Cumulative distribution function estimation under interval censoring case 1

We consider projection methods for the estimation of the cumulative distribution function under interval censoring, case 1. Such censored data also known as current status data, arise when the only information available on the variable of interest is whether it is greater or less than an observed random time. Two types of adaptive estimators are investigated. The first one is a two-step estimator built as a quotient estimator. The second estimator results from a mean square regression contrast. Both estimators are proved to achieve automatically the standard optimal rate associated with the unknown regularity of the function, but with some restriction for the quotient estimator. Simulation experiments are presented to illustrate and compare the methods.Comment: Published in at http://dx.doi.org/10.1214/08-EJS209 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Transmuted Lindley-Geometric Distribution and its applications

A functional composition of the cumulative distribution function of one probability distribution with the inverse cumulative distribution function of another is called the transmutation map. In this article, we will use the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distributions taking Lindley geometric distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility. It will be shown that the analytical results are applicable to model real world data.Comment: 20 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1309.326

Partial stochastic dominance for the multivariate Gaussian distribution

Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the partial stochastic dominance result that the cumulative distribution function of the maximum of a multivariate normal random vector, with positive intraclass correlation coefficient, intersects the cumulative distribution function of a standard normal random variable at most once. This result can be applied to the Bayesian design of a clinical trial in which several experimental treatments are compared to a single control.Comment: 7 page

Statistical tests for whether a given set of independent, identically distributed draws does not come from a specified probability density

We discuss several tests for whether a given set of independent and identically distributed (i.i.d.) draws does not come from a specified probability density function. The most commonly used are Kolmogorov-Smirnov tests, particularly Kuiper's variant, which focus on discrepancies between the cumulative distribution function for the specified probability density and the empirical cumulative distribution function for the given set of i.i.d. draws. Unfortunately, variations in the probability density function often get smoothed over in the cumulative distribution function, making it difficult to detect discrepancies in regions where the probability density is small in comparison with its values in surrounding regions. We discuss tests without this deficiency, complementing the classical methods. The tests of the present paper are based on the plain fact that it is unlikely to draw a random number whose probability is small, provided that the draw is taken from the same distribution used in calculating the probability (thus, if we draw a random number whose probability is small, then we can be confident that we did not draw the number from the same distribution used in calculating the probability).Comment: 18 pages, 5 figures, 6 table

A comparison of the accuracy of saddlepoint conditional cumulative distribution function approximations

Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic R. The standard normal approximation to the conditional distribution of R typically has error of order O(n^{-1/2}), where n is the sample size. There are several modifications for R, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations.Comment: Published at http://dx.doi.org/10.1214/074921707000000193 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic inference for spatial cumulative distribution function

Random-field models with continuous spatial index are commonly used to model spatially and temporally dependent data in environmental studies. One quantity that captures the variation of a random field is the spatial cumulative distribution function (SCDF). The empirical distribution function, based on finite samples from a regular grid, can be used as a predictor of the SCDF. In this dissertation, existing methods are adapted and new methods are developed for making inference about the spatial and temporal changes in the SCDFs;The inference of the SCDF over one region at one point in time is extended to the inference across regions and over time by comparison of the invariant cumulative distribution functions and prediction of quantities that characterize the differences between two SCDFs. The quantiles of the test statistic are estimated and prediction intervals are constructed by resampling methods: subsampling for the comparison over time and bootstrapping for the comparison across regions. Next, the inference of the SCDF of a random field with an additive trend in the mean structure is examined. Large-sample properties of the empirical-distribution-function predictor are established. Finally, since a complete regular grid of data is rarely available in practice, imputations are often used to fill in the missing data values. A simulation study is conducted to assess the effect of different imputation methods on the SCDF prediction;The approach in this work is to derive weak convergence for the predictor based on the empirical distribution function. The asymptotic structure is nonstandard it is a mixture of increasing domain asymptotics and infill asymptotics . Further, suitable assumptions of stationarity and dependence structures are made on the random fields. For illustration, these methods are applied to both simulated data and forest health data in the state of Maine in the early 1990s

Analysis of Cumulative Distribution Function of 2-year Rainfall Measurements in Ogbomoso, Nigeria

Abstract: The conversion of most available hourly rainfall data to 1-minute integration time rain rate statistic is imperative for accurate estimation of attenuation due to rain employed in the design of both terrestrial and earth-to-space microwave systems. 2-year rainfall data collected at Ogbomoso, South-west region of Nigeria, between the periods of 2009 and 2010 was used in the analysis. Result shows that a power law relationship exists between the equiprobable rain rates of two different integration times. The regression coefficients a and b obtained are slightly different from the ITU-R recommendation. The conversion factor obtained at Ogbomoso is lower compared to Ile-Ife, in the South-west region of the country. The disagreement is attributed to the effect of global warming hitting the whole universe most especially the tropical regions. This study also reveals that different conversion factors are required for different locations even within the same climatic region