A Kernel Goodness-of-fit Test for Maximum Likelihood Density Estimates of Normal Mixtures

2020 ◽  
pp. 67-76
Author(s):  
Dimitrios Bagkavos ◽  
Prakash N. Patil
Keyword(s):  
Maximum Likelihood ◽  
Goodness Of Fit ◽  
Goodness Of Fit Test ◽  
Density Estimates ◽  
Normal Mixtures

scholarly journals Asymptotic Distribution of Cramer-von Mises Statistic When Contamination Exists

2015 ◽  
Vol 5 (1) ◽  
pp. 90
Author(s):  
Mayumi Naka ◽  
Ritei Shibata
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Distribution ◽  
Minimum Distance ◽  
Goodness Of Fit ◽  
Likelihood Estimator ◽  
Goodness Of Fit Test ◽  
Minimum Distance Estimator ◽  
Von Mises Statistic ◽  
Von Mises ◽  
Distance Estimator

In this paper, asymptotic distribution of Cram\'er-von Mises goodness-of-fit test statistic is investigated when contamination exists.<br />We first derive the asymptotic distribution of the Cram\'er-von Mises statistic when the observations are contaminated with noise as a mixture.<br />The result is extended to the case where the parameters are estimated by the minimum distance estimator,<br />which minimizes the Cram\'er-von Mises statistic.<br />In both cases the asymptotic distribution of the Cram\'er-von Mises statistic is given by that of the weighted infinite sum of non-central $\chi^2_1$ variables and the effect of contamination appears only in the non-centrality of the variables.<br />We also demonstrate the robustness of the goodness-of-fit test by Monte Carlo simulations when the parameters are estimated<br />by the minimum distance estimator and the maximum likelihood estimator.<br />Numerical experiments indicate that the use of the minimum distance estimator makes the test insensitive to contamination whereas the power is retained almost the same as that of the maximum likelihood estimator.

scholarly journals Maximum Likelihood Estimation for the Generalized Pareto Distribution and Goodness-of-Fit Test with Censored Data

2019 ◽  
Vol 17 (2) ◽  
Author(s):  
Minh H. Pham ◽  
Chris Tsokos ◽  
Bong-Jin Choi
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Censored Data ◽  
Goodness Of Fit ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Likelihood Estimation ◽  
R Package ◽  
Goodness Of Fit Test ◽  
Generalized Pareto

The generalized Pareto distribution (GPD) is a flexible parametric model commonly used in financial modeling. Maximum likelihood estimation (MLE) of the GPD was proposed by Grimshaw (1993). Maximum likelihood estimation of the GPD for censored data is developed, and a goodness-of-fit test is constructed to verify an MLE algorithm in R and to support the model-validation step. The algorithms were composed in R. Grimshaw’s algorithm outperforms functions available in the R package ‘gPdtest’. A simulation study showed the MLE method for censored data and the goodness-of-fit test are both reliable.

A guided nonparametric goodness-of-fit test with application to income distributions

2019 ◽  
Vol 22 (3) ◽  
pp. 207-222
Author(s):  
Kuangyu Wen ◽  
Ximing Wu
Keyword(s):  
Goodness Of Fit ◽  
Bias Reduction ◽  
Density Estimator ◽  
Consistent Estimate ◽  
Finite Sample ◽  
Goodness Of Fit Test ◽  
Normal Mixtures ◽  
Parametric Density ◽  
Income Distributions ◽  
Goodness Of Fit Tests

Summary We have developed a customizable goodness-of-fit test of a parametric density based on its distance to a consistently estimated density. This consistent estimate is obtained via a nonparametric density estimator with a parametric start, wherein the start is set to be the hypothesized parametric density. To cope with the influence of nonparametric estimation bias, nonparametric goodness-of-fit tests have resorted to remedies such as undersmoothing or convolution of the hypothesized density. Our test requires no such devices and possesses enhanced powers against alternative densities because the guided density estimator is free of the typical nonparametric bias under the null hypothesis and attains bias reduction when the underlying density is in a broad nonparametric neighborhood of the hypothesized density. Here, we establish the statistical properties of our test and use Monte Carlo simulations to demonstrate its finite sample performance. We use this test to examine the goodness-of-fit of normal mixtures to the distributions of log income of U.S. states. Although normality is rejected decisively, our results suggest that normal mixtures with two or three components suffice for all but one state.

