Extreme Value Distribution Of Type Research Articles

Maximum-type tests for high-dimensional regression coefficients using Wilcoxon scores

In this article, we develop new maximum-type tests to infer the overall significance of coefficients in high-dimensional linear models based on the Wilcoxon scores. The proposed testing procedures are free of error variance estimation and robust to heavy-tailed distributions and outliers, making them widely applicable in practice. We incorporate the dependence structure among predictors in the test statistics to enhance their powers. The limiting null distributions of the test statistics are derived to be the extreme value distribution of type I under regularity conditions. To reduce the size distortion, we further propose a multiplier bootstrap method based on the high-dimensional Gaussian approximations, which does not impose any structural assumptions on the unknown covariance matrices. We also evaluate the powers of proposed tests theoretically in comparison with two existing methods. The effectiveness of our proposed tests in the finite samples is illustrated through simulation studies and a real data application.

High-dimensional two-sample mean vectors test and support recovery with factor adjustment

Testing the equality of two mean vectors is a classical problem in multivariate analysis. In this article, we consider the test in the high-dimensional setting. Existing tests often assume that the covariance matrix (or its inverse) of the underlying variables is sparse, which is rarely true in social science due to the existence of latent common factors. In the article, we introduce a maximum-type test statistic based on the factor-adjusted data. The factor-adjustment step increases the signal-to-noise ratio and thus results in more powerful test. We obtain the limiting null distribution of the maximum-type test statistic, which is the extreme value distribution of type I. To overcome the well-known slow convergence rate of the test statistic’s distribution to the limiting extreme value distribution, we also propose a multiplier bootstrap method to improve the finite-sample performance. In addition, a multiple testing procedure with false discovery rate (FDR) control is proposed for identifying specific locations that differ significantly between the two groups. Thorough numerical studies are conducted to show the superiority of the test over other state-of-the-art tests. The performance of the test is also assessed through a real stock market dataset.

Estimation of probable maximum precipitation for tropical catchment

Probable Maximum Precipitation (PMP) is the maximum precipitation depth for specific region or station within a certain time. The main purpose of PMP estimation is calculate the Probable Maximum Flood (PMF). The PMF is considered necessary for design and manage the hydraulic structures. PMP can be estimate using two methods, either using a physical method or by using statistical method. In this study, statistical approach was used to estimate the PMP for Temengor catchment in Perak state, Malaysia. Extreme value type-1 distribution (EV1) is adopted to estimate the extreme rainfall and Hershfeid method was used to estimate PMP value. Also, intensity duration curve (IDC) was derived for 1, 2 and 3 days storm duration with return period 5, 10, 50, 100, 500 years. The results showed that the values of PMP for 1000 return period are 222.361mm, 311.847mm and 348.307mm for 1, 2 and 3 days respectively.

Modelling maximum daily yearly rainfall in northern Algeria using generalized extreme value distributions from 1936 to 2009

ABSTRACTThe generalized extreme value distribution is generally used to model annual maximum daily precipitation, and it then allows the calculation of the return values of this phenomenon. The distribution was fitted for selected stations in the north of Algeria; it was found that the generalized extreme value distribution of type I or the Gumbel distribution is more suitable for the Algiers and Miliana stations and the Fréchet distribution is more appropriate for the Oran station. The parameters were estimated using the maximum likelihood method, and the return levels were calculated at selected return periods T; for instance, about 100 years must elapse to record a level of 181.9 mm of rainfall per day in Algiers, 173 mm in Miliana and 109.54 mm in Oran.

STATISTICAL ANALYSIS OF RAINFALL DATA AND ESTIMATION OF PEAK FLOOD DISCHARGE FOR UNGAUGED CATCHMENTS

Estimation of Peak Flood Discharge (PFD) at a desired location on a river is important for planning, design and management of hydraulic structures. For ungauged catchments, rainfall depth becomes an important input in derivation of PFD. So, rainfall depth can be estimated through statistical analysis by fitting probability distribution to the rainfall data. In the present study, the series of annual 1-day maximum rainfall derived from the daily rainfall data recorded at Manjuhi Khad is used for estimation of 1-day extreme rainfall adopting Extreme Value Type-1 (EV1) distribution. Maximum Likelihood Method (MLM) is used for determination of parameters of the distribution. Anderson-Darling is applied for checking the adequacy of fitting of the distribution to the recorded data. The estimated 1-day extreme rainfall obtained from EV1 distribution is used to compute the 1-hour maximum value of distributed rainfall that is considered as an input to estimate the PFD by rational formula. The study suggests the estimated PFD could be used for design of flood protection measures for ungauged catchments of Bhul and Manjuhi Khad, Himachal Pradesh.

