George Seber

University of Auckland · Department of Statistics

One of the main methods of adaptive sampling is adaptive cluster sampling. As it involves unequal probability of sampling, standard Horvitz-Thompson and Hansen-Hurwitz estimators can be modified to provide unbiased estimates of finite population parameters along with unbiased variance estimators. These estimators are compared with each other and wi...

This book endeavors to provide a concise and integrated overview of hypothesis testing in four important subject areas, namely linear and nonlinear models, multivariate analysis, and large sample theory. The approach used is a geometrical one based on the concept of projections and their associated idempotent matrices, thus avoiding getting too inv...

Many people today believe that Christianity will not stand up to a scientific or intellectual investigation, and that science has all the answers. Such an attitude shows an ignorance of the wealth of available philosophical arguments and scientific information that Dr Seber taps into in this book. Initially he shows that mathematics and science are...

Least Squares EstimationProperties of Least Squares EstimatesUnbiased Estimation of σ2Distribution TheoryMaximum Likelihood EstimationOrthogonal Columns in the Regression MatrixIntroducing Further Explanatory VariablesEstimation with Linear RestrictionsDesign Matrix of Less Than Full RankGeneralized Least SquaresCentering and Scaling the Explanator...

Based on some theoretical results, we recommend a new algorithm for estimating the total and mean of a subpopulation variable for the case of a known subpopulation size, which is different from the algorithm recommended by most of sampling books. The latter usually recommend the multiplication of the subpopulation sample mean by the subpopulation s...

An in-depth study of the effects of alcohol on users, the community, the nation of New Zealand, and globally.

Capture–recapture methods have a long history and in this introductory chapter we begin by briefly describing some of the defining research papers up to about 1950. The rest of the book takes us up to the beginning of 2019. Model building plays an essential part in this book and some basic ideas are described relating to fixed and random sample siz...

This short stand-alone chapter is introduced mainly for historical reasons, though it has some interesting ideas and has had a resurgence for closed populations along with some new R software. For open and closed populations, the method consists of taking logarithms of the expected values of random variables, and then applying a two-way analysis of...

This chapter could occupy several different places in this book as it is linked to both dead recovery models and continuous models. The latter have already been mentioned in earlier chapters where natural and exploitation mortalities are modeled using Poisson processes with instantaneous rates. The chapter begins with laying out the general theory...

This short chapter does some crystal ball gazing about the future of capture–recapture. Clearly, computer software will continue to develop along with increasingly sophisticated computers. As models get more complicated, the number of unknown parameters goes up steeply. In contrast, even if capture–recapture, resighting, and dead recovery data are...

In previous chapters, we have considered three methods of obtaining recapture data: dead recoveries, resightings, and live recaptures. As the models for all three methods have a similar structure, except for different definitions of the population parameters, it is not surprising we can combine three different pairs of methods or all three into com...

In the previous chapter, we introduced some models that have formed a historical basis for developments in capture–recapture, with some extensions. The subject has now taken off in a number of different directions that will be considered in this and the following chapters. Although the split with Chap. 5 is somewhat arbitrary, we begin this chapter...

In designing a capture–recapture experiment, a substantial check list of questions that need answers is provided, as well as references to other aspects of design (e.g., designs using radio telemetry). Test methods for model fitting are briefly described before moving into a section on various measures of model comparison such as AIC, TIC, incorpor...

Although Bayesian methods are sprinkled throughout some of the previous chapters, we now give the topic our full attention and add some extensions. Their particular advantage arises from being able to apply Markov chain Monte Carlo techniques along with so-called reversible jump methods to sample from posterior distributions. We then consider model...

Capture–recapture has made considerable advances because of progress in the development of extensive computer packages (a list is given in the Appendix). In the past, the emphasis has been on obtaining explicit maximum likelihood estimates and using mathematically derived formulae for asymptotic variances and covariances to obtain standard errors....

This scenario arises when tagged animals die through natural mortality and possibly exploitation (e.g., hunting and fisheries), and their tags are recovered. A wide variety of models for both exploited and unexploited populations are considered depending on what underlying population assumptions are made, the first being time dependence for both th...

This is a large and complex chapter which, with recent computational developments and matrix methods, is becoming a fundamental model as it includes many of the previous models as special cases (e.g., the dead recovery and CJS models). A multisite model involves applying the capture–recapture method to a number of interconnecting geographical areas...

We consider two basic live-recapture models referred to as the CJS model (after Cormack–Jolly–Seber) that models the tagged data, and the JS model (after Jolly and Seber in Biometrika, 52:225–247, 1965), which also includes the untagged data. Likelihood methods as originally used are described, while Bayesian and random-effects methods have since b...

