We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.

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Section (1) is devoted to a discussion of the model-fitting problem, which finds its explicit solution in equation (1.13). In section (2) the maximum likelihood, (ML), estimates of the model parameters are investigated, and for the class of series considered shown to possess the same optimum properties as in the case of independent series. Next, the covariance matrix of the parameter estimates is expressed in terms of the spectral function of the generating process (eq. 3.7). The last section is concerned with certain working approximations to the ML statistics.(1) The ultimate objects of any time series analysis are rarely more than two in number: firstly, to estimate, for its own interest, the stochastic relation generating the terms of the series, and secondly, to obtain a forecast by the use of this relation. If the spectrum of the process is known, then both of these problems may be solved by existing methods, under the assumption that the stochastic relation is a linear one (refs. 3, 10). Thus, if we limit ourselves to the case of linearity, the analysis is reduced to the estimation of the spectrum. The word "estimation" is here used in a fairly wide sense, since we require to estimate a function, not merely ~ finite set of parameters. In general we must specify a definite kind of function, and it is this necessity which compels the analyst to use some sort of test or decision procedure. However, we shall consider this aspect of the problem only in passing, since it has already been treated in ref. 9.Suppse, then, that we have a time series of N equidistant observations; xl, x2 .... XN, forming an N • 1 vector X. We shall assume that these observations constitute a part realisation of a real, discrete, stationary process, and our aim is then to obtain the best possible estimate of this process. As above, the generating process will for our purposes be considered as determined when we known its spectrum, F(y), defined by (see ref. 2) (1.1) If the spectrum is differentiable, we may define the spectral ]unction A (z) by A (e ~v) OF (y) _ ~ vs e isv.

The conventional linear/quadratic/Gaussian assumptions are modified in that minimisation of the expectation of cost G defined by (2) is replaced by minimisation of the criterion function (5). The scalar –θ is a measure of risk-aversion. It is shown that modified versions of certainty equivalence and the separation theorem still hold, that optimal control is still linear Markov, and state estimate generated by a version of the Kalman filter. There are also various new features, remarked upon in Sections 5 and 7. The paper generalises earlier work of Jacobson.

Summary A plausible conjecture (C) has the implication that a relationship (12) holds between the maximal expected rewards for a multi‐project process and for a one‐project process (F and φi respectively), if the option of retirement with reward M is available. The validity of this relation and optimality of Gittins' index rule are verified simultaneously by dynamic programming methods. These results are partially extended to the case of so‐called “bandit superprocesses”.

We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.

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