Browsing by Author "Puri, Madan L."

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Actuarial and Financial Risks: Models, Statistical Inference, and Case Studies
(Journal of Probability and Statistics, 2010-06) Zitikis, Ričardas; Furman, Edward; Necir, Abdelhakim; Nešlehová, Johanna; Puri, Madan L.
Understanding actuarial and financial risks poses major challenges. The need for reliable approaches to risk assessment is particularly acute in the present context of highly uncertain financial markets. New regulatory guidelines such as the Basel II Accord for banking and Solvency II for insurance are being implemented in many parts of the world. Regulators in various countries are adopting risk-based approaches to the supervision of financial institutions.
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.
Asymptotic Behavior of the Universally Consistent Conditional U-Statistics for Nonstationary and Absolutely Regular Processes
(Physica-Verlag HD, 2010-09) Elharfaoui, Echarif; Puri, Madan L.; Harel, Michel
A general class of conditional U-statistics was introduced by W. Stute as a generalization of the Nadaraya–Watson estimates of a regression function. It was shown that such statistics are universally consistent. Also, universal consistencies of the window and $k_n$-nearest neighbor estimators (as two special cases of the conditional U-statistics) were proved. Later, (Harel and Puri, Ann Inst Stat Math 56(4):819–832, 2004) extended his results from the i.i.d. case to the absolute regular case. In this paper, we extend these results from the stationary case to the nonstationary case.
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.
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.
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 Concept of Normality for Fuzzy Random Variables
(The Annals of Probability, 1985-11) Puri, Madan L.; Ralescu, Dan A.
In this paper we define the concept of a normal fuzzy random variable and we prove the following representation theorem: Every normal fuzzy random variable equals the sum of its expected value and a mean zero random vector.
Convergence and Remainder Terms in Linear Rank Statistics
(The Annals of Statistics, 1977-07) Bergstrom, Harald; Puri, Madan L.
A new approach to the asymptotic normality of simple linear rank statistics for the regression case studied earlier by Hajek (1968) is provided along with the estimation of the remainder term in the approximation to normality.
Cramér Type Large Deviations for Generalized Rank Statistics
(The Annals of Probability, 1985-02) Puri, Madan L.; Ralescu, Stefan S.; Seoh, Munsup
A Cramér type large deviation theorem is proved under alternatives as well as under hypothesis for the generalized linear rank statistic which includes as special cases (unsigned) linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, and a rank combinatorial statistic.
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.
L-functions, processes, and statistics in measuring economic inequality and actuarial risks
(Statistics and Its Interface, 2009) Zitikis, Ričardas; Greselin, Francesca; Puri, Madan L.
$L$-statistics play prominent roles in various research areas and applications, including development of robust statistical methods, measuring economic inequality and insurance risks. In many applications the score functions of $L$-statistics depend on parameters (e.g., distortion parameter in insurance, risk aversion parameter in econometrics), which turn the $L$-statistics into functions that we call $L$-functions. A simple example of an $L$-function is the Lorenz curve. Ratios of $L$-functions play equally important roles, with the Zenga curve being a prominent example. To illustrate real life uses of these functions/curves, we analyze a data set from the Bank of Italy year 2006 sample survey on household budgets. Naturally, empirical counterparts of the population $L$-functions need to be employed and, importantly, adjusted and modified in order to meaningfully capture situations well beyond those based on simple random sampling designs. In the processes of our investigations, we also introduce the $L$-process on which statistical inferential results about the population $L$-function hinges. Hence, we provide notes and references facilitating ways for deriving asymptotic properties of the $L$-process.
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.
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.
Linear Serial Rank Tests for Randomness Against Arma Alternatives
(The Annals of Statistics, 1985-09) Ingenbleek, Jean-Francois; Puri, Madan L.; Hallin, Marc
In this paper we introduce a class of linear serial rank statistics for the problem of testing white noise against alternatives of ARMA serial dependence. The asymptotic normality of the proposed statistics is established, both under the null as well as alternative hypotheses, using LeCam's notion of contiguity. The efficiency properties of the proposed statistics are investigated, and an explicit formulation of the asymptotically most efficient score-generating functions is provided. Finally, we study the asymptotic relative efficiency of the proposed procedures with respect to their normal theory counterparts based on sample autocorrelations.
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.
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.
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.
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].
Nonparametric density estimators based on nonstationary absolutely regular random sequences
(Journal of Applied Mathematics and Stochastic Analysis, 1996) Puri, Madan L.; Harel, Michel
In this paper, the central limit theorems for the density estimator and for the integrated square error are proved for the case when the underlying sequence of random variables is nonstationary. Applications to Markov processes and ARMA processes are provided.
Normal Approximation of U-Statistics in Hilbert Space
(Theory of Probability & Its Applications, 1997) Borovskikh, Yu. V.; Puri, Madan L.; Sazonov, V. V.
Let $\{U_n\}$, $n=1,2,...,$ be Hilbert space H-valued U-statistics with kernel $\Phi(\cdotp,\cdot)$, corresponding to a sequence of observations (random variables) $X_1,X_2,\ldots\ $. The rate of convergence on balls in the central limit theorem for $\{U_n\}$ is investigated. The obtained estimate is of order $n^{-1/2}$ and depends explicitly on $E\|\Phi(X_1,X_2)\|^3$ and on the trace and the first nine eigenvalues of the covariance operator of $E(\Phi(X_1,X_2)|X_1)$.