Flood frequency analysis using method of moments and L-moments of probability distributions

AbstractEstimation of maximum flood discharge (MFD) at a desired location on a river is important for planning, design and management of hydraulic structures. This can be achieved using deterministic models with extreme storm events or through frequency analysis by fitting of probability distributions to the recorded annual maximum discharge data. In the latter approach, suitable probability distributions and associated parameter estimation methods are applied. In the present study, method of moments and L-moments (LMO) are used for determination of parameters of six probability distributions. Goodness-of-Fit tests such as Chi-square and Kolmogorov–Smirnov are applied for checking the adequacy of fitting of probability distributions to the recorded data. Diagnostic test of D-index is used for the selection of a suitable distribution for estimation of MFD. The study reveals that the Extreme Value Type-1 distribution (using LMO) is better suited amongst six distributions used in the estimation of MFD at Mal...

High-dimensional sparse MANOVA

This paper considers testing the equality of multiple high-dimensional mean vectors under dependency. We propose a test that is based on a linear transformation of the data by the precision matrix which incorporates the dependence structure of the variables. The limiting null distribution of the test statistic is derived and is shown to be the extreme value distribution of type I. The convergence to the limiting distribution is, however, slow when the number of groups is relatively large. An intermediate correction factor is introduced which significantly improves the accuracy of the test. It is shown that the test is particularly powerful against sparse alternatives and enjoys certain optimality. A simulation study is carried out to examine the numerical performance of the test and compare with other tests given in the literature. The numerical results show that the proposed test significantly outperforms those tests against sparse alternatives.

Wind climate modeling using Weibull and extreme value distribution

It is very much important to fit wind speed data into some suitable statistical model for two aspects. One is fatigue failure due to periodic vortex shedding and the other is to estimate the wind energy potential of a particular location. For the fatigue failure due to periodic vortex shedding, it is important to analyse the load cycle. The expected number of stress cycles in the projected working life of a structure is related to the expected number of hours in the critical wind speed range and wind climate modelling is required to know this. The most popular model for this purpose is Weibull distribution. Again, wind energy is proportional to the cube of the wind speed and the same Weibull model is appropriate for cubic wind speed with different parameters. But the problem with Weibull distribution is that it fails to describe the upper tail. The aim of this paper is to determine the technique for finding the wind speed range till which Weibull model is appropriate and also to fit upper wind speed data in a suitable statistical distribution like extreme value distribution of type I (Gumbel). The hourly mean wind speed data of Ahmadabad has been used to validate the procedure.

Estimation of the Extreme Value Type I Distribution by the Method of LQ-Moments

Problem statement: The study evaluated the effectiveness of the various quantile estimators of the LQ-moments method for estimating parameters of the Extreme Value Type 1 (EV1) distribution. Approach: The performances of the LQ-moments were analyzed and compared against a widely used method of L-moments by using simulated samples of both EV1 and generalized Lambda distribution, focusing on small and moderate sample sizes. Results: The analysis results showed that LQMOM method wais more efficient in many cases especially for the upper tails of the distribution and for various sample sizes. Conclusion: This study demonstrated that conventional LMOM was not optimal for the estimation of the EV1 distribution.

Evolution of biological sequences implies an extreme value distribution of type I for both global and local pairwise alignment scores

BackgroundConfidence in pairwise alignments of biological sequences, obtained by various methods such as Blast or Smith-Waterman, is critical for automatic analyses of genomic data. Two statistical models have been proposed. In the asymptotic limit of long sequences, the Karlin-Altschul model is based on the computation of a P-value, assuming that the number of high scoring matching regions above a threshold is Poisson distributed. Alternatively, the Lipman-Pearson model is based on the computation of a Z-value from a random score distribution obtained by a Monte-Carlo simulation. Z-values allow the deduction of an upper bound of the P-value (1/Z-value2) following the TULIP theorem. Simulations of Z-value distribution is known to fit with a Gumbel law. This remarkable property was not demonstrated and had no obvious biological support.ResultsWe built a model of evolution of sequences based on aging, as meant in Reliability Theory, using the fact that the amount of information shared between an initial sequence and the sequences in its lineage (i.e., mutual information in Information Theory) is a decreasing function of time. This quantity is simply measured by a sequence alignment score. In systems aging, the failure rate is related to the systems longevity. The system can be a machine with structured components, or a living entity or population. "Reliability" refers to the ability to operate properly according to a standard. Here, the "reliability" of a sequence refers to the ability to conserve a sufficient functional level at the folded and maturated protein level (positive selection pressure). Homologous sequences were considered as systems 1) having a high redundancy of information reflected by the magnitude of their alignment scores, 2) which components are the amino acids that can independently be damaged by random DNA mutations. From these assumptions, we deduced that information shared at each amino acid position evolved with a constant rate, corresponding to the information hazard rate, and that pairwise sequence alignment scores should follow a Gumbel distribution, which parameters could find some theoretical rationale. In particular, one parameter corresponds to the information hazard rate.ConclusionExtreme value distribution of alignment scores, assessed from high scoring segments pairs following the Karlin-Altschul model, can also be deduced from the Reliability Theory applied to molecular sequences. It reflects the redundancy of information between homologous sequences, under functional conservative pressure. This model also provides a link between concepts of biological sequence analysis and of systems biology.

