Statistical Inference for Stochastic ProcessesAuthor(s): Mark BrownSource: Advances in Applied Probability, Vol. 5, No. 1 (Apr., 1973), pp. 6-8Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1425944 .

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to + oo the limit behaviour of Mn is similar to that of Sn, and we obtain a weak-convergence theorem to this effect, thereby extending earlier work by Heyde. When oscillation occurs, Problems I and II are substantially (though not entirely) equivalent, the connection being provided by the continuous map- ping theorem of weak convergence theory; again there are connections with earlier work by Heyde.

A complete solution to Problem II is obtained when X is spectrally negative, by means of results of Zolotarev and the author (this contains the usual solu- tion for the Wiener case). In general one cannot obtain even the one-dimen- sional distributions of Y explicitly, but we obtain complete asymptotic in- formation on Y and Z. We find explicitly the process X(Z) giving the place of first exit from an interval. We also discuss the method of ladder-points in this context, note some open problems, and discuss some connections with related work.

Statistical inference for stochastic processes

MARK BROWN, Cornell University

The purpose of the paper is to describe some techniques and results in the general area of statistical inference for stochastic processes.

Second order and Gaussian processes. Consider the model Y(t)= X(t) + m(t), t e T; X is a zero mean stochastic process with known covariance kernel R and m is a deterministic unknown function. It is sometimes assumed that X (and thus Y) is Gaussian. In a series of papers, [10], Parzen has developed an approach to statistical inference for this model based on reproducing kernels, and has obtained generalizations of several classical results derived for a finite index set.

Markov processes. Large sample theory for the parametric i.i.d. case is based on analysis of the Taylor series expansion of the log likelihood as a function of 0 in shrinking (with n) neighbourhoods of the true parameter. The classical results relate to the asymptotic efficiency of the maximum likeli- hood estimator and asymptotically optimal hypothesis tests based on maximum likelihood estimators. In an elegant monograph, [3], Billingsley has extended these results to discrete time Markov processes and continuous time countable state Markov processes by developing a machinery for handling the terms in the Taylor series. Billingsley gives many interesting applications to specific

to + oo the limit behaviour of Mn is similar to that of Sn, and we obtain a weak-convergence theorem to this effect, thereby extending earlier work by Heyde. When oscillation occurs, Problems I and II are substantially (though not entirely) equivalent, the connection being provided by the continuous map- ping theorem of weak convergence theory; again there are connections with earlier work by Heyde.

A complete solution to Problem II is obtained when X is spectrally negative, by means of results of Zolotarev and the author (this contains the usual solu- tion for the Wiener case). In general one cannot obtain even the one-dimen- sional distributions of Y explicitly, but we obtain complete asymptotic in- formation on Y and Z. We find explicitly the process X(Z) giving the place of first exit from an interval. We also discuss the method of ladder-points in this context, note some open problems, and discuss some connections with related work.

Statistical inference for stochastic processes

MARK BROWN, Cornell University

The purpose of the paper is to describe some techniques and results in the general area of statistical inference for stochastic processes.

Second order and Gaussian processes. Consider the model Y(t)= X(t) + m(t), t e T; X is a zero mean stochastic process with known covariance kernel R and m is a deterministic unknown function. It is sometimes assumed that X (and thus Y) is Gaussian. In a series of papers, [10], Parzen has developed an approach to statistical inference for this model based on reproducing kernels, and has obtained generalizations of several classical results derived for a finite index set.

Markov processes. Large sample theory for the parametric i.i.d. case is based on analysis of the Taylor series expansion of the log likelihood as a function of 0 in shrinking (with n) neighbourhoods of the true parameter. The classical results relate to the asymptotic efficiency of the maximum likeli- hood estimator and asymptotically optimal hypothesis tests based on maximum likelihood estimators. In an elegant monograph, [3], Billingsley has extended these results to discrete time Markov processes and continuous time countable state Markov processes by developing a machinery for handling the terms in the Taylor series. Billingsley gives many interesting applications to specific

6 6 MARK BROWN MARK BROWN

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Statistical inference for stochastic processes

Markov chains. In particular the chi-square goodness of fit test which tests whether a submodel based on fewer parameters than the assumed model is reasonable, seems to have significant potential for applications. Billingsley's regularity conditions can be weakened by following an approach developed by LeCam, [8], and Hajek, [7], (see Roussas and Johnson, [11]).

General large sample theory. LeCam, [8], has developed an approach to large sample theory which in principal can be applied in problems in which Radon-Nikodym derivatives are computable. His conditions involve the local behaviour of the log likelihood. In addition he has demonstrated that in some cases the verification of his conditions need not involve existence of more than one derivative. This was also noted by Hajek, [7]. Weiss and Wolfowitz have developed an approach to large sample theory based on maximum probability estimators. They define a notion of efficiency which is stronger than the classical and show that maximum probability estimators have this efficiency property. Their approach is to demonstrate that maximum prob- ability estimators are asymptotically Bayes under a sequence of uniform priors in shrinking neighbourhoods of the true parameter. Their results and conditions apply also to non i.i.d. cases. Their work which appears in several papers will be published in a research monograph [12].

Point processes. Many interesting procedures for statistical inference for point processes can be found in the monograph of Cox and Lewis, [4]. Also see the recent paper of Cox and Lewis on multiple processes, [5], and papers of Cox, Lewis and others in [9]. Some problems of interest in this area have been:

(i) inference on the intensity of a homogeneous Poisson process ([4] Chapter 2);

(ii) testing a Poisson process for homogeneity against parametric and non-parametric alternatives ([4] Chapter 3);

(iii) testing whether a renewal process with increasing failure rate is Poisson ([1] p. 232-236, [2]);

(iv) spectral analysis of stationary point processes ([4] Chapter 5); (v) doubly stochastic Poisson processes ([6]). In this paper we have not discussed time-series analysis. Several books

on various levels are available in this field.

