C. Radhakrishna Rao’s research while affiliated with Pennsylvania State University and other places

The earliest method of estimation of statistical parameters is the method of least squares due to Markoff. A set of observations whose expectations are liner of a number of unknown parameters being given the problem which Markoff posed for solution is to find out a linear function of observations whose expectation is an assigned linear function of the unknown parameters and whose variance is a minimum.

MSC: primary 60E05 secondary 62E10 Keywords: Generalized negative binomial distributions Generalized Poisson distributions Infinitely divisible distributions Log-extreme stable laws Mixtures of exponential distributions Mixtures of geometric distributions Moment problem Wiener–Hopf factorization In the course of studying the moment sequence {n n : n = 0, 1, . . .}, Eaton et al. [1971. On ex-treme stable laws and some applications. J. Appl. Probab. 8, 794–801] have shown that this sequence, which is, indeed, the moment sequence of a log-extreme stable law with charac-teristic exponent = 1, corresponds to a scale mixture of exponential distributions and hence to a distribution with decreasing failure rate. Following essentially the approach of Shanbhag et al. [1977. Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc. 82, 289–295] we show that, under certain conditions, log-extreme stable laws with characteristic exponent ∈ [1, 2) are scale mixtures of exponential distributions and hence are infinitely divisible and have decreasing failure rates. In addition, we study the moment problem as-sociated with the log-extreme stable laws with characteristic exponent ∈ (0, 2] and throw further light on the existing literature on the subject. As a by-product, we show that general-ized Poisson and generalized negative binomial distributions are mixed Poisson distributions. Finally, we address some relevant questions on structural aspects of infinitely divisible distri-butions, and make new observations, including in particular that certain results appearing in Steutel and van Harn [2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York] have links with the Wiener–Hopf factorization met in the theory of random walk.

In a recent paper, Scobey (1975) observed that the usual least squares theory can be applied even when the covariance matrix σ2V of Y in the linear model Y = Xβ + e is singular by choosing the Moore-Penrose inverse (V+XX′)+ instead of V-1 when V is nonsingular. This result appears to be wrong. The appropriate treatment of the problem in the singular case is described.

Consider the linear regression model, yi = xiβ0 + ei, i = l,…,n, and an M-estimate β of βo obtained by minimizing Σρ(yi — xiβ), where ρ is a convex function. Let Sn = ΣXiXiXi and rn = Sn½ (β — β0) — Sn2 Σxih(ei), where, with a suitable choice of h(.), the expression Σ xix(e,) provides a linear representation of β. Bahadur (1966) obtained the order of rn as n→ ∞ when βo is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of rn as n → ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.Soient p une fonction convexe et β un M-estimateur de βo obtenu en minimisant Σρ(yi-xiβ)xiβ) dans le cadre du modéle de regression lineaire yi=xiβ+ ei,i=[,…,n. En introduisant une fonction h(.) convenable, il est possible de reprdsenter β sous la forme ΣXih(ei(ei). Soient alors Sn =ΣXiXiet rn=Sn½(β — β0) — Sn2Σxih(ei)Σ xih(ei)Dans le cas particulier où β0est un paramétre de localisation unidimensionnel représentant une médiane, l'ordre asymptotique de rna ete etabli par Bahadur (1966) et un résultat semblable a été demontré par Babu (1989) pour un paramétre de régression plus général estimé par la methode des moindres écarts absolus. Cet article a pour objectif d'etendre ces resultats en etablissant l'ordre stochastique de rnlorsque n → ∞ dans le cadre abstrait décrit ci-haut.

This paper studies the problem of estimating the number of clusters in the context of logistic regression clustering. The classi.cation likelihood approach is employed to tackle this problem. An information theoretic criterion for selecting the number of logistic curves is proposed in the sequel and then its asymptotic property is considered. The paper is arranged as follows: In Section 2, some notations are given and an information theoretic criterion is proposed for estimating the number of clusters. In Section 3, the small sample performance of the proposed criterion is studied by Monte Carlo simulation. In Section 4, the asymptotic property of the criterion proposed in Section 2 is investigated. Some lemmas needed in Section 4 are given in the appendix.

Different ways of characterizing antieigen and antisingularvalues are considered. Sev- eralmatrixinequalities, theirgeneralizationsandapplicationstoproblemsinmultivariatestatistics are given. A new concept of homologous canonical correlations is introduced and applied to a problem in genetics.

The general form of a matrix which appears in the normal equation for estimating parameters in the Gauss-Markoff linear model has been obtained.

A dermal patch is designed to activate some targeted receptors. On Day 1, the patch releases one dose of medicine, which latches onto a receptor and makes it active. On Day 2, the patch releases two doses of medicine, which latch onto two receptors, one dose per receptor. If the receptor is already active, the new dose makes it inactive. If the receptor is inactive, the new dose makes it active. On Day 3, the patch releases three doses of medicine, which latch onto three receptors, one dose per receptor. This continues for ten days with the patch releasing a total of 55 doses progressively. In this paper, we obtain the distribution of the number of receptors active at the end of Day 10.

Let Z=(Z(1),Z(2)) be an s-variate random vector partitioned into r- and q-variate subvectors whose distribution depends on an s-variate location parameter θ=(θ(1),θ(2)) partitioned in the same way as Z. For the s×s matrix I of Fisher information on θ contained in Z and r×r and q×q matrices I1 and I2 of Fisher information on θ(1) and θ(2) in Z(1) and Z(2), it is proved that . The inequality is similar to Carlen's superadditivity but has a different statistical meaning: it is a large sample version of an inequality for the covariance matrices of Pitman estimators. If the distribution of Z depends also on an m-variate nuisance parameter η (of a general nature) and and are the efficient matrices of information on θ,θ(1),θ(2) in Z,Z(1) and Z(2), respectively, then .

In this paper, some new observations concerning the problem of solving two or more integral equations of the type in Ramachandran, Gu and Lau (1988) and Rao and Shanbhag (1989b) are made by reducing it to the one of solving the integral equation of the type in Davies and Shanbhag (1987). Some applications of the results on integral equations are also given.