Madan Lal Puri Collection

Permanent link for this collectionhttps://hdl.handle.net/2022/21643

Professor Emeritus
College of Arts and Sciences Distinguished Research Scholar

Department of Mathematics
Indiana University Bloomington

Puri was awarded his B.A. degree in 1948 from the Punjab University in India and continued to study there for a master's degree which was awarded in 1950. In January 1951 he was appointed as a lecturer in mathematics at Punjab University of India and he taught mathematics at several different colleges between January 1951 and August 1957. In September 1957 Puri moved to the United States being appointed as an instructor and graduate student at the University of Colorado in Boulder. After a year he moved to the University of California at Berkeley where he was a research assistant in statistics. Puri was awarded his doctorate in 1962 after submitting his thesis Asymptotic Efficiency of a Class of C-Sample Tests. After the award of his Ph.D., Puri was appointed as an assistant professor at the Courant Institute of Mathematical Sciences in New York. Puri came to Indiana University as a full professor in 1968.

Puri is considered one of the world’s most versatile, prolific researchers and influential contributors to theoretical statistics over more than four decades. His research areas include nonparametric statistics, limit theory under mixing, time series, tests of normality, generalized inverses of matrices, stochastic processes, statistics of directional data and fuzzy sets and fuzzy measures. His work on rank-based methods has driven the frontier of the subject forward, and his fundamental contributions in developing rank-based methods and precise evaluation of the standard procedures, asymptotic expansions of distributions of rank statistics and large deviation results concerning them, span various areas such as analysis of variance, analysis of covariance, multivariate analysis and time series.

Puiri has been honored by universities in Australia, New Zealand, Europe and the Middle East as visiting fellow, chair professor, guest professor and distinguished visitor. In 1974, he was invited by the Japanese Society for the Promotion of Sciences to visit Japan under its Visiting Professorship Program to conduct cooperative research with Japanese scientists. Puri is an elected fellow of the Royal Statistical Society, the Institute of Mathematical Statistics and the American Statistical Association; an elected member of the International Statistical Institute and the New York Academy of Sciences; and an Honorary Fellow of the International Indian Statistical Society. In 1975 the Punjab University in India awarded him a D.Sc. Puri twice received the Senior U.S. Scientist Award from the Alexander von Humboldt Foundation in Germany, in 1974 and in 1983. In 1974 he was honoured by the German government "in recognition of past achievements in research and teaching." He was also a Distinguished Visitor at the London School of Economics and Political Science in 1991. As Indiana University College of Arts and Sciences Distinguished Research Scholar and Professor Emeritus of Mathematics Madan Puri also some of the American Statistical Association’s most prestigious honors, including the Gottfried E. Noether Senior Scholar Award in 2008 and the Samuel S. Wilks Award in 2014.

