Celebratio Mathematica

M. Rosenblatt: “Density estimates and Markov sequences,” pp. 199–213 in Nonparametric techniques in statistical inference (Bloomington, IN, 1–6 June 1969). Edited by M. L. Puri. Cambridge University Press (London), 1970. MR 278468 incollection

Estimates of the density function of a population based on a sample of independent observations have been considered in a number of papers [Bartlett 1963; Parzen 1962; Rosenblatt 1956]. Questions of bias, variance and asymptotic distribution of the estimates have been dealt with at greatest length. Our object is to look at such estimates of the density function when the observations are dependent. The results will not be dealt with in the most general context or under very general conditions. To obtain results in a simple and readily understandable form, the observations are assumed to be sampled from a stationary Markov sequence with a fairly strong condition on the Markov transition operator. However, the extent to which some of the conditions can be obviously relaxed will be indicated.

It should be noted that the asymptotic results we obtain in the case of dependent observations are essentially the same as those in the case of independent observations. This initially is surprising because it is certainly not true when estimating the distribution function by means of the sample distribution function. The more complicated nature of asymptotic results for this problem in the case of dependence can be seen in [Billingsley 1968]. However, the happy fact that the results we obtain in estimating the density have the same character as in the case of independence is due to the local character of the estimates.

K.-S. Lii and M. Rosenblatt: “Asymptotic results on a spline estimate of a probability density,” pp. 77–85 in Statistical inference and related topics (Bloomington, IN, 31 July–9 August 1974), vol. 2. Edited by M. L. Puri. Academic Press (New York), 1975. Volume dedicated to Z. W. Birnbaum. MR 373139 Zbl 0341.62033 incollection

Zygmunt William Birnbaum	Related
Madan Lal Puri	Related
Keh-Shin Lii	Related

M. Bhattacharjee, R. Lockhart, J. Rolph, G. Roussas, H. Tucker, R. J.-B. Wets, P. Bickel, T. S. Ferguson, A. Lo, M. L. Puri, S. Stigler, W. Sudderth, Y. Yatracos, D. Brillinger, L. A. Goodman, J. Shaffer, H. Chernoff, P. Diaconis, M. Rosenblatt, and F. J. Samaniego: “A tribute to David Blackwell.” Edited by G. G. Roussas. Notices Am. Math. Soc. 58 : 7 (2011), pp. 912–928. MR 2850553 Zbl 1225.01082 article

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Francisco J. Samaniego	Related
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William D. Sudderth	Related
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@article {key2850553m,
	AUTHOR = {Bhattacharjee, Manish and Lockhart,
		 Richard and Rolph, John and Roussas,
		 George and Tucker, Howard and Wets,
		 Roger J.-B. and Bickel, Peter and Ferguson,
		 Thomas S. and Lo, Albert and Puri, Madan
		 L. and Stigler, Stephen and Sudderth,
		 W. and Yatracos, Yannis and Brillinger,
		 David and Goodman, Leo A. and Shaffer,
		 Juliet and Chernoff, Herman and Diaconis,
		 Persi and Rosenblatt, Murray and Samaniego,
		 Francisco J.},
	TITLE = {A tribute to {D}avid {B}lackwell},
	JOURNAL = {Notices Am. Math. Soc.},
	FJOURNAL = {Notices of the American Mathematical
		 Society},
	VOLUME = {58},
	NUMBER = {7},
	YEAR = {2011},
	PAGES = {912--928},
	URL = {http://www.ams.org/notices/201107/rtx110700912p.pdf},
	NOTE = {Edited by G. G. Roussas.
		 MR:2850553. Zbl:1225.01082.},
	ISSN = {0002-9920},
	CODEN = {AMNOAN},
}

year	title	people

			clear

Murray Rosenblatt

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