Interpretation of Graphs that Compare the Distribution Functions of Estimators

1990 ◽  
Vol 6 (1) ◽  
pp. 97-102 ◽  
Author(s):  
Brent R. Moulton
Keyword(s):  
Cumulative Distribution Function ◽  
Absolute Error ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Limited Information ◽  
One Dimensional ◽  
Error Loss ◽  
Least Squares Estimators ◽  
Squared Error ◽  
Exact Distributions

In this paper I examine graphical comparisons of one-dimensional (or marginal) distribution functions of alternative estimators. It is shown that areas under the c.d.f. (cumulative distribution function) curve can be given a decision-theoretic interpretation as risk under a bounded absolute-error loss function. I also show that by a simple rescaling of the graph's axes, graphical areas are created which can be interpreted as risk under bounded squared-error loss. The bounded loss functions are applied to compare graphically and numerically the risk of exact distributions of the limited-information maximum likelihood and two-stage least-squares estimators in a simultaneous equations model.

scholarly journals Minimax estimation of a bivariate cumulative distribution function

Metrika ◽  
2019 ◽  
Vol 83 (5) ◽  
pp. 597-615
Author(s):  
Rafał Połoczański ◽  
Maciej Wilczyński
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Decision Rules ◽  
Cumulative Distribution ◽  
Squared Error Loss ◽  
Error Loss ◽  
Squared Error ◽  
Von Mises

Abstract The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed.

scholarly journals Exact distributions of order statistics from ln,p-symmetric sample distributions

2017 ◽  
Vol 5 (1) ◽  
pp. 221-245 ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Distribution Function ◽  
Finite Number ◽  
Order Statistics ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Symmetric Distributions ◽  
Sample Distribution ◽  
Special Cases ◽  
Exact Distributions ◽  
Gaussian Sample

Abstract We derive the exact distributions of order statistics from a finite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end,we first review the special cases of order statistics fromspherical aswell as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure.We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ {1, . . . , n − 1}.

scholarly journals A New Extended Cosine—G Distributions for Lifetime Studies

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2758
Author(s):  
Mustapha Muhammad ◽  
Rashad A. R. Bantan ◽  
Lixia Liu ◽  
Christophe Chesneau ◽  
Muhammad H. Tahir ◽  
...  
Keyword(s):  
Weibull Distribution ◽  
Residual Life ◽  
Series Representation ◽  
Absolute Error ◽  
Quantile Function ◽  
Least Square ◽  
Cumulative Distribution ◽  
Reliability Parameter ◽  
Asymptotic Results ◽  
Error Loss

In this article, we introduce a new extended cosine family of distributions. Some important mathematical and statistical properties are studied, including asymptotic results, a quantile function, series representation of the cumulative distribution and probability density functions, moments, moments of residual life, reliability parameter, and order statistics. Three special members of the family are proposed and discussed, namely, the extended cosine Weibull, extended cosine power, and extended cosine generalized half-logistic distributions. Maximum likelihood, least-square, percentile, and Bayes methods are considered for parameter estimation. Simulation studies are used to assess these methods and show their satisfactory performance. The stress–strength reliability underlying the extended cosine Weibull distribution is discussed. In particular, the stress–strength reliability parameter is estimated via a Bayes method using gamma prior under the square error loss, absolute error loss, maximum a posteriori, general entropy loss, and linear exponential loss functions. In the end, three real applications of the findings are provided for illustration; one of them concerns stress–strength data analyzed by the extended cosine Weibull distribution.

scholarly journals Non-uniform Estimates in the Approximation by the Irwin Law

2016 ◽  
Vol 55 (1) ◽  
pp. 112-118
Author(s):  
Kazimieras Padvelskis ◽  
Ruslan Prigodin
Keyword(s):  
Distribution Function ◽  
Remainder Term ◽  
Cumulative Distribution Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Type Condition ◽  
Uniform Estimates ◽  
Cumulative Distribution Functions ◽  
Cumulant Method ◽  
The Difference

We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).

scholarly journals Bias Correction of Historical and Future Simulations of Precipitation and Temperature for China from CMIP5 Models

2018 ◽  
Vol 19 (3) ◽  
pp. 609-623 ◽  
Author(s):  
X. Yang ◽  
E. F. Wood ◽  
J. Sheffield ◽  
L. Ren ◽  
M. Zhang ◽  
...  
Keyword(s):  
Cumulative Distribution Function ◽  
Absolute Error ◽  
Mapping Method ◽  
Cumulative Distribution ◽  
Cmip5 Models ◽  
Monthly Precipitation ◽  
Significance Level ◽  
Jackknife Method ◽  
Precipitation And Temperature ◽  
Slight Advantage