Browse

Now showing 1 - 20 of 37

Strong solutions of stochastic differential equations for multiparameter processes
(Stochastics, 1986) Puri, Madan L.; Dozzi, Markus
We consider the stochastic differential equation (SDE) $X_t = V_t \int_{[0,t]}f(s,\omega, X.(\omega)),dZ_5(\omega)$, where $V$ and $Z$ are vector valued process indexed by $t\varepsilon\Re^p_+$. The assumptions we make on $Z$ and on the increasing process controlling $Z$ are satisfied by certain classes of square integrable martingales, by processes of finite variation and by mixtures of these types of processes. Existence, uniqueness and the possibility of explosions of a strong solution $X$ are investigated under Lipschitz conditions on $f$. A well-known sufficient condition for non-explosion is shown to work also in the multiparameter case and stability of $X$ under perturbation of $V$, $f$ and $Z$ is proved. Finally more special SDE without Lipschitz conditions are considered, including a class of SDE of the Tsirel'son type.
A multivariate Wald‐Wolfowitz rank test against serial dependence
(Canadian Journal of Statistics, 1995-03) Puri, Madan L.; Hallin, Marc
Rank‐based cross‐covariance matrices, extending to the case of multivariate observed series the (univariate) rank autocorrelation coefficients introduced by Wald and Wolfowitz (1943), are considered. A permutational central limit theorem is established for the joint distribution of such matrices, under the null hypothesis of (multivariate) randomness as well as under contiguous alternatives of (multivariate) ARMA dependence. A rank‐based, permutationaily distribution‐free test of the portmanteau type is derived, and its asymptotic local power is investigated. Finally, a modified rank‐based version of Tiao and Box's model specification procedure is proposed, which is likely to be more reliable under non‐Gaussian conditions, and more robust against gross errors.
Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter
(Canadian Journal of Statistics, 1984-09) Puri, Madan L.; Wu, Tiee‐Jian
In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.
Asymptotic Normality of Nearest Neighbor Regression Function Estimates Based on Nonstationary Dependent Observations
(American Journal of Mathematical and Management Sciences, 1995) Harel, Michel; Puri, Madan L.
In this paper the convergence of the regression function estimators and the central limit theorem for these estimators are proved for the case when the underlying sequence of random variables is dependent and nonstationary.
Some distribution-free k-sample rank tests of homogeneity against ordered alternatives
(Communications on Pure and Applied Mathematics, 1965-05) Puri, Madan L.
A problem which occurs frequently in statistical analysis is that of deciding whether several samples should be regarded as coming from the same population. This problem, usually referred to as the k-sample problem, when expressed formally is stated as follows: Let $X_{ii} , j = 1, · · · , m_i, i = 1, · · · , k,$ be a set of independent random variables and let $F_i(x)$ be the probability distribution function of $X_{ii}$ . The set of admissible hypotheses designates that each $F_i$ belongs to some class of distribution functions $\Omega$.
Law of the iterated logarithm for perturbed empirical distribution functions evaluated at a random point for nonstationary random variables
(Journal of Theoretical Probability, 1994-10) Harel, Michel; Puri, Madan L.
We consider perturbed empirical distribution functions $\hat{F}_n (x) = 1/n\sum^n_{i=1} G_n (x − X_i)$ , where {Gi$nn$, n≥1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and ${X_i, i≥1}$ is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic $\hat{F}_n (U_n)$ where $U_n$ is a $U$-statistic based on $X_1, ... , X_n$. The results obtained extend or generalize the results of Nadaraya,$^{(7)}$ Winter,$^{(16)}$ Puri and Ralescu,$^{(9,10)}$ Oodaira and Yoshihara,$^{(8)}$ and Yoshihara,$^{(19)}$ among others.
Strong Law of Large Numbers for Banach Space Valued Fuzzy Random Variables
(Journal of Theoretical Probability, 2002-04) Proske, Frank N.; Puri, Madan L.
In this paper we prove a strong law of large numbers for Borel measurable nonseparably valued random elements in the case of Banach space valued fuzzy random variables.
Adaptive Nonparametric Procedures and Applications
(Journal of the Royal Statistical Society, Series C (Applied Statistics), 1988) Puri, Madan L.; Hill, N.J.; Padmanabhan, A. R.
Two adaptive nonparametric procedures are proposed for multiple comparisons and testing for ordered alternatives in the one-way ANOVA model. The first procedure resembles a proposal of Hogg, Fisher and Randles (for hypothesis testing) while the second is a variation of the first. Applications to data on lung cancer illustrate the theory.The supremacy of these procedures over the parametric normal theory procedures and the rank-based procedures is established. Monte Carlo studies show that these procedures can be safely applied when the size of each sample is at least 20.
A new semigroup technique in poisson approximation
(Semigroup Forum, 1989-12) Puri, Madan L.; Deheuvels, Paul; Pfeifer, Dietmar
We present a unified and self-contained approach to Poisson approximation problems for independent Bernoulli summands with respect to several metrics by a general semigroup technique, expanding and completing earlier work on this subject by the first two authors [4], [5], [6].
Maximum likelihood estimation for stationary point processes
(Proceedings of the National Academy of Sciences, 1986-02) Puri, Madan L.; Tuan, Pham D.
In this paper we derive the log likelihood function for point processes in terms of their stochastic intensities by using the martingale approach. For practical purposes we work with an approximate log likelihood function that is shown to possess the usual asymptotic properties of a log likelihood function. The resulting estimates are strongly consistent and asymptotically normal (under some regularity conditions). As a by-product, a strong law of large numbers and a central limit theorem for martingales in continuous times are derived.
Locally asymptotically rank-based procedures for testing autoregressive moving average dependence
(Proceedings of the National Academy of Sciences, 1988-04) Hallin, Marc; Puri, Madan L.
