Table for Estimating the Goodness of Fit of Empirical Distributions
01 Jun 1948-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 19, Iss: 2, pp 279-281
About: This article is published in Annals of Mathematical Statistics.The article was published on 1948-06-01 and is currently open access. It has received 1660 citations till now. The article focuses on the topics: Cramér–von Mises criterion & Minimum distance estimation.
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TL;DR: In this paper, the maximum difference between an empirical and a hypothetical cumulative distribution is calculated, and confidence limits for a cumulative distribution are described, showing that the test is superior to the chi-square test.
Abstract: The test is based on the maximum difference between an empirical and a hypothetical cumulative distribution. Percentage points are tabled, and a lower bound to the power function is charted. Confidence limits for a cumulative distribution are described. Examples are given. Indications that the test is superior to the chi-square test are cited.
5,143 citations
Cites methods from "Table for Estimating the Goodness o..."
...The values in Table 1 for N ? 35 were computed by the procedure described in [5]; those for N>35 are from Smirnov's table [10]....
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...A table of the limiting distribution was given by Smirnov [10]....
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TL;DR: Enrichr is an easy to use intuitive enrichment analysis web-based tool providing various types of visualization summaries of collective functions of gene lists, and can be embedded into any tool that performs gene list analysis.
Abstract: System-wide profiling of genes and proteins in mammalian cells produce lists of differentially expressed genes/proteins that need to be further analyzed for their collective functions in order to extract new knowledge. Once unbiased lists of genes or proteins are generated from such experiments, these lists are used as input for computing enrichment with existing lists created from prior knowledge organized into gene-set libraries. While many enrichment analysis tools and gene-set libraries databases have been developed, there is still room for improvement. Here, we present Enrichr, an integrative web-based and mobile software application that includes new gene-set libraries, an alternative approach to rank enriched terms, and various interactive visualization approaches to display enrichment results using the JavaScript library, Data Driven Documents (D3). The software can also be embedded into any tool that performs gene list analysis. We applied Enrichr to analyze nine cancer cell lines by comparing their enrichment signatures to the enrichment signatures of matched normal tissues. We observed a common pattern of up regulation of the polycomb group PRC2 and enrichment for the histone mark H3K27me3 in many cancer cell lines, as well as alterations in Toll-like receptor and interlukin signaling in K562 cells when compared with normal myeloid CD33+ cells. Such analyses provide global visualization of critical differences between normal tissues and cancer cell lines but can be applied to many other scenarios. Enrichr is an easy to use intuitive enrichment analysis web-based tool providing various types of visualization summaries of collective functions of gene lists. Enrichr is open source and freely available online at: http://amp.pharm.mssm.edu/Enrichr
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4,713 citations
Cites methods from "Table for Estimating the Goodness o..."
...The authors of this seminal publication developed a statistical test that is based on the Kolmogorov-Smirnov test [4] as well as developed a database of gene-set libraries called MSigDB [5]....
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TL;DR: In this article, a general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations.
Abstract: The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty
3,082 citations
TL;DR: In this article, the authors derived the distributions of the least-squares residuals under a variety of specification errors, including omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity.
Abstract: SUMMARY The effects on the distribution of least-squares residuals of a series of model mis-specifications are considered. It is shown that for a variety of specification errors the distributions of the least-squares residuals are normal, but with non-zero means. An alternative predictor of the disturbance vector is used in developing four procedures for testing for the presence of specification error. The specification errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. THE objectives of this paper are two. The first is to derive the distributions of the classical linear least-squares residuals under a variety of specification errors. The errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. It is assumed that the disturbance terms are independently and normally distributed. It will be shown that the effect of the specification errors considered above is, with the exception of the error of heteroskedasticity, to yield residuals which though normally distributed do not have zero means, so that the distribution of the squared residuals is non-central x2. The second objective is to derive procedures to test for the presence of the specification errors considered in the first part of the paper. The tests are developed by comparing the distribution of residuals under the hypothesis that the specification of the model is correct to the distribution of the residuals yielded under the alternative hypothesis that there is a specification error of one of the types considered in the first part of the paper. As a preliminary step to deriving the test procedures the classical least-squares residual vector is transformed to a sub-vector which has more desirable properties for testing the null hypothesis that the specification of the model is correct. Also, under certain assumptions, with respect to the alternative hypothesis, it is shown that the mean vector of the residuals can be approximated by a linear sum of vectors qj,
2,269 citations
TL;DR: In this paper, the authors explored several approaches for testing the hypothesis that no switch has occurred in the true values of the parameters of a linear regression system, and the distribution of the relevant likelihood ratio λ was analyzed on the basis of the empirical distribution resulting from some sampling experiments.
Abstract: Several approaches are explored for testing the hypothesis that no switch has occurred in the true values of the parameters of a linear regression system. The distribution of the relevant likelihood ratio λ is analyzed on the basis of the empirical distribution resulting from some sampling experiments. The hypothesis that −2 log λ has the χ2 distribution with the appropriate degrees of freedom is rejected and an empirical table of percentage points is obtained. Finally some small sample tests are suggested.
1,229 citations
Cites background from "Table for Estimating the Goodness o..."
..., "K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-v. Mises Tests," Annals of Mathematical Statistics, 30 (1959), 420-47....
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