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Likelihood Interval
In statistics, suppose that we have been given some data, and we are selecting a statistical model for that data. The relative likelihood compares the relative plausibilities of different candidate models or of different values of a parameter of a single model. Relative likelihood of parameter values Assume that we are given some data for which we have a statistical model with parameter . Suppose that the maximum likelihood estimate for is \hat. Relative plausibilities of other values may be found by comparing the likelihoods of those other values with the likelihood of \hat. The ''relative likelihood'' of is defined to be :\frac where \mathcal(\theta \mid x) denotes the likelihood function. Thus, the relative likelihood is the likelihood ratio with fixed denominator \mathcal(\hat \mid x). The function :\theta \mapsto \frac is the ''relative likelihood function''. Likelihood region A ''likelihood region'' is the set of all values of whose relative likelihood is greater th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coverage Probability
In statistics, the coverage probability is a technique for calculating a confidence interval which is the proportion of the time that the interval contains the true value of interest. For example, suppose our interest is in the mean number of months that people with a particular type of cancer remain in remission following successful treatment with chemotherapy. The confidence interval aims to contain the unknown mean remission duration with a given probability. This is the "confidence level" or "confidence coefficient" of the constructed interval which is effectively the "nominal coverage probability" of the procedure for constructing confidence intervals. The "nominal coverage probability" is often set at 0.95. The ''coverage probability'' is the actual probability that the interval contains the true mean remission duration in this example. If all assumptions used in deriving a confidence interval are met, the nominal coverage probability will equal the coverage probability (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model Validation
In statistics, model validation is the task of evaluating whether a chosen statistical model is appropriate or not. Oftentimes in statistical inference, inferences from models that appear to fit their data may be flukes, resulting in a misunderstanding by researchers of the actual relevance of their model. To combat this, model validation is used to test whether a statistical model can hold up to permutations in the data. This topic is not to be confused with the closely related task of model selection, the process of discriminating between multiple candidate models: model validation does not concern so much the conceptual design of models as it tests only the consistency between a chosen model and its stated outputs. There are many ways to validate a model. Residual plots plot the difference between the actual data and the model's predictions: correlations in the residual plots may indicate a flaw in the model. Cross validation is a method of model validation that iteratively ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model Specification
In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, given personal income y together with years of schooling s and on-the-job experience x, we might specify a functional relationship y = f(s,x) as follows: : \ln y = \ln y_0 + \rho s + \beta_1 x + \beta_2 x^2 + \varepsilon where \varepsilon is the unexplained error term that is supposed to comprise independent and identically distributed Gaussian variables. The statistician Sir David Cox has said, "How hetranslation from subject-matter problem to statistical model is done is often the most critical part of an analysis". Specification error and bias Specification error occurs when the functional form or the choice of independent variables poorly represent relevant aspects of the true data-generating process. In particular, bias (the expected value of the di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model Selection
Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is well-suited to the problem of model selection. Given candidate models of similar predictive or explanatory power, the simplest model is most likely to be the best choice (Occam's razor). state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, has said, "How hetranslation from subject-matter problem to statistical model is done is often the most critical part of an analysis". Model selection may also refer to the problem of selecting a few representative models from a large set of computational models for the purpose of decision making or optimization under uncertainty. Introduction In its most basic forms, model selection is one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Function
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more ''likely'' than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as \mathcal(\theta\mid X) instead of P(X \mid \theta), to emphasize that it is to be understood as a function of the parameters \theta instead of the random variable X. In maximum likelihood estimation, the arg max of the likelihood function serves as a point estimate for \theta, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akaike Information Criterion
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model. In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting. The Akaike information criterion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akaike Information Criterion
The Akaike information criterion (AIC) is an estimator of prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. AIC is founded on information theory. When a statistical model is used to represent the process that generated the data, the representation will almost never be exact; so some information will be lost by using the model to represent the process. AIC estimates the relative amount of information lost by a given model: the less information a model loses, the higher the quality of that model. In estimating the amount of information lost by a model, AIC deals with the trade-off between the goodness of fit of the model and the simplicity of the model. In other words, AIC deals with both the risk of overfitting and the risk of underfitting. The Akaike information criterion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi-squared Distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral chi-squared distribution. The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilks' Theorem
In statistics Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals for maximum-likelihood estimates or as a test statistic for performing the likelihood-ratio test. Statistical tests (such as hypothesis testing) generally require knowledge of the probability distribution of the test statistic. This is often a problem for likelihood ratios, where the probability distribution can be very difficult to determine. A convenient result by Samuel S. Wilks says that as the sample size approaches \infty, the distribution of the test statistic -2 \log(\Lambda) asymptotically approaches the chi-squared (\chi^2) distribution under the null hypothesis H_0. Here, \Lambda denotes the likelihood ratio, and the \chi^2 distribution has degrees of freedom equal to the difference in dimensionality of \Theta and \Theta_0, where \Theta is the full parameter space and \Theta_0 is the subset of the parameter space assoc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Probability
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Credible Interval
In Bayesian statistics, a credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals and confidence regions in frequentist statistics, although they differ on a philosophical basis: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value. Also, Bayesian credible intervals use (and indeed, require) knowledge of the situation-specific prior distribution, while the frequentist confidence intervals do not. For example, in an experiment that determines the distribution of possible values of the parameter \mu, if the subjective probability that \mu lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