The problem of testing a given autoregressive moving average (ARMA) model (in which the density of the generating white noise is unspecified) against other ARMA models is considered. A distribution-free asymptotically most powerful test, based on a generalized linear serial rank statistic, is provided against contiguous ARMA alternatives with specified coefficients. In the case in which the ARMA model in the alternative has unspecified coefficients, the asymptotic sufficiency (in the sense of Hájek) of a finite-dimensional vector of rank statistics is established. This asymptotic sufficiency is used to derive an asymptotically maximin most powerful test, based on a generalized quadratic serial rank statistic. The asymptotically maximin optimal test statistic can be interpreted as a rank-based, weighted version of the classical Box-Pierce portmanteau statistic, to which it reduces, in some particular problems, under gaussian assumptions.
Limit theorems for fuzzy random variables
(Proceedings of the Royal Society of London, 1986-09) Ralescu, Dan A.; Puri, Madan L.; Klement, E.P.
A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.
Statistical inference based on incomplete blocks designs
(Cambridge University Press, 1970) Puri, Madan L.; Shane, Harold D.
In an earlier paper (Shane and Puri, 1969), the authors developed a class of asymptotically nonparametric tests for a bivariate paired comparison model. This paper unifies and complements the results of the previous paper by deriving a class of genuinely distribution free tests for the same problem but under the more general framework of $p(\ge2)$-variate situations. This is done by exploiting the theory of permutation distribution under sign invariant transformations to a class of rank order statistics. Asymptotic properties of these permutation rank order tests are studied and certain stochastic equivalence relationship with a similar class of multisample extensions of the $p$-variate one sample rank order tests proposed by Sen and Puri (1967) are derived. The asymptotic power properties of these tests are also studied.
On the Asymptotic Normality of One Sample Rank Order Test Statistics
(Theory of Probability & Its Applications, 1969) Puri, Madan L.; Sen, P. K.
The asymptotic normality of a class of one sample rank order test statistics is established. This class includes among other test statistics the well-known normal scores test of symmetry developed by Fraser [2] and the Wilcoxon paired comparison test [8].
Centering of Signed Rank Statistics with a Continuous Score-Generating Function
(Theory of Probability & Its Applications, 1985) Puri, Madan L.; Ralescu, Stefan S.
For a continuous score generating function, Hájek [2] established the asymptotic normality of a simple linear rank statistic $S_N $ with natural parameters $({\bf E}S_N ,{\operatorname{Var}}S_N )$ as well as $({\bf E}S_N ,\sigma _N^2 )$, where $\sigma _N^2 $ is some constant. The permissibility of replacing ${\bf E}S_N $ by a simpler constant $\mu _N $ was shown by Hoeffding [4] under conditions slightly stronger than Hájek’s. Following Hájek’s methods, Hušková [5] derived the asymptotic normality of a simple signed rank statistic $S_N^ + $ with parameters $({\bf E}S_N^ + ,{\operatorname{Var}}S_N^ + )$ as well as $({\bf E}S_N^2 ,\sigma _N^2 )$ and left open the problem of the replacement of ${\bf E}S_N^ + $ by some simpler constant. In this note we close this problem of the replacement of ${\bf E}S_N^ + $ by a simpler constant $\mu _N^ + $. The solution is a follow-up of Hoeffding [4]. We also provide a slight generalization with regard to the choice of scores.
The Order of Normal Approximation for Signed Linear Rank Statistics
(Theory of Probability & Its Applications, 1987) Puri, Madan L.; Wu, Tiee-Jian
The rate of convergence of the cdf (cumulative distribution function) of the signed linear rank statistics to the normal one is investigated. Under suitable assumptions, it is shown that the convergence rate is of order $O(N^{-1/2+\delta})$ for any $\delta > 0$.
Gaussian Random Sets in Banach Space
(Theory of Probability & Its Applications, 1987) Puri, Madan L.; Ralescu, Dan A.; Ralescu, Stefan S.
We define a Gaussian random set in a Banach space, and we prove the following characterization theorem: Every Gaussian random set can be represented as the sum of its expected value and a Gaussian mean zero random element.
On the Rate of Convergence in Normal Approximation and Large Deviation Probabilities for a Class of Statistics
(Theory of Probability & Its Applications, 1988) Puri, Madan L.; Seoh, M.
A new class of statistics is introduced to include, as special cases, unsigned linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, linear functions of concomitants of order statistics, and a rank combinatorial statistic. For this class, the rate of convergence to normality and Cramér’s type large deviation probabilities are investigated. Under the assumption that underlying observations are only independent, it is shown that this rate is $O(N^{ - {\delta / 2}} \log N)$ if the first derivative of the score generating function $\varphi $ satisfies Lipschitz’s condition of order $\delta $, $0 < \delta \leqq 1$, and it is $O(N^{ - {1 / 2}} )$ if $\varphi ''$ satisfies Lipschitz’s condition of order $\delta \geqq \frac{1}{2}$; and that Cramér’s large deviation theorem holds in the optimal range $0 < x \leqq \rho _N N^{{1 / 6}} $, $\rho _N = o(1)$. The results obtained provide new results and extend as well as generalize a number of known results obtained in this direction.
On Hilbert-Space-Valued U-Statistics
(Theory of Probability & Its Applications, 1991) Puri, Madan L.; Sazonov, V. V.
In this note we show that an elementary argument similar to the one used in [1] for real-valued $U$-statistics can be applied to Hilbert-space-valued $U$-statistics to obtain the rate of convergence of their distributions to their Gaussian limits.
Weak Convergence of the Simple Linear Rank Statistic under Mixing Conditions in the Nonstationary Case
(Theory of Probability & Its Applications, 1993) Puri, Madan L.; Harel, Michel
The asymptotic distribution theory of simple linear rank statistics for the case when the underlying random variables are nonstationary is studied both for the $\varphi $-mixing and strong mixing cases.

Browse

Recent Submissions