A tribute to P.R. Krishnaiah

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Abstract

The authors reminisce on their association with P.R. Krishnaiah, renowned professor of statistics at the University of Pittsburgh and founding editor of the Journal of Multivariate Analysis. They recount their individual associations with him, mainly involving the behavior of eigenvalues of random matrices, and outline two areas of applied work he performed with one of the authors.

Introduction

Paruchuri R. (P.R.) Krishnaiah was considered a great leader in the field of statistics, especially multivariate analysis. He devoted his academic career in the pursuit of knowledge and was a major influence in this area of statistics. He founded the famous Center for Multivariate Analysis at the Department of Mathematics and Statistics, University of Pittsburgh in 1982 and in 1971 created the prestigious Journal of Multivariate Analysis as the founding editor in chief. Based on his wide knowledge in statistics and familiarity of the literature, his activities were spread throughout the world. He has contributed a great deal, including the area involving limit theorems on the eigenvalues of random matrices as the dimensions increase. The authors of this paper became directly associated with Professor Krishnaiah’s work on large dimensional random matrices. They pay tribute to this wonderful man by first recalling their fond remembrance of him, and then outlining significant contributions he gave to image reconstruction and to general information criteria (GIC) in model selection (note: “ZB” will refer to the first author, “JS” to the second author).

Section snippets

Association with Professor Krishnaiah

The recollection spans the years 1978–1987, beginning in the winter of 1978 when JS visited the Department of Mathematics and Statistics at the University of Pittsburgh for a job interview. He gave a talk on his work on the eigenvalues and eigenvectors of random matrices, the former stemming from his Ph.D. thesis and appearing in print [11]. Professor Krishnaiah was interested in his work and gave him a copy of a recent manuscript by Dag Jonsson to review for possible publication in JMVA [13].

Contributions to image reconstruction

The CT scanner was invented by Godfrey Newbold Hounsfield 1n 1972 and a Nobel prize was awarded to him in 1979. The principle of the machine is simple, just an inverse-Fourier transform. Consider a cross-section of a human’s body, a two-dimensional distribution density, It can be easily reconstructed and visualized if its Fourier transform can be observed. Suppose the density is p(x,y). Then the Fourier transform is g(t,s)=ei(tx+sy)p(x,y)dxdy. Given a direction θ, write t=ρcosθ and s=ρsinθ.

Contributions to GIC of model selection

It is well known that if a model is under-specified the data would not be well fitted by the model and if the model is over specified the efficiency becomes low because much information would be wasted by the estimation of extra parameters. In 1972, H. Akaike [1] proposed his famous model selection criterion, AIC. The AIC is minus two times the logarithm of the maximum likelihood (the information) plus two times the number of parameters (the penalty). The AIC received much attention, resulting

Acknowledgment

Zhidong Bai acknowledges support from NSF China , Grant No. 12171198.

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