Communication: Reply to Fox and Schruben
In the recent issue of Management Science Fox Fox, B. 1978. Estimation and simulation. Management Sci.24 8, April 860--861. expressed doubts about the relevance of steady-state behavior. In a reply Schruben Schruben, L. W. 1978. Reply to Fox. Management Sci.24 8, April 862. disagreed on this point. With Fox he was in favor of publishing failures of variance reduction techniques VRT's. I would like to add the following remarks.
On the Choice of Alternative Measures in Importance Sampling with Markov Chains
In the simulation of Markov chains, importance sampling involves replacing the original transition matrix, say P, with a suitably chosen transition matrix Q that tends to visit the states of interest more frequently. The likelihood ratio of P relative to Q is an important random variable in the importance sampling method. It always has expectation one, and for any interesting pair of transition matrices P and Q, there is a sample path length that causes the likelihood ratio to be close to zero with probability close to one. This may cause the variance of the importance sampling estimator to be larger than the variance of the traditional estimator. We develop sufficient conditions for ensuring the tightness of the distribution of the logarithm of the likelihood ratio for all sample path lengths, and we show that when these conditions are satisfied, the likelihood ratio is approximately lognormally distributed with expected value one. These conditions can be used to eliminate some choices of the alternative transition matrix Q that are likely to result in a variance increase. We also show that if the likelihood ratio is to remain well behaved for all sample path lengths, the alternative transition matrix Q has to approach the original transition matrix P as the sample path length increases. The practical significance of this result is that importance sampling can be difficult to apply successfully in simulations that involve long sample paths.
Lake Muzahi, Rwanda: limnological features and phytoplankton production
Lake Muhazi, a small lake of Rwanda (East Africa) was studied from 1986 to 1990. A dramatic decrease of the catch of Oreochromis niloticus (350 T y−1 in the fifties vs 30 T y−1 in 1982) suggested a loss of productivity or overfishing. In the same period, other ecological changes occurred: the submerged macrophytes regressed and there was a decrease in Secchi depth (0.65 m in 1987 vs 1.5 m in the fifties). Compared to other lakes of the same area, the plankton production seemed low. The results of the present study characterize lake Muhazi as a shallow lake with a rather unstable diurnal stratification and with slight differences in mixing regime between its eastern, deepest part and its western, shallowest part. Secchi disk depth does not vary seasonally to a large extent. The water has a rather high mineral content (conductivity of about 500 µS cm−1 at 25 °C) and low concentrations of dissolved N and P, except in the hypolimnion, where NHinf4sup+-N can be high.Two species, Microcystis aeruginosa and Ceratium hirundinella, account for most of the phytoplankton biomass, which is about 50–80 mg chlorophyll a m−2 in the euphotic zone, usually with little seasonal variation. Daily gross production estimates amount to about 6 to 9.5 g O2 m−2 d−1 with a significant difference between the two parts of the lake. Data on C:N and C:P ratio in the phytoplankton suggest that some N deficiency might occur in the eastern part. Moreover, the Zm:Zc ratio could also lead to rather low net production rates (0.21–0.25 d−1 for a mixed layer of 4 m)In conclusion, the primary production of lake Muhazi is medium for African lakes and the hypothesis that decreased planktonic production could account for a reduced fish production should be discarded. Whereas the present yield of the fishery is only 20 kg ha−1 y−1, the yield estimated from primary production ranges between 46 and 64 kg ha−1 y−1. This could be reached through proper management. Finally, some hypotheses are given to explain the ecological changes which occurred in the lake.