Fitting The Best–Fit Distribution For The Hourly Rainfall Amount In The Wilayah Persekutuan

2012 ◽  
Author(s):  
Fadhilah Y. ◽  
Zalina Md. ◽  
Nguyen V–T–V. ◽  
Suhaila S. ◽  
Zulkifli Y.
Keyword(s):  
Maximum Likelihood ◽  
Goodness Of Fit ◽  
Rainfall Amount ◽  
Likelihood Method ◽  
Exceedance Probability ◽  
Parameter Estimates ◽  
Goodness Of Fit Test ◽  
Hourly Rainfall ◽  
Best Fit ◽  
Mixed Exponential Distribution

Dalam mengenal pasti model yang terbaik untuk mewakili taburan jumlah hujan bagi data selang masa satu jam di 12 stesen di Wilayah Persekutuan empat taburan digunakan iaitu Taburan Eksponen, Gamma, Weibull dan Gabungan Eksponen. Parameter–parameter dianggar menggunakan kaedah kebolehjadian maksimum. Model yang terbaik dipilih berdasarkan nilai minimum yang diperolehi daripada ujian–ujian kebagusan penyuaian yang digunakan dalam kajian ini. Ujian ini dipertahankan lagi dengan plot kebarangkalian dilampaui. Taburan Gabungan Eksponen di dapati paling baik untuk mewakili taburan jumlah hujan dalam selang masa satu jam. Daripada anggaran parameter bagi taburan Gabungan Eksponen ini, boleh diterjemah bahawa jumlah hujan tertinggi yang direkodkan diperolehi daripada hujan yang dikategorikan sebagai hujan lebat, walaupun hujan renyai–renyai berlaku lebih kerap. Kata kunci: Jumlah hujan dalam selang masa sejam, ujian kebagusan penyuaian, kebolehjadian maksimum In determining the best–fit model for the hourly rainfall amounts for the twelve stations in the Wilayah Persekutuan, four distributions namely, the Exponential, Gamma, Weibull and Mixed–Exponential were used. Parameters for each distribution were estimated using the maximum likelihood method. The best–fit model was chosen based upon the minimum error produced by the goodness–offit tests used in this study. The tests were justified further by the exceedance probability plot. The Mixed–Exponential was found to be the most appropriate distribution in describing the hourly rainfall amounts. From the parameter estimates for the Mixed–Exponential distribution, it could be implied that most of the hourly rainfall amount recorded were received from the heavy rainfall even though there was a high occurrences of light rainfall. Key words: Hourly rainfall amount, goodness-of-fit test, exceedance probability, maximum likelihood

Research on Complex Equipment Reliability Growth AMSAA-ELP Model Based on Explore Learning Promotion

2014 ◽  
Vol 940 ◽  
pp. 531-534 ◽  
Author(s):  
Na Zhang
Keyword(s):  
Maximum Likelihood ◽  
Goodness Of Fit ◽  
Likelihood Estimation ◽  
Trend Test ◽  
Reliability Growth ◽  
Goodness Of Fit Test ◽  
Model Based ◽  
Estimation Formula ◽  
Equipment Reliability ◽  
Complex Equipment

According to learning other models equipment test information of complex equipment in the development process, and making their own systems to improve the reliability of the case, a complex equipment reliability growth AMSAA-ELP model based on explore learning promotion was developed, and the property was analyzed from different parameter values. Additionally, the trend test was also presented. Secondly, maximum likelihood estimation formula of the parameters was given under time censored and failure censored test of AMSAA-ELP model, and point out that there are multiple poles value of the maximum likelihood estimate can use pseudo-Monte-Carlo method parameter calculation. Additionally, the model's goodness of fit test was also given. Finally, the combination of complex equipment with engine failure data was analyzed. The results shows that the AMSAA-ELP model is prefer to AMSAA model intended to test data, and the AMSAA-ELP model is suitable to the engineering applications.