This topic needs particular emphasis in research. Departures considered include the initial effect of tagging, tag-loss models (including telemetry data), heterogeneity, and catchability dependence, along with appropriate test procedures. Mixtures model can be used here together with incorporating age dependency in the JS (Jolly-Seber) model. State...

With new developments in technology such as with cameras and radio telemetry, we describe a wide variety of tags and markers used for uniquely identifying animals. Examples are various types of attached tags, natural markers on animals (e.g., tiger stripes, tail fins), radio tags, PIT tags that use an integrated circuit chip, water acoustic tags, g...

Instead of dead recoveries, we can obtain recapture information through simply resighting live tagged individuals using a time-specific model. This was described by Richard Cormack in 1964. Underlying assumptions are considered and tests are provided.

A range of models for closed populations are described that allow for such things as heterogeneity and variable catchability that are difficult to deal with in open populations. Pollock, in 1982, introduced the simple but elegant idea of combining models for both closed and open populations (the so-called “robust” design) by dividing the study peri...

Describes what happens when person is dying, e.g., gasping may be just trying to talk. Also considers what need s to be done after the death a loved plus some counselling.

This comprehensive book, rich with applications, offers a quantitative framework for the analysis of the various capture-recapture models for open animal populations, while also addressing associated computational methods. The state of our wildlife populations provides a litmus test for the state of our environment, especially in light of global wa...

An adaptive sample involves modifying the sampling design on the basis of information obtained during the survey while remaining in the probability sampling framework. Complete allocation sampling is an efficient and easily implemented 2-phase adaptive sampling design that targets field effort to rare species and is logistically feasible. The popul...

Up till now we have been considering various univariate linear models of the form \(y_{i} =\theta _{i} +\varepsilon _{i}\) (i = 1, 2, …, n), where \(E[\varepsilon _{i}] = 0\) and the \(\varepsilon _{i}\) are independently and identically distributed. We assumed G that \(\boldsymbol{\theta }\in \varOmega\), where Ω is a p-dimensional vector space in...

Nonlinear models arise when E[y] is a nonlinear function of unknown parameters. Hypotheses about these parameters may be linear or nonlinear. Such models tend to be used when they are suggested by theoretical considerations or used to build non-linear behavior into a model. Even when a linear approximation works well, a nonlinear model may still be...

Apart from Chap. 8 on nonlinear models we have been considering linear models and hypotheses. We now wish to extend those ideas to non-linear hypotheses based on samples of n independent observations \(x_{1},x_{2},\ldots,x_{n}\) (these may be vectors) from a general probability density function \(f(x,\boldsymbol{\theta })\), where \(\boldsymbol{\th...

We assume the model \(\mathbf{y} =\boldsymbol{\theta } +\boldsymbol{\varepsilon }\), \(G:\boldsymbol{\theta }\in \varOmega\), a p-dimensional vector space in \(\mathbb{R}^{n}\), and \(H:\boldsymbol{\theta }\in \omega\), a p − q dimensional subspace of Ω; \(\boldsymbol{\varepsilon }\) is \(N_{n}[\mathbf{0},\sigma ^{2}\mathbf{I}_{n}]\). To test H we...

Suppose we have the model \(\mathbf{y} =\boldsymbol{\theta } +\boldsymbol{\varepsilon }\), where \(\mathrm{E}[\boldsymbol{\varepsilon }] = \mathbf{0}\), \(\mathrm{Var}[\boldsymbol{\varepsilon }] =\sigma ^{2}\mathbf{I}_{n}\), and \(\boldsymbol{\theta }\in \varOmega\), a p-dimensional vector space. One reasonable estimate of \(\boldsymbol{\theta }\)...

Sometimes after a linear model has been fitted it is realized that more explanatory (x) variables need to be added, as in the following examples. In an industrial experiment in which the response (y) is the yield and the explanatory variables are temperature, pressure, etc., we may wish to determine what values of the x-variables are needed to prod...

Given the model \(\mathbf{y} \sim N_{n}(\boldsymbol{\theta },\sigma ^{2}\mathbf{I}_{n})\) and assumption G that \(\boldsymbol{\theta }\in \varOmega\), a p-dimensional subspace of \(\mathbb{R}^{n}\), we wish to test the linear hypothesis \(H:\boldsymbol{\theta }\in \omega\), where ω is a p − q dimensional subspace of Ω.

In this chapter we assume once again that \(\boldsymbol{\theta }\in W\). However our hypothesis H now takes the form of freedom equations, namely \(\boldsymbol{\theta }=\boldsymbol{\theta } (\boldsymbol{\alpha })\), where \(\boldsymbol{\alpha }= (\alpha _{1},\alpha _{2},\ldots,\alpha _{p-q})'\). We require the following additional notation. Let \(\...

