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Type 1 And Type 2 Errors In Statistics Pdf And Cdf

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The Wrapped package computes the probability density function, cumulative distribution function, quantile function and also generates random samples for many univariate wrapped distributions. It also computes maximum likelihood estimates, standard errors, confidence intervals and measures of goodness of fit for nearly fifty univariate wrapped distributions.

Numerical illustrations of the package are given. This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication. Competing interests: The authors have declared that no competing interests exist. Circular data are data recorded in degrees or radii. They arise in a wide variety of scientific areas.

Some published applications involving real circular data include: data from fibre composites and from ceramic foams [ 1 ]; skeletal representations in medical image analysis and biomechanical gait analysis of the knee joint [ 2 ]; worldwide earthquakes with magnitude greater than or equal to 7.

Wrapping is a popular method for constructing distributions for circular data. Let g denote a valid probability density function PDF defined on the real line. Let G denote the corresponding cumulative distribution function CDF. So, we stick to 1 and 2. Many wrapped distributions have been proposed and studied in the literature.

These include the wrapped normal distribution [ 6 ], wrapped Cauchy distribution [ 7 ], wrapped skew normal distribution [ 8 ], wrapped exponential and Laplace distributions [ 9 ], wrapped stable distribution [ 10 ], wrapped gamma distribution [ 11 ], wrapped t distribution [ 12 ], wrapped lognormal and Weibull distributions [ 13 ], wrapped skew Laplace distribution [ 14 ], wrapped weighted exponential distribution [ 15 ], wrapped hypo exponential distribution [ 16 ], wrapped geometric distribution [ 17 ], wrapped Poisson distribution [ 18 ], wrapped zero inflated Poisson distribution [ 19 ] and wrapped Lindley distribution [ 20 ].

There are also a number of R [ 21 ] packages developed to implement wrapped distributions including: NPCirc [ 22 ] giving procedures for wrapped Cauchy, wrapped normal and wrapped skew normal distributions; wle [ 23 ] giving procedures for the wrapped normal distribution; circular [ 24 ] giving procedures for wrapped normal, wrapped Cauchy and wrapped stable distributions; CircStats [ 25 ] giving procedures for wrapped normal, wrapped Cauchy and wrapped stable distributions; BAMBI [ 26 ] giving procedures for wrapped normal and wrapped normal mixtures distributions; movehMM [ 27 ] giving procedures for the wrapped Cauchy distribution; SpaDES [ 28 ] giving procedures for the wrapped normal distribution.

But we are not aware of an R package applicable for computing 1 and 2 for given parametric forms for g and G.

The aim of this paper is to introduce an R package developed by the authors that is applicable for computing many wrapped distributions. The package is named Wrapped [ 29 ]. The package performs the following:. A description of the program structure of the package is given in the next section. Some numerical illustrations of the package are given in the following section. The paper concludes with a discussion section.

The g must be specified through the string spec. For both dwrappedg and pwrappedg , x can be a scalar or a vector. The random numbers are cos rspec n,… , cos rspec n,…. To use dwrappedg , pwrappedg and rwrappedg , one must have the functions dg , pg , qg and rg available in the base package of R or one of its contributed packages. In the latter case, the relevant contributed package s must be first downloaded.

For example, one must have the functions dnorm , pnorm , qnorm and rnorm available for computing the wrapped normal distribution. These functions are indeed available in the base package of R. The following choices are possible for g :. These were computed using the R package AdequacyModel due to [ 78 ].

There are other packages for fitting univariate distributions, for example, the R package fitdistrplus due to [ 79 ]. But none of these packages give as much output as [ 78 ] gives. The wrapped distribution must be specified by g and K as explained before. Here, we provide several illustrations of the practical use of the package Wrapped. The first illustration plots the PDFs of the wrapped beta normal, wrapped skew normal, wrapped asymmetric Laplace and wrapped skew t type 3 distributions for selected parameter values.

The second illustration plots the CDFs of the wrapped beta normal, wrapped skew normal, wrapped asymmetric Laplace and wrapped skew t type 3 distributions for selected parameter values. The third illustration plots the quantile functions of the wrapped beta normal, wrapped skew normal, wrapped asymmetric Laplace and wrapped skew t type 3 distributions for selected parameter values. The fourth illustration plots the histograms of the radii of random numbers generated from the wrapped beta normal, wrapped skew normal, wrapped asymmetric Laplace and wrapped skew t type 3 distributions for selected parameter values.