scholarly journals Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1949
Author(s):  
Mukhtar M. Salah ◽  
M. El-Morshedy ◽  
M. S. Eliwa ◽  
Haitham M. Yousof
Keyword(s):  
Maximum Likelihood ◽  
Goodness Of Fit ◽  
Estimation Method ◽  
Least Square ◽  
Estimation Methods ◽  
Goodness Of Fit Test ◽  
Least Square Estimation ◽  
Chi Square ◽  
Anderson Darling ◽  
The Right

The extreme value theory is expanded by proposing and studying a new version of the Fréchet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, different non-Bayesian estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation method, Anderson–Darling estimation method, ordinary least square estimation method, Cramér–von-Mises estimation method, weighted least square estimation method, left-tail Anderson–Darling estimation method, and right-tail Anderson–Darling estimation method. Numerical simulations were performed for comparing the estimation methods using different sample sizes for three different combinations of parameters. The Barzilai–Borwein algorithm was employed via a simulation study. Three applications were presented for measuring the flexibility and the importance of the new model for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. The modified goodness-of-fit statistic test was applied for the right censored real data set, called leukemia free-survival times for autologous transplants. Based on the maximum likelihood estimators on initial data, the modified goodness-of-fit test recovered the loss in information while the grouping data and followed chi-square distributions. All elements of the modified goodness-of-fit criteria tests are explicitly derived and given.

A viable method for goodness-of-fit test in maximum likelihood fit

2011 ◽  
Vol 35 (6) ◽  
pp. 580-584
Author(s):  
Feng Zhang ◽  
Yuan-Ning Gao ◽  
Lei Huo
Keyword(s):  
Maximum Likelihood ◽  
Goodness Of Fit ◽  
Goodness Of Fit Test ◽  
Viable Method

scholarly journals Goodness-of-fit Testing for Left-truncated Two-parameter Weibull Distributions with Known Truncation Point

2016 ◽  
Vol 45 (3) ◽  
pp. 15-42 ◽  
Author(s):  
Ayse KIZILERSU ◽  
Markus Kreer ◽  
Anthony W. Thomas
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Goodness Of Fit ◽  
Brownian Bridge ◽  
Critical Values ◽  
Test Statistic ◽  
Goodness Of Fit Test ◽  
Weibull Parameters ◽  
Kolmogorov Smirnov ◽  
Power Testing

The left-truncated Weibull distribution is used in life-time analysis, it has many applications ranging from financial market analysis and insurance claims to the earthquake inter-arrival times.We present a comprehensive analysis of the left-truncated Weibull distribution when the shape, scale or both parameters are unknown and they are determined from the data using the maximum likelihood estimator. We demonstrate that if both the Weibull parameters are unknown then there are sets of sample configurations, with measure greater than zero, for which the maximum likelihood equations do not possess non trivial solutions. The modified critical values of the goodness-of-fit test from the Kolmogorov-Smirnov test statistic when the parameters are unknown are obtained from a quantile analysis. We find that the critical values depend on sample size and truncation level, but  not on the actual Weibull parameters.  Confirming this behavior, we present a complementary analysis using the Brownian bridge approach as an asymptotic limit of the Kolmogorov-Smirnov statistics and find that both approaches are in good agreement. A power testing is performed for our Kolmogorov-Smirnov goodness-of-fit test and  the issues related to the left-truncated data are discussed. We conclude the paper by showing the importance of left-truncated Weibulldistribution hypothesis testing on  the duration times of failed marriages in the US, worldwide terrorist attacks, waiting times between stock market orders, and time intervals of radioactive decay.

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