In this chapter we consider asymptotic theory for the multinomial distribution, which is defined below. Although the distribution used is singular, the approximating linear theory can still be used.

Let \(\boldsymbol{\theta }\) be an unknown vector parameter, let G be the hypothesis that \(\boldsymbol{\theta }\in \varOmega\), a p-dimensional vector space in \(\mathbb{R}^{n}\), and assume that \(\mathbf{y} \sim N_{n}[\boldsymbol{\theta },\sigma ^{2}\mathbf{I}_{n}]\).

Ini this chapter we consider a number of linear hypotheses before giving a general definition. Our first example is found in regression analysis.

Linear algebra is used extensively throughout this book and those topics particularly relevant to the development in this monograph are given within the chapters; other results are given in the Appendix. References to the Appendix are labeled with a prefix “A”, for example A.3 is theorem 3 in the Appendix. Vectors and matrices are denoted by boldfa...

This very large 95-chapter book on linear algebra contains three chapters on probability and statistics. It just lists results and some examples

We establish the asymptotic equivalence of several test procedures for testing hypotheses about the Multinomial distribution, namely the Likehood-ratio, Wald, Score, and Pearson’s goodness-of-fit tests. Particular emphasis is given to contingency tables, especially \(2\times 2\) tables where exact and approximate test methods are given, including m...

We discuss the Multi-hypergeometric and Multinomial distributions and their properties with the focus on exact and large sample inference for comparing two proportions or probabilities from the same or different populations. Relative risks and odds ratios are also considered. Maximum likelihood estimation, asymptotic normality theory, and simultane...

The Binomial distribution and its properties are discussed in detail including maximum likelihood estimation of the probability \(p\). Exact and approximate hypothesis tests and confidence intervals are provided for \(p\). Inverse sampling and the Negative Binomial Distribution are also considered.

This chapter focusses on the problem of estimating a population proportion using random sampling with or without replacement, or inverse sampling. Exact and approximate confidence intervals are discussed using the Hypergeometric distribution. Applications to capture-recapture models are given.

In this chapter we consider log-linear and logistic models for handling contingency tables, multinomial distributions, and binomial data. The role of the deviance in hypothesis testing is discussed. The log-linear model is applied to an epidemiological problem involving the merging of incomplete lists.

It covers all the main issues that can arise in the counseling room. Chapter headings are: Brain matters, Wholeness, Anger, Guilt and shame, Stress, Anxiety and fear, Anxiety disorders, Compulsive disorders, Depression, Suicide risk, Grief and loss, Addictions: general, Substance addictions, Addictions: behavioral, Adults abused as children, Abused...

Models are given for inference about proportions and probabilities including single, paired, and multiple comparisons. Multinomial, log-linear and logistic models are considered briefly

Inverse sampling is an adaptive method whereby it is the sample size that is adaptive. On the basis of a new proof, Murthy’s estimator can now be applied with or without adaptive cluster sampling to inverse sampling to provide unbiased estimators of the mean and variance of the mean estimator. A number of sequential plans along with parameter estim...

This chapter summarizes some foundational theory for adaptive sampling methods. The Rao-Blackwell theorem can be applied to unbiased estimators to provide more efficient estimators. Closed form expressions for these and related estimators are discussed. The theory is also applied to selecting networks without replacement, and the question of ignori...

In adaptive sampling, the sampling units can sometimes be divided into primary and secondary units. After a sample of primary units is taken, adaptive cluster sampling can be carried out within each primary unit selected using either a sample or all of its secondary units. Primary units can be a variety of shapes such as strip transects or Latin sq...

Adaptive allocation is a form of sampling whereby information from an initial phase of sampling is use to allocate the effort for further sampling, usually referred to as the the second phase. The material in this chapter is an extension of the material of the previous chapter with its emphasis on stratified sampling and two-stage sampling. A numbe...

In this chapter we consider the problem of estimating such quantities as the number of objects, the total biomass, or total ground cover in a finite population from a sample. Various traditional methods of sampling such as sampling with or without replacement, inverse sampling, and unequal probability sampling are often inadequate when the populati...