We see that a variety of symmetric and asymmetric shapes are possible for the PDFs, CDFs, quantile functions and histograms. The data used are 30 cross-beds azimuths of palaeocurrents from [ 80 ]. For each of the five fitted wrapped distributions, the output gives the parameter estimates, standard errors, 95 percent confidence intervals, value of Akaike Information Criterion, value of Consistent Akaike Information Criterion, value of Bayesian Information Criterion, value of Hannan Quinn Information Criterion, Cramer von Misses statistic value, Anderson Darling statistic value, minimum value of the negative log likelihood, Kolmogorov Smirnov statistic value, its p value and convergence status of the minimization of the negative log likelihood.

The output for the fitted wrapped normal distribution is as follows. The output for the fitted wrapped logistic distribution is as follows. The output for the fitted wrapped Gumbel distribution is as follows.

The output for the fitted wrapped Laplace distribution is as follows. The standard errors appear small compared to the parameter estimates for each fitted distribution. Also the p -value for each fitted distribution appears acceptable at the five percent significance level. The wrapped normal distribution gives the smallest values for the Cramer von Misses statistic, Anderson Darling statistic, Akaike Information Criterion, Consistent Akaike Information Criterion, Bayesian Information Criterion, Hannan Quinn information criterion and the minimum of the negative log likelihood function.

But the wrapped logistic distribution gives the smallest Kolmogorov Smirnov test statistic and the largest p -value. We now check to the goodness of these approximations and recommend a value for K.

Thereafter it decreases approximately linearly. Thereafter they increase approximately linearly. The figures were similar for other wrapped distributions and a wide range of parameter values. Furthermore, the variation with respect to K was always approximately linear. We do not claim that our package is an umbrella for other packages that analyze wrapped distributions. But other packages in R only implement the wrapped Cauchy, wrapped normal, wrapped skew normal, wrapped stable and wrapped normal mixtures distributions.

Our package can compute the pdf, cdf, quantile function and random samples for any given parametric forms for g and G that is, parametric forms for which the functions dg , pg , qg and rg are available in the base package of R or one of its contributed packages. Our package can also compute the following for 41 different wrapped distributions: maximum likelihood estimates of the parameters, standard errors, 95 percent confidence intervals, value of Cramer von Misses statistic, value of Anderson Darling statistic, value of Kolmogorov Smirnov test statistic and its p -value, value of Akaike Information Criterion, value of Consistent Akaike Information Criterion, value of Bayesian Information Criterion, value of Hannan Quinn Information Criterion, minimum value of the negative log likelihood function and convergence status when some data are fitted by the wrapped distribution.

Hence, our package is a lot more applicable. If the chosen g and G do not belong to one of the 41 distributions mentioned here, then our package will need updating to allow performing estimation. Nevertheless, the pdf, cdf, quantile function and random samples of the wrapped distribution can still be computed for the chosen g and G as long as the functions dg , pg , qg and rg are available in the base package of R or one of its contributed packages.

A future work is to develop similar R packages for bivariate and multivariate wrapped distributions. Another future work is to extend the package to cases when g is defined on domains different from the entire real line or when g is the probability mass function of a discrete random variable.

The authors would like to thank the two referees and the Editor for careful reading and comments which greatly improved the paper. Browse Subject Areas?

Click through the PLOS taxonomy to find articles in your field. Abstract The Wrapped package computes the probability density function, cumulative distribution function, quantile function and also generates random samples for many univariate wrapped distributions. Data Availability: All relevant data are within the paper. Funding: The authors received no specific funding for this work. Introduction Circular data are data recorded in degrees or radii.

The package performs the following: i computes 1 , 2 and the corresponding quantile function for given parametric forms for g and G. The wrapped distribution must be one of those mentioned in iii. The contributed R package evd due to [ 33 ] is used to compute this PDF g. The contributed R package sn due to [ 37 ] is used to compute this PDF g.

The contributed R package ald due to [ 40 ] is used to compute this PDF g. The contributed R package ald due to [ 42 ] is used to compute this PDF g. The contributed R package glogis due to [ 44 ] is used to compute this PDF g. The contributed R package sld due to [ 46 ] is used to compute this PDF g. The contributed R package normalp due to [ 47 ] is used to compute this PDF g.

