Introduction to the Theory of Statistical Inference

Liero, Hannelore; Zwanzig, Silvelyn

doi:10.1201/9781466503205

Cited by 10 publications

(15 citation statements)

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“…If it can be stated that the distribution belongs to an exponential family one might be able to reduce the statistic to a lower dimension without the risk of losing information. [1] Definition 1.1. A statistical model is a class of probability measures…”

Section: Exponential Familiesmentioning

confidence: 99%

“…The exponential expression of the probability function will determine the statistical properties. The functions η = (η 1 , .., η k ) and T = (T 1 , ..., T k ) and k are not uniquely determined but we refer to equation (1) as a k-parameter exponential family. Example 1.1 (Normal distribution).…”

Section: Definition 13 (Exponential Family)mentioning

confidence: 99%

“…This makes it a strong property since it can be viewed as a way of data reduction, where all the important information in the sample is condensed into the statistic. [1] Definition 3.1. (Sufficient Statistic) A statistic T is said to be sufficient for the statistical model P = {P θ : θ ∈ Θ} of X if the conditional distribution of X given T = t is independent of θ for all t.…”

Section: Sufficiencymentioning

confidence: 99%

“…The previous sections leads up to these theorems. [1] Theorem 5.1 (Rao-Blackwell). Let T be a sufficient statistic for the statistical model P, and letγ be an unbiased estimator for the parameter γ = g(θ) ∈ R k .…”

Section: The Rao-blackwell and Lehmann-scheffé Theoremsmentioning

confidence: 99%

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Curved Exponential Families

Sundberg

2019

Statistical Modelling by Exponential Families

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This paper covers theory of inference statistical models that belongs to curved exponential families. Some of these models are the normal distribution, binomial distribution, bivariate normal distribution and the SSR model. The purpose was to evaluate the belonging properties such as sufficiency, completeness and strictly k-parametric. Where it was shown that sufficiency holds and therefore the Rao Blackwell Theorem but completeness does not so Lehmann Scheffé Theorem cannot be applied.

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Section: Exponential Familiesmentioning

confidence: 99%

Section: Definition 13 (Exponential Family)mentioning

confidence: 99%

Section: Sufficiencymentioning

confidence: 99%

Section: The Rao-blackwell and Lehmann-scheffé Theoremsmentioning

confidence: 99%

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Curved Exponential Families

Sundberg

2019

Statistical Modelling by Exponential Families

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“…Moreover, the bias of the maximum likelihood estimator is sensitive to the distributional assumption, censoring percentage and the type of the statistic one wishes to estimate, whereas the rROS, GROS, and Kaplan-Meier estimators have the same magnitude of bias. In the statistical literature, the bias ofû Ã can be accepted only if it tends to vanish as the sample size increases (for example, see Liero and Zwanzig [24]). This property is defined as asymptotic unbiasedness where theû Ã approaches the true value u, as the sample size n tends to 1.…”

Section: Uncertainty and Approximated Bias Of The Estimatesmentioning

confidence: 99%

An investigation of the impact of left-censored soil contamination data on the uncertainty of descriptive statistical parameters

Shoari¹,

Dubé²

2016

Environmental Toxicology and Chemistry

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Left-censored concentration data are frequently encountered because measuring instruments cannot detect concentrations below the instrument detection limit. For statistical analysis of left-censored data, the environmental literature mainly refers to the following methods: maximum likelihood estimator, regression on order statistics using log-normal and gamma assumption (rROS and GROS, respectively), and Kaplan-Meier. A number of simulation experiments examined the performance of these methods in terms of bias and/or mean square error. However, no matter which method is adopted, some uncertainty is introduced into outcomes because all that is known about a left-censored observation is that the concentration falls between 0 and the detection limit. The data used in the present study come from analysis of soil samples collected for a site characterization in Montreal, Canada. Employing nonparametric bootstrap, the authors quantify the uncertainty and bias in the mean and standard deviation estimates obtained by the maximum likelihood estimation (under log-normal, Weibull, and gamma distributions), rROS, GROS, and Kaplan-Meier methods. First, the authors demonstrate that the highest uncertainty is associated with the maximum likelihood estimator under log-normality and Weibull assumptions, whereas a gamma assumption leads to estimates with less uncertainty. Second, the authors show that although an increase in sample size improves the uncertainty, it reduces the bias only in the rROS, GROS, and Kaplan-Meier methods. Finally, comparing percentage uncertainty in the mean of contaminant data, the authors illustrate that adopting an inappropriate estimator results in large uncertainties. Environ Toxicol Chem 2016;35:2623-2631. © 2016 SETAC.

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High‐precision target echo generation technology based on one‐dimensional linear array radar

Peng

Lan²,

et al. 2019

J. eng.

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Introduction to the Theory of Statistical Inference

Cited by 10 publications

References 0 publications

Curved Exponential Families

Curved Exponential Families

An investigation of the impact of left-censored soil contamination data on the uncertainty of descriptive statistical parameters

High‐precision target echo generation technology based on one‐dimensional linear array radar

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