IntroductionWhy Select?Choosing the Best SubsetStepwise MethodsShrinkage MethodsBayesian Methods Effect of Model Selection on InferenceComputational ConsiderationsComparison of Methods

IntroductionLikelihood Ratio TestF-TestMultiple Correlation CoefficientCanonical Form for HGoodness-of-Fit TestF-Test and Projection Matrices

Polynomials in One VariablePiecewise Polynomial FittingPolynomial Regression in Several Variables

IntroductionBiasIncorrect Variance MatrixEffect of OutliersRobustness of the F-Test to NonnormalityEffect of Random Explanatory VariablesCollinearity

Simultaneous Interval EstimationConfidence Bands for the Regression SurfacePrediction Intervals and Bands for the ResponseEnlarging the Regression Matrix

IntroductionDirect Solution of the Normal EquationsQR DecompositionSingular Value DecompositionWeighted Least SquaresAdding and Deleting Cases and VariablesCentering the DataComparing Methods Rank-Deficient CaseComputing the Hat Matrix DiagonalsCalculating Test StatisticsRobust Regression Calculations

IntroductionResiduals and Hat Matrix DiagonalsDealing with CurvatureNonconstant Variance and Serial CorrelationDepartures from NormalityDetecting and Dealing with OutliersDiagnosing Collinearity

The Straight LineStraight Line through the OriginWeighted Least Squares for the Straight LineComparing Straight LinesTwo-Phase Linear RegressionLocal Linear Regression

IntroductionOne-Way ClassificationTwo-Way Classification (Unbalanced)Two-Way Classification (Balanced)Two-Way Classification (One Observation per Mean)Higher-Way Classifications with Equal Numbers per MeanDesigns with Simple Block StructureAnalysis of Covariance

Density FunctionMoment Generating FunctionsStatistical IndependenceDistribution of Quadratic Forms

NotationStatistical ModelsLinear Regression ModelsExpectation and Covariance OperatorsMean and Variance of Quadratic FormsMoment Generating Functions and Independence

Introduction Estimation Testing for the dean Linear Constraints on the Mean Inference for the Dispersion Matrix Comparing Two Normal Populations

Introduction One-way Classification Randomized Block Design Two-way Classification With Equal Observations per Mean Analysis of Covariance Multivariate Cochran's Theorem on Quadratics Growth Curve Analysis

Computational Techniques Log-Linear Models for Binary Data Incomplete Data

Least Squares Estimation Properties of Least Squares Estimates Least Squares With Linear Constraints Distribution Theory Analysis of Residuals Hypothesis Testing A Generalized Linear Hypothesis Step-Down Procedures Multiple Design Models

Notation What Is Multivariate Analysis Expectation and Covariance Operators Sample Data Mahalanobis Distances and Angles Simultaneous Inference Likelihood Ratio Tests

General Definitions Some Continuous Univariate Distributions Glossary of Notation

Limits Sequences Asymptotically Equivalent Sequences Series Matrix Functions Matrix Exponentials

Unknown Vector Unknown Matrix

Cauchy-Schwarz inequalities Hölder's Inequality and Extensions Minkowski's Inequality and Extensions Weighted Means Quasilinearization (Representation) Theorems Some Geometrical Properties Miscellaneous Inequalities

Kronecker Product Vec Operator Vec-Permutation (Commutation) Matrix Generalized Vec-Permutation Matrix Vech Operator Star Operator Hadamard Product Rao-Khatri Product

Some General Properties Matrix Products Matrix Cancellation Rules Matrix Sums Matrix Differences Partitioned and Patterned Matrices Maximal and Minimal Ranks Matrix Index

Introduction Generalized Quadratic Forms Random Samples Multivariate Linear Model Dimension Reduction Techniques Procrustes Analysis (Matching Configurations) Some Specific Random Matrices Allocation Problems Matrix-Variate Distributions Matrix Ensembles

Schur Complement Inverses Determinants Positive and Non-negative Definite Matrices Eigenvalues Generalized Inverses Miscellaneous partitions

Complex Matrices Hermitian Matrices Skew-Hermitian Matrices Complex Symmetric Matrices Real Skew-Symmetric Matrices Normal Matrices Quaternions

Stationary Values Using Convex and Concave Functions Two General Methods Optimizing a Function of a Matrix Optimal Designs

Introduction Non-negative Definite Matrices Positive Definite Matrices Pairs of Matrices

Introduction Spectral Radius Canonical Form of a Non-negative Matrix Irreducible Matrices Leslie Matrix Stochastic Matrices Doubly Stochastic Matrices

Definitions Weak Inverses Other Inverses Moore-Penrose (g1234) Inverse Group Inverse Some General Properties of Inverses

Many scientific problems involve modeling the relationship between variables. Regression analysis is concerned with modeling the behavior of a variable Y (the response or dependent variable) in terms of a set x1, …, xp of explanatory variables (also called covariates, regressors, and independent variables). The model takes the form E(Y) = f (x1, ,...

Not having a variance estimator is a seriously weak point of a sampling design from a practical perspective. This paper provides unbiased variance estimators for several sampling designs based on inverse sampling, both with and without an adaptive component. It proposes a new design, which is called the general inverse sampling design, that avoids...

IntroductionTerminologySecond-Derivative (Modified Newton) MethodsFirst-Derivative Methods Methods without DerivativesMethods for Nonsmooth FunctionsSummary