The contributed R package gamlss. The contributed R package cubfits due to [ 62 ] is used to compute this PDF g. The contributed R package lqmm due to [ 64 ] is used to compute this PDF g. The contributed R package ordinal due to [ 77 ] is used to compute this PDF g. Illustrations Here, we provide several illustrations of the practical use of the package Wrapped.

Download: PPT. Fig 1. PDFs of the wrapped beta normal four plots in the top left , wrapped skew normal four plots in the top right wrapped asymmetric Laplace four plots in the bottom left and wrapped skew t type 3 four plots in the bottom right distributions for selected parameter values.

Fig 2. CDFs of the wrapped beta normal four plots in the top left , wrapped skew normal four plots in the top right wrapped asymmetric Laplace four plots in the bottom left and wrapped skew t type 3 four plots in the bottom right distributions for selected parameter values.

Fig 3. Qunatile functions of the wrapped beta normal four plots in the top left , wrapped skew normal four plots in the top right wrapped asymmetric Laplace four plots in the bottom left and wrapped skew t type 3 four plots in the bottom right distributions for selected parameter values. Fig 4. Histograms of random numbers generated from the wrapped beta normal four plots in the top left , wrapped skew normal four plots in the top right wrapped asymmetric Laplace four plots in the bottom left and wrapped skew t type 3 four plots in the bottom right distributions for selected parameter values.

Discussion We do not claim that our package is an umbrella for other packages that analyze wrapped distributions. Acknowledgments The authors would like to thank the two referees and the Editor for careful reading and comments which greatly improved the paper. References 1. Scandinavian Journal of Statistics.

Errors in Statistical Inference Under Model Misspecification: Evidence, Hypothesis Testing, and AIC

We now give some examples of how to use the binomial distribution to perform one-sided and two-sided hypothesis testing. Determine whether the die is biased. We use the following null and alternative hypotheses:. Example 2 : We suspect that a coin is biased towards heads. When we toss the coin 9 times, how many heads need to come up before we are confident that the coin is biased towards heads? INV 9,.

As we learned from our work in the previous lesson, whenever we perform a hypothesis test, we should make sure that the test we are conducting has sufficient power to detect a meaningful difference from the null hypothesis. Is there instead a K -test or a V -test or you-name-the-letter-of-the-alphabet-test that would provide us with more power? A very important result, known as the Neyman Pearson Lemma, will reassure us that each of the tests we learned in Section 7 is the most powerful test for testing statistical hypotheses about the parameter under the assumed probability distribution. Before we can present the lemma, however, we need to:. That is, the joint p. Any hypothesis that is not a simple hypothesis is called a composite hypothesis.

In probability theory , a normal or Gaussian or Gauss or Laplace—Gauss distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. It states that, under some conditions, the average of many samples observations of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors , often have distributions that are nearly normal.


Type II Error – failing to reject the null when it is false. The probability of a Type I Error in hypothesis testing is predetermined by the significance level.


Introduction to Type I and Type II errors

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Introduction to power in significance tests.

The Wrapped package computes the probability density function, cumulative distribution function, quantile function and also generates random samples for many univariate wrapped distributions. It also computes maximum likelihood estimates, standard errors, confidence intervals and measures of goodness of fit for nearly fifty univariate wrapped distributions. Numerical illustrations of the package are given.

The methods for making statistical inferences in scientific analysis have diversified even within the frequentist branch of statistics, but comparison has been elusive. We approximate analytically and numerically the performance of Neyman-Pearson hypothesis testing, Fisher significance testing, information criteria, and evidential statistics Royall, This last approach is implemented in the form of evidence functions: statistics for comparing two models by estimating, based on data, their relative distance to the generating process i.

Type I and Type II Errors

Computational Intelligence Methods for Brain-Machine Interfacing or Brain-Computer Interfacing

In this tutorial, we discuss many, but certainly not all, features of scipy. The intention here is to provide a user with a working knowledge of this package. We refer to the reference manual for further details. There are two general distribution classes that have been implemented for encapsulating continuous random variables and discrete random variables. Over 80 continuous random variables RVs and 10 discrete random variables have been implemented using these classes. Besides this, new routines and distributions can be easily added by the end user. If you create one, please contribute it.

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Characteristics of the standard normal distribution. The normal distribution is centered at the mean, μ. The degree to which population data values.

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