References [1] BARLOW, R. E. AND PROSCHAN, F. (1965) Mathematical Theory of Reliability. John

Wiley, New York. [2] BICKEL, P. J. AND DOKSUM, K. S. (1969) Tests for monotone failure rate based on

normalized spacings. Ann. Math. Statist. 40, 1216-1235. [3] BILLINGSLEY, P. (1961) Statistical Inference for Markov Processes. University of

Chicago Press.

7

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[4] Cox, D. R. AND LEWIS, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.

[5] Cox, D. R. AND LEWIS, P. A. W. (1972) Multivariate point processes, Proc. 6th Berkeley Symposium Math. Statist. Prob.

[6] GRANDELL, J. (1972) Statistical inference for doubly stochastic Poisson processes. [9] (below) 67-90.

[7] HAJEK, J. (1962) Asymptotically most powerful rank order tests. Ann. Math. Statist. 33, 1124-1147.

[8] LECAM, L. (1960) Locally Asymptotically Normal Families of Distribution. University of California Press.

[9] LEWIS, P. A. W. (1972) Stochastic Point Processes. John Wiley, New York. [10] PARZEN, E. (1967) Time Series Analysis Papers. Holden Day, San Francisco. [11] RoussAS, G. G. AND JOHNSON, R. A. (1969) Asymptotically most powerful tests in

Markov processes. Ann. Math. Statist. 40, 1207-1215. [12] WEISS, L. AND WOLFOWITZ, J. Asymptotic Methods in Statistics. In preparation.

Asymptotic relations in queueing theory

J. W. COHEN, Technological University, Delft

It may be said that until the late fifties research in Queueing Theory was mainly directed to the derivation of analytic expressions for the distributions of quantities like the waiting time, the queue length, the busy period and so on, for a great variety of queueing models. Very often the investigator had to be satisfied with the Laplace-Stieltjes transform or generating function of the distributions he intended to find. Quite often indeed the researcher could be proud with the results, since they could be only obtained at the cost of hard labour and much ingenuity. It was certainly not unjustified that Kendall in his 1963 review of queueing theory made mention of the Laplacian curtain in queueing theoretic results, although even at that time the curtain had many (la)places with holes to peep through.

The appearance of Kingman's paper in 1961 may be considered as a turning point; from that time on asymptotic methods played an important role in research of queueing theory and provided a large number of applicable results.

Queueing theory offers a large field for the application of asymptotic methods. We mention here:

(i) asymptotics with respect to the time parameter for t -s oo, and the

speed of convergence (the relaxation time); (ii) asymptotics related to the nearly saturated system, e.g., for the single

server model, traffic intensity approaching 1, heavy traffic theory; (iii) models with restricted accessibility if the restriction becomes weak,

e.g., models with finite number of waiting places K if K becomes large; (iv) the approximation of the actual process by a diffusion process;

[4] Cox, D. R. AND LEWIS, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.

[5] Cox, D. R. AND LEWIS, P. A. W. (1972) Multivariate point processes, Proc. 6th Berkeley Symposium Math. Statist. Prob.

[6] GRANDELL, J. (1972) Statistical inference for doubly stochastic Poisson processes. [9] (below) 67-90.

[7] HAJEK, J. (1962) Asymptotically most powerful rank order tests. Ann. Math. Statist. 33, 1124-1147.

[8] LECAM, L. (1960) Locally Asymptotically Normal Families of Distribution. University of California Press.

[9] LEWIS, P. A. W. (1972) Stochastic Point Processes. John Wiley, New York. [10] PARZEN, E. (1967) Time Series Analysis Papers. Holden Day, San Francisco. [11] RoussAS, G. G. AND JOHNSON, R. A. (1969) Asymptotically most powerful tests in

Markov processes. Ann. Math. Statist. 40, 1207-1215. [12] WEISS, L. AND WOLFOWITZ, J. Asymptotic Methods in Statistics. In preparation.

Asymptotic relations in queueing theory

J. W. COHEN, Technological University, Delft

It may be said that until the late fifties research in Queueing Theory was mainly directed to the derivation of analytic expressions for the distributions of quantities like the waiting time, the queue length, the busy period and so on, for a great variety of queueing models. Very often the investigator had to be satisfied with the Laplace-Stieltjes transform or generating function of the distributions he intended to find. Quite often indeed the researcher could be proud with the results, since they could be only obtained at the cost of hard labour and much ingenuity. It was certainly not unjustified that Kendall in his 1963 review of queueing theory made mention of the Laplacian curtain in queueing theoretic results, although even at that time the curtain had many (la)places with holes to peep through.

The appearance of Kingman's paper in 1961 may be considered as a turning point; from that time on asymptotic methods played an important role in research of queueing theory and provided a large number of applicable results.

Queueing theory offers a large field for the application of asymptotic methods. We mention here:

(i) asymptotics with respect to the time parameter for t -s oo, and the

speed of convergence (the relaxation time); (ii) asymptotics related to the nearly saturated system, e.g., for the single

server model, traffic intensity approaching 1, heavy traffic theory; (iii) models with restricted accessibility if the restriction becomes weak,

e.g., models with finite number of waiting places K if K becomes large; (iv) the approximation of the actual process by a diffusion process;

8 8 J. W. COHEN J. W. COHEN

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