Extreme value models with application to drought data

The exact distributions of the productXY are derived whenX andY are independent random variables and come from the extreme value distribution of Type I, the extreme value distribution of Type II or the extreme value distribution of Type III. Of the, six possible combinations, only three yield closed-form expressions for the distribution ofXY. A detailed application of the results is provided to drought data from Nebraska.

Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théorique

The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100–154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.

Predicting the mean annual flood and flood quantiles for ungauged catchments in Greece

Abstract The spatial variation of the mean annual flood of both mean daily and instantaneous extremes and of the parameters of the Extreme Value Type 1 (EV1) distribution for catchments in the northwest and west regions of Greece are significantly explained in terms of physiographic and climatological characteristics of the catchments by using multiple regression techniques. The catchment characteristics used in the study are the following: drainage area, mean annual areal precipitation, stream frequency, main stream slope and length, intensity of the one-day rainfall of five-year return period, and a soil type index. The EV1 distribution has been found to describe adequately the annual frequency distributions of the daily and of the standardised by-their-mean-value daily extremes of the catchments. Based on the regional models for the parameters of the distribution, annual flood frequency curves, and thus flood quantiles of the assumed distribution, can be derived. The developed regional models are succe...

Seismic Reliability of Rigid Frame Structures

A reliability analysis method has been developed for plane‐frame structures with shear walls subjected to earthquake ground accelerations. The structural reliability is measured in terms of the probability that throughout the structure's lifetime, its response will remain linear. The earthquake ground acceleration is assumed to be a stationary and Gaussian random process with mean zero and a Kanai‐Tajimi spectrum. To describe the statistical variation of an individual earthquake's intensity, an extreme value distribution of Type II is employed, while the Poisson arrival law is used to model the earthquake occurrences. The modal analysis approach is taken to map the safe domain associated with individual critical member forces in the (transformed) generalized coordinate vector space. Then, the global reliability is obtained by estimating the probability that fhe generalized coordinate vector will remaifn in the domain representing the interaction of these safe domains throughout the structure's lifetime. N...

A frequency study of rainfall for the Periyar basin of Kerala

A rainfall depth-duration-frequency study was carried out for the Periyar river basin of Kerala using average sixty years of data from fourteen raingauge stations distributed in an area of 5398 km'. The annual maximum rainfall depths of 1, 2, 3, 7 and 10 days durations were analysed. The extreme value Type-1 distribution fitted with maximum likelihood estimates. was utilized to .derive the re9uired relationships. Analyses were carried out for the Individual stations and later regional relationships were obtained based on the station-year concept. The regions were demarcated considering rainfall and topographical characteristics. Satisfactory results were obtained. The results obtained from the regional study are expected to be useful for design engineers and agricultural scientists.

Modelling extreme rain sequences in Zambia

The annual maximum daily rainfall sequences and rainfall intensities ranging from 5 min to 24 h duration were subjected to statistical analyses. The annual maximum daily rainfall sequences can be modelled satisfactorily using the extreme value type 1 (EV1) distribution in order to obtain extreme daily rainfall magnitude versus return period relationships. The annual maximum rainfall intensities for varying durations were found to fit satisfactorily the EV1 distribution. The rainfall intensity-duration-return period interactions fitted a power law relationship.

Sampling properties of the maximum entropy estimators for the extreme-value type-1 distribution

The extreme-value type-1 (EV1) distribution can be viewed as the distribution that satisfies two specified expected values. These expected values give rise to a method of parameter estimation referred to as the method of maximum entropy (MME). The main purpose of this note is to provide a scheme to estimate the variances and covariance of the MME estimators. As a by-product of the simulation runs used, some useful sampling properties of the MME estimators are obtained. These clearly show that the MME is a good method for fitting the EV1 distribution, and the approximations obtained analytically for the variance of estimates of the T-year event are of sufficient accuracy.

Frequency analysis of short‐duration rainfalls

Abstract An investigation was carried out to determine the appropriateness of a probability distribution and fitting technique commonly used in Canada for rainfall frequency analysis. The extreme value type 1 (EV1) distribution was assessed for three long‐term Canadian stations: Victoria, St Thomas and Quebec. The EV1 distribution appears to provide a reasonable fit for durations varying from 5 min to 6 h, but is not clearly superior to another two‐parameter distribution, the lognormal. The fitting technique, known as modified moments or regression, was assessed by comparing it with three other fitting techniques: moments, maximum likelihood and adjusted maximum likelihood. This comparison was carried out using Monte Carlo simulation techniques over the parameter space deemed to be representative of short duration rainfall data for Canada. In terms of estimating rainfall amounts of specified return period, the modified moments technique was the poorest with regard to both bias and efficiency. In general, ...

The extreme-value type-1 distribution and the principle of maximum entropy—Comments

The extreme-value type-1 distribution and the principle of maximum entropy—Comments

The extreme-value type-1 distribution and the principle of maximum entropy—Reply

The extreme-value type-1 distribution and the principle of maximum entropy—Reply