the Fredholm Approach derived by considering a differential equation with boundary conditions followed by some properties of the Fredholm Determinent of the Kernel, some classic examples of which are statistics by Cramér von Mises, Lvy’s stochastic area, Anderson Darling, Watson’s Statistic.

Having solved the unit root problem he then tackled the case of the near unit root distribution which was then illustrated.  This was followed by a discussion of computation of limiting distributions via characteristic functions.  The Fredholm approach was then applied to other models involving trend and unit root MA(1) error structure and subsequent testing.  Spurious regression and also cointegration were briefly discussed. This led to a test of no cointegration and a discussion of panel data ensued.

In Katsuto’s concluding remarks  he pointed out that nonstationarity and noninvertibility of time series bring about nonstandard aspects of estimators and test statistics, like non-normality. The asymptotic distributions of nonstandard statistics are functionals of the Brownian motion, for which  presented the method for computing distribution functions.  He also elaborated on the Ornstein-Uhlenbeck process in his talk and alluded to the fractional O-U process involving fractional Brownian motion.

The audience was taken aback with the extent of the mathematics involved especially when it was relayed that this is in the first chapter of his book.  Several people joined Katsuto at the Malaysian-Chinese restaurant in the local suburb of Bateman after the talk.  

Brenton R Clarke

Dinner after the Meeting with from left to right: Berwin Turlach, John Henstridge, Katsuto Tanaka, Stephen Clarke (Brentons son),
Brenton Clarke, Nicola Armstrong, Devindri Perera, Barbara Kachigunda, and Mario D’Antuono. Photo courtesy of Katsuto’s camera.