Explicit Maximum Likelihood Estimates Research Articles

Statistical inference for models driven by 𝑛-th order fractional Brownian motion

We consider the following stochastic integral equation X ( t ) = μ t + σ ∫ 0 t φ ( s ) d B H n ( s ) X(t)=\mu t + \sigma \int _0^t \varphi (s) dB_H^n(s) , t ≥ 0 t\geq 0 , where φ \varphi is a known function and B H n B^n_H is the n n -th order fractional Brownian motion. We provide explicit maximum likelihood estimators for both μ \mu and σ 2 \sigma ^2 , then we formulate explicitly a least squares estimator for μ \mu and an estimator for σ 2 \sigma ^2 by using power variations method. The consistency and asymptotic normality are established for those estimators when the number of observations or the time horizon is sufficiently large.

Design considerations for estimating survival rates with standing age structures

ABSTRACTSurvival rate estimates are critical to understanding the dynamics and status of a population, and they are often inferred from samples of the population's age structure. A recently developed method uses time series of standing age‐structure data with information about population growth rate or fecundity to provide explicit maximum likelihood estimators of age‐specific survival rates, without assuming population stability or stationarity. We explored properties of these estimators and determined sample size requirements for the estimators to achieve desired levels of precision, limit bias, and limit the probability a rate will be inestimable or its estimate inadmissible (>1). We show that estimating combined rates for adjacent age classes is an effective method of overcoming sensitivity to sampling noise in situations where collecting a larger sample of data is not feasible. Published 2018. This article is a U.S. Government work and is in the public domain in the USA.

Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models

Cyclic models are a subclass of graphical Markov models with simple, undirected probability graphs that are chordless cycles. In general, all currently known distributions require iterative procedures to obtain maximum likelihood estimates in such cyclic models. For exponential families, the relevant conditional independence constraint for a variable pair is given all remaining variables, and it is captured by vanishing canonical parameters involving this pair. For Gaussian models, the canonical parameter is a concentration, that is, an off‐diagonal element in the inverse covariance matrix, while for Ising models, it is a conditional log‐linear, two‐factor interaction. We give conditions under which the two different likelihood functions, that is, one for continuous and one for binary variables, permit nevertheless explicit maximum likelihood estimates, and we show that their estimated correlation matrices are identical, provided the relevant starting correlation matrices coincide. Copyright © 2017 John Wiley & Sons, Ltd.

Estimating discount factors for public and private goods and testing competing discounting hypotheses

The observation of declining discount rates in experimental settings has led many to promote hyperbolic discounting over standard exponential discounting as the preferred descriptive model of intertemporal choice. I develop a new framework, consistent with the random utility model, which directly models the intertemporal utility function and produces explicit maximum likelihood estimates of discounting parameters. I apply this estimation method to a stated-preference survey of river basin cleanup options and revealed-preference lottery payment choices. Formal statistical tests fail to find evidence in support of hyperbolic or quasi-hyperbolic discounting. Annual discount rates range from ten to fourteen percent across the data sets and empirical specifications.

A two-step PLS inspired method for linear prediction with group effect

In this article, we consider prediction of a univariate response from background data. The data may have a near-collinear structure and additionally group effects are assumed to exist. A two-step method is proposed. The first step summarizes the information in the predictors via a bilinear model. The bilinear model has a Krylov structured within individual design matrix, which is the link to classical partial least squares (PLS) analysis and a between-individual design matrix which handles group effects. The second step is the prediction step where a conditional expectation approach is used. The two-step method gives new insight into PLS. Explicit maximum likelihood estimators of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variables is known. It is shown that for within-sample prediction the mean squared error of the two-step method is always smaller than PLS.

On estimation in multilevel models with block circular symmetric covariance structure

In this article we consider a multilevel model with block circular symmetric covariance structure. Maximum likelihood estimation of the parameters of this model is discussed. We show that explicit maximum likelihood estimators of variance components exist under certain restrictions on the parameter space.

Estimating survival rates with time series of standing age‐structure data

It has long been recognized that age-structure data contain useful information for assessing the status and dynamics of wildlife populations. For example, age-specific survival rates can be estimated with just a single sample from the age distribution of a stable, stationary population. For a population that is not stable, age-specific survival rates can be estimated using techniques such as inverse methods that combine time series of age-structure data with other demographic data. However, estimation of survival rates using these methods typically requires numerical optimization, a relatively long time series of data, and smoothing or other constraints to provide useful estimates. We developed general models for possibly unstable populations that combine time series of age-structure data with other demographic data to provide explicit maximum likelihood estimators of age-specific survival rates with as few as two years of data. As an example, we applied these methods to estimate survival rates for female bison (Bison bison) in Yellowstone National Park, USA. This approach provides a simple tool for monitoring survival rates based on age-structure data.