ABSTRACT In this study, the equidistant cumulative distribution function (EDCDF) quantile-based mapping method was used to develop bias-corrected and downscaled monthly precipitation and temperature for China at 0.5° × 0.5° spatial resolution for the period 1961–2099 for eight CMIP5 GCM simulations. The downscaled dataset was constructed by combining observations from 756 meteorological stations across China with the monthly GCM outputs for the historical (1961–2005) and future (2006–99) periods for the lower (RCP2.6), medium (RCP4.5), and high (RCP8.5) representative concentration pathway emission scenarios. The jackknife method was used to cross validate the performance of the EDCDF method and was compared with the traditional quantile-based matching method (CDF method). This indicated that the performance of the two methods was generally comparable over the historic period, but the EDCDF was more efficient at reducing biases than the CDF method across China. The two methods had similar mean absolute error (MAE) for temperature in January and July. The EDCDF method had a slight advantage over the CDF method for precipitation, reducing the MAE by about 0.83% and 1.2% at a significance level of 95% in January and July, respectively. For future projections, both methods exhibited similar spatial patterns for longer periods (2061–90) under the RCP8.5 scenario. However, the EDCDF was more sensitive to a reduction in variability.

scholarly journals Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]

2021 ◽  
Vol 54 (1) ◽  
pp. 85-109
Author(s):  
Allison Byars ◽  
Evan Camrud ◽  
Steven N. Harding ◽  
Sarah McCarty ◽  
Keith Sullivan ◽  
...  
Keyword(s):  
Cumulative Distribution Function ◽  
Invariant Measures ◽  
Borel Probability Measure ◽  
Iterated Function Systems ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Cumulative Distribution ◽  
Sampling Schemes ◽  
Iterated Function ◽  
Borel Probability

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

scholarly journals A New One-term Approximation to the Standard Normal Distribution

2021 ◽  
pp. 381-385
Author(s):  
Ahmad Hanandeh ◽  
Omar Eidous
Keyword(s):  
Normal Distribution ◽  
Cumulative Distribution Function ◽  
Absolute Error ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Maximum Absolute Error ◽  
Practical Applications ◽  
Form Representation ◽  
Standard Normal ◽  
Term Approximation

This paper deals with a new, simple one-term approximation to the cumulative distribution function (c.d.f) of the standard normal distribution which does not have closed form representation. The accuracy of the proposed approximation measured using maximum absolute error (M.S.E) and the same criteria is used to compare this approximation with the existing one-term approximation approaches available in the literature. Our approximation has a maximum absolute error of about 0.0016 and this accuracy is sufficient for most practical applications.

scholarly journals Beta-Burr Type X Distribution with Application to Luteinizing Hormone Data

2021 ◽  
pp. 121-129
Author(s):  
A A D Ikram ◽  
S Nurrohmah ◽  
I Fithriani
Keyword(s):  
Luteinizing Hormone ◽  
Cumulative Distribution Function ◽  
Maximum Likelihood Method ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Parameter Distribution ◽  
Composition Distribution ◽  
Mean And Variance ◽  
Probability Density Function Pdf

Beta-Burr Type X distribution is a three parameter distribution and can model right skewed, left skewed, and symmetric data. Beta-Burr Type X distribution is the result of composition distribution functions of beta distribution and Burr Type X distribution. In this study, the characteristics such as probability density function (PDF), cumulative distribution function (CDF), the r-th moment, mean, and variance are presented. The maximum likelihood method is used to estimate the parameters of Beta-Burr Type X distribution, and the solution is obtained using a numerical method. As an illustration, Beta-Burr Type X distribution is used to model the data of luteinizing hormone in female blood samples.    

scholarly journals Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 772 ◽  
Author(s):  
Valeriy A. Naumov ◽  
Yuliya V. Gaidamaka ◽  
Konstantin E. Samouylov
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Cumulative Distribution Function ◽  
Queueing Systems ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Queuing Systems ◽  
Cumulative Distribution Functions ◽  
Resource Requirements ◽  
Convolution Powers

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements.

scholarly journals A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2605
Author(s):  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet
Keyword(s):  
Mean Squared Error ◽  
Kernel Method ◽  
Numerical Study ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Gaussian Kernel ◽  
Inverse Gaussian ◽  
Inverse Gamma ◽  
Squared Error ◽  
Asymmetric Kernel

In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [0,∞), namely the Gamma, inverse Gamma, LogNormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum–Saunders and Weibull kernel c.d.f. estimators from Mombeni, Masouri and Akhoond. By using the same experimental design, we show that the LogNormal and Birnbaum–Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method from C. Tenreiro.

Sign in / Sign up

Export Citation Format

Share Document