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Register a new userBreaking Bivariate Records
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James Allen Fill
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We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.
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The analysis of record-breaking events is of interest in fields such as climatology, hydrology, economy or sports. In connection with the record occurrence, we propose three distribution-free statistics for the changepoint detection problem. They are CUSUM-type statistics based on the upper and/or lower record indicators which occur in a series. Using a version of the functional central limit theorem, we show that the CUSUM-type statistics are asymptotically Kolmogorov distributed. The main results under the null hypothesis are based on series of independent and identically distributed random variables, but a statistic to deal with series with seasonal component and serial correlation is also proposed. A Monte Carlo study of size, power and changepoint estimate has been performed. Finally, the methods are illustrated by analyzing the time series of temperatures at Madrid, Spain. The $textsf{R}$ package $texttt{RecordTest}$ publicly available on CRAN implements the proposed methods.
We consider the connections among `clumped residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a `disordered GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of `stick breaking processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.
A new bivariate copula is proposed for modeling negative dependence between two random variables. We show that it complies with most of the popular notions of negative dependence reported in the literature and study some of its basic properties. Specifically, the Spearmans rho and the Kendalls tau for the proposed copula have a simple one-parameter form with negative values in the full range. Some important ordering properties comparing the strength of negative dependence with respect to the parameter involved are considered. Simple examples of the corresponding bivariate distributions with popular marginals are presented. Application of the proposed copula is illustrated using a real data set.
Univariate Weibull distribution is a well-known lifetime distribution and has been widely used in reliability and survival analysis. In this paper, we introduce a new family of bivariate generalized Weibull (BGW) distributions, whose univariate marginals are exponentiated Weibull distribution. Different statistical quantiles like marginals, conditional distribution, conditional expectation, product moments, correlation and a measure component reliability are derived. Various measures of dependence and statistical properties along with ageing properties are examined. Further, the copula associated with BGW distribution and its various important properties are also considered. The methods of maximum likelihood and Bayesian estimation are employed to estimate unknown parameters of the model. A Monte Carlo simulation and real data study are carried out to demonstrate the performance of the estimators and results have proven the effectiveness of the distribution in real-life situations
We consider a classical model related to an empirical distribution function $ F_n(t)=frac{1}{n}sum_{k=1}^nI_{{xi_kle t}}$ of $(xi_k)_{ige 1}$ -- i.i.d. sequence of random variables, supported on the interval $[0,1]$, with continuous distribution function $F(t)=mathsf{P}(xi_1le t)$. Applying ``Stopping Time Techniques, we give a proof of Kolmogorovs exponential bound $$ mathsf{P}big(sup_{tin[0,1]}|F_n(t)-F(t)|ge varepsilonbig)le text{const.}e^{-ndelta_varepsilon} $$ conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of $$ mathsf{P}big(sup_{tin[0,1]}n^alpha|F_n(t)-F(t)|ge varepsilonbig) $$ with rate $ frac{1}{n^{1-2alpha}} $ slower than $frac{1}{n}$ for any $alphainbig(0,{1/2}big)$.
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