Estimating Discount Factors for Public and Private Goods and Testing Competing Discounting Hypotheses

The observation of declining discount rates in experimental settings has led many to promote hyperbolic discounting over standard exponential discounting as the preferred descriptive model of intertemporal choice. I develop a new framework, consistent with the random utility model, which directly models the intertemporal utility function and produces explicit maximum likelihood estimates of discounting parameters. I apply this estimation method to a stated-preference survey of river basin cleanup options and revealed-preference lottery payment choices. Formal statistical tests fail to find evidence in support of hyperbolic or quasi-hyperbolic discounting. Annual discount rates range from eight to thirteen percent across the data sets and empirical specifications.

Maximum Likelihood Estimators in a Two Step Model for PLS

Univariate partial least squares regression (PLS1) is a method of modeling relationships between a response variable and explanatory variables, especially when the explanatory variables are almost collinear. The purpose is to predict a future response observation, although in many applications there is an interest to understand the contributions of each explanatory variable. It is an algorithmic approach. In this article, we are going to use the algorithm presented by Helland (1988). The population PLS predictor is linked to a linear model including a Krylov design matrix and a two-step estimation procedure. For the first step, the maximum likelihood approach is applied to a specific multivariate linear model, generating tools for evaluating the information in the explanatory variables. It is shown that explicit maximum likelihood estimators of the dispersion matrix can be obtained where the dispersion matrix, besides representing the variation in the error, also includes the Krylov structured design matrix describing the mean.

Density of tiger and leopard in a tropical deciduous forest of Mudumalai Tiger Reserve, southern India, as estimated using photographic capture–recapture sampling

Density of tiger Panthera tigris and leopard Panthera pardus was estimated using photographic capture–recapture sampling in a tropical deciduous forest of Mudumalai Tiger Reserve, southern India, from November 2008 to February 2009. A total of 2,000 camera trap nights for 100 days yielded 19 tigers and 29 leopards within an intensive sampling area of 107 km2. Population size of tiger from closed population estimator model Mb Zippin was 19 tigers (SE = ±0.9) and for leopards Mh Jackknife estimated 53 (SE = ±11) individuals. Spatially explicit maximum likelihood and Bayesian model estimates were 8.31 (SE = ±2.73) and 8.9 (SE = ±2.56) per 100 km2 for tigers and 13.17 (SE = ±3.15) and 13.01 (SE = ±2.31) per 100 km2 for leopards, respectively. Tiger density for MMDM models ranged from 6.07 (SE = ±1.74) to 9.72 (SE = ±2.94) per 100 km2 and leopard density ranged from 13.41 (SE = ±2.67) to 28.91 (SE = ±7.22) per 100 km2. Spatially explicit models were more appropriate as they handle information at capture locations in a more specific manner than some generalizations assumed in the classical approach. Results revealed high density of tiger and leopard in Mudumalai which is unusual for other high density tiger areas. The tiger population in Mudumalai is a part of the largest population at present in India and a source for the surrounding Reserved Forest.

Estimating Discount Factors within a Random Utility Theoretic Framework

Choices involving tradeoffs of benefits and costs over time are pervasive in our everyday lives. The observation of declining discount rates in experimental settings has led many to promote hyperbolic discounting over standard exponential discounting as the preferred descriptive model of intertemporal choice. In this paper, I develop a new framework that directly models the intertemporal utility function associated with an intertemporal outcome. This random utility model produces explicit maximum likelihood estimates of the discounting parameters. The main benefit of this approach is that I am able to perform formal statistical tests of quasi-hyperbolic and hyperbolic discounting, which has not been done previously in the economics literature. I apply this estimation method to two original data sets, a stated-preference survey of cleanup options for the Minnesota River Basin and revealed-preference choices of lottery payment options, in addition to one published data set. Formal statistical tests fail to find evidence in support of hyperbolic or quasi-hyperbolic discounting. Constant (exponential) annual discount rates range from eight to eleven percent over the three data sets, which are lower than those usually found in experimental studies but consistent with interest rates found in capital markets. I propose that confounding experimental artifacts may be responsible for previous evidence in favor of hyperbolic discounting. Specifically, uncertainty in future rewards, perceived future transaction costs, and subadditive discounting may confound estimates of rates of time preference (discount rates) from previous experimental designs.

Inadmissibility of the Maximum Likekihood Estimator of Normal Covariance Matrices with the Lattice Conditional Independence

Lattice conditional independence (LCI) models introduced by S. A. Andersson and M. D. Perlman (1993, Ann. Statist.21, 1318–1358) have the pleasant feature of admitting explicit maximum likelihood estimators and likelihood ratio test statistics. This is because the likelihood function and parameter space for a LCI model can be factored into products of conditional likelihood functions and parameter spaces, where the standard multivariate techniques can be applied. In this paper we consider the problem of estimating the covariance matrices under LCI restriction in a decision theoretic setup. The Stein loss function is used in this study and, using the factorization mentioned above, minimax estimators are obtained. Since the maximum likelihood estimator has constant risk and is different from the minimax estimator, this shows that the maximum likelihood estimator under LCI restriction inadmissible. These results extend those obtained by W. James and C. Stein (1960, in “Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics, and Probability,” Vol. 1, pp. 360–380, Univ. of California Press, Berkeley, CA) and D. K. Dey and C. Srinivasan (1985, Ann. Statist.13, 1581–1591) for estimating normal covariance matrices to the LCI models.

Joint analysis of mark—recapture, resighting and ring-recovery data with age-dependence and marking-effect

Using the joint analysis of live-recapture and ring-recovery data described by Burnham in 1993, we generalize it to the case where resightings of live birds are also reported between capture periods. Explicit estimators under 'random' temporary emigration correspond to those suggested by Jolly in 1965. The model is further generalized to admit age-dependence and a temporary marking effect. Explicit maximum-likelihood estimators under random emigration are reported and also contingency table tests for age-dependence and temporary marking effect. The new analysis is illustrated using a five-year subset of data from a 25-year ringing programme for Goldeneye Ducks Bucephala clangula¸. Alternative forms of emigration from the study area are also discussed.

INFERENCE OF SPERM COMPETITION FROM BROODS OF FIELD-CAUGHT DROSOPHILA.

In field studies of multiple mating and sperm competition there typically is no experimental control over the number of times that a female mates, the interval between matings, or the genetic identity of multiple fathers contributing to a brood. Irrespective of this complexity, high-resolution molecular markers can be used to assign paternity with considerable confidence. This study employed two highly heterozygous microsatellite loci to assess multiple paternity and sperm displacement in a sample of broods taken from a natural population of Drosophila melanogaster. The large number of alleles present at each of the loci makes it difficult to derive explicit maximum-likelihood estimates for multiple paternity and sperm displacement from brood samples. Monte Carlo simulations were used to estimate maximum-likelihood parameters for the distribution of female remating frequency and the proportion of offspring sired by the second or subsequent mating males. Estimates were made based on genotypes scored at two distinct marker loci because they were found to give statistically homogeneous results. Fitting a Poisson distribution of number of matings, the mean number of males mated by a female was 1.82. The sperm displacement parameter estimated from doubly mated females were 0.79 and 0.86 for the two loci (0.83 for the joint estimate). The overall probability that a multiply mated female will be misclassified as singly mated was only 0.006, which indicates that microsatellites can provide excellent resolution for identifying multiple mating. In addition, microsatellites can be used to generate relatively precise estimates of sperm precedence in brood-structured samples from a natural population.

A characterization of Markov equivalence classes for acyclic digraphs

Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.

Catch-Effort Estimation of Population Parameters under the Robust Design

Catch-effort models, as well as models for many other animal abundance estimation techniques, are distinctly divided in their application to either closed or open populations. Pollock (1982, Journal of Wildlife Management 46, 752-757) first attempted combining open-population and closed-population models in the capture-recapture literature and constructed what is now known as the robust design. In light of the advantages of the capture-recapture robust design, a similar sampling scheme is of interest in the catch-effort estimation framework. Here a description of the robust design for catch-effort methods is provided, complete with explicit maximum likelihood estimators for a range of open models. The presence of mortality and recruitment are handled in the model development sequentially. Monte Carlo simulations were used to evaluate the performance of maximum likelihood estimators under the robust design in comparison with a previously defined regression estimator. The robust design provided for greater model flexibility and in almost all circumstances produced maximum likelihood estimators that were superior to those estimated by regression methods. The advantages of the robust design are discussed for a variety of modeling scenarios.

A Note on Calculating the π* Index of Fit for the Analysis of Contingency Tables

Rudas, Clogg, and Lindsay (1994) proposed a mixture index-of-fit approach for evaluating goodness of fit in the analysis of contingency tables. Clogg, Rudas, and Xi (1995) applied this approach to the analysis of models for mobility tables. The maximum likelihood estimate of the mixture index of fit π* can be obtained by the expectation and maximization (EM) algorithm. The objective of this article is to describe an alternative algorithm, using sequential quadratic programming (SQP). Two major advantages SQP holds over the EM algorithm are the following: (1) The speed of convergence is greatly increased and (2) SQP is more general than the EM algorithm of Rudas et al. and does not require explicit maximum likelihood estimates. The SQP algorithm is available in the MATLAB package; a sample code is provided in the appendix.

Bayesian model averaging and model selection for markov equivalence classes of acyclic digraphs

Acyclic digraphs (ADGs) are widely used to describe dependences among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. There may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Recent results have shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself Markov-equivalent simultaneously to all ADGs in the equivalence class. Here we propose t...

A Note on Seber's Model for Ring-Recovery Data

We provide simple explicit maximum-likelihood estimates for Seber's (1971) model for ring-recovery data, obtaining both the general solution, and the particular estimate satisfying Seber's constraint.

A note on Seber's model for ring-recovery data

SUMMARY We provide simple explicit maximum-likelihood estimates for Seber's (1971) model for ringrecovery data, obtaining both the general solution, and the particular estimate satisfying Seber's constraint.