Explicit Maximum Likelihood Estimates Research Articles

On maximum likelihood estimators for the multisite lag‐one streamflow model: Complete and incomplete data cases

Two practical problems may arise when using the traditional estimators of the parameters in the multisite lag‐one streamflow model: (1) the estimated covariance matrix may not be positive definite thereby preventing its decomposition, a necessary step for synthetic streamflow generation, and (2) if there are missing observations, streamflow data must be truncated to the shortest record resulting in a loss of useful information. This note draws attention to the existence of maximum likelihood estimators which overcome both problems. In the complete data case, explicit maximum likelihood estimators exist. However, in the incomplete data case, iterative maximum likelihood procedures must be used. In particular, the expectation maximization (EM) algorithm is considered in a state‐space framework compatible with the multisite streamflow model; this algorithm has robust convergence properties, is simple to implement, and produces smoothed estimates of the missing data. An example is presented illustrating problems with decomposition of the traditional covariance matrix estimate and also illustrating application of the EM algorithm.

Relative efficiencies of estimates using patterned covariances or correlations in the multivariate normal estimation problem

The relative efficiency of maximum likelihood estimates is studied when taking advantage of underlying linear patterns in the covariances or correlations when estimating covariance matrices. We compare the variances of estimates of the covariance matrix obtained under two nested patterns with the assumption that the more restricted pattern is the true state. Formulas for the asymptotic variances are given which are exact for linear covariance patterns when explicit maximum likelihood estimates exist. Several specific examples are given using complete symmetry, circular symmetry and general covariance patterns as well as an example involving a covariance matrix with a linear pattern in the correlations.

Finding maximum likelihood estimates of patterned covariance matrices by the EM algorithm

Some patterned covariance matrices used to model multivariate normal data that do not have explicit maximum likelihood estimates can be viewed as submatrices of larger patterned covariance matrices that do have explicit maximum likelihood estimates. In such cases, the smaller covariance matrix can be viewed as the covariance matrix for observed variables and the larger covariance matrix can be viewed as the covariance matrix for both observed and missing variables. The advantage of this perspective is that the EM algorithm can be used to calculate the desired maximum likelihood estimates for the original problem. Two examples are presented.

Assumptions of Modern Band-Recovery Models, with Emphasis on Heterogeneous Survival Rates

The assumptions inherent in modern band-recovery models are reviewed with particular attention to homogeneity of survival and band-recovery rates for all individuals in the population. If this assumption fails, the following implications emerge: (1) the models only enable estimation of average annual survival and band-recovery rates; (2) estimators of these averages probably give underestimates; (3) the degree of underestimation in practical studies is difficult to assess, but may sometimes be important for survival estimates; (4) if sampling is nonrandom and heterogeneity of survival and recovery rates is present in the population, then any estimates could be misleading; and (5) if survival rates are homogeneous but recovery rates are heterogeneous (due perhaps to geographical variation in hunting pressure and reporting rates), then there is no bias in survival estimates. An example where data from neck-collared birds showed heterogeneity in segments of a Canada goose (Branta canadensis) population is discussed. We believe that the practical limitations of bird-banding studies deserve careful review by population biologists and managers. J. WILDL. MANAGE. 46(1):88-98 Band-recovery data have long been used to estimate mortality rates in exploited migratory bird populations. However, it is only in recent years that methods of analysis of banding data have been rigorously considered. By making certain assumptions, Seber (1970, 1971, 1972, 1973) formulated a rigorous statistical model that gave rise to explicit maximum-likelihood estimators of survivaland recovery-rate parameters. Extensions allowing for age dependence of survival and recovery rates were made by Johnson (1974), Brownie and Robson (1976), and Brownie et al. (1978), who, in a comprehensive monograph, detailed a variety of age-dependent models. Development of modern methods clearly identified the weaknesses of some of the older procedures such as the composite dynamic life table. Burnham and Anderson (1979) analyzed 45 substantial data sets from migratory waterfowl band recoveries and demonstrated that the 1 Present address: Department of Statistics, North Carolina State University, Box 5457, Raleigh, NC 27650. composite dynamic life table should no longer be used, as only 2 data sets (4%) fit the model at a = 0.05. Burnham and Anderson (1979) also found that modern methods are not necessarily always adequate for analyzing banding data, as 14 of 45 data sets (30%) did not fit. In this paper we further examine the assumptions of the modern methods. An important assumption of all recovery models is that all banded individuals of an identifiable class (e.g., by species, age, sex) in the sample have the same annual survival and recovery rates (Brownie et al. 1978:6). We consider what happens when this assumption fails, and the importance of random sampling in conjunction with homogeneity of survival and recovery rates. First, we consider a specific model (Model 1: Adults Only Banded) when the homogeneity assumption holds. We shall refer to this as the Homogeneous Population Model. This will be followed by the same model when the homogeneity assumption fails, the Heterogeneous Population Model. We discuss a populaJ. Wildl. Manage. 46(1):1982 88 HETEROGENEOUS SURVIVAL RATES Pollock and Raveling 89 tion of Canada geese for which segments with heterogeneous survival and recovery rates are known to exist from the use of individually identifiable neck-collared birds (Raveling 1978). Finally, we provide a general discussion of all the assumptions underlying modern band-recovery models, with an emphasis on the practical implications of assumption failure. We thank D. R. Anderson, D. W. Anderson, K. P. Bumham, D. S. Gilmer, D. H. Johnson, and R. M. McLandress for helpful comments on an earlier version of the manuscript. HOMOGENEOUS POPULATION MODEL Here we consider the situation where all birds are banded as adults and all birds have the same survival and recovery rates in a particular year (Model 1 of Brownie et al. 1978:15). For simplicity, we will also assume the number of years of recoveries is the same as the number of years of banding. The data matrix for observed recoveries is given in Table 1. Under this model, each row of recoveries follows a multinomial distribution, and the expected recoveries take the form given in Table 2. The maximum-likelihood estimators of the survivaland recovery-rate parameters are given by the following equations:

RELATIVE EFFICIENCIES OF ESTIMATES USING PATTERNED COVARIANCES OR CORRELATIONS IN THE MULTIVARIATE NORMAL ESTIMATION PROBLEM

ABSTRACTThe relative efficiency of maximum likelihood estimates is studied when taking advantage of underlying linear patterns in the covariances or correlations when estimating covariance matrices. We compare the variances of estimates of the covariance matrix obtained under two nested patterns with the assumption that the more restricted pattern is the true state. Formulas for the asymptotic variances are given which are exact for linear covariance patterns when explicit maximum likelihood estimates exist. Several specific examples are given using complete symmetry, circular symmetry and general covariance patterns as well as an example involving a covariance matrix with a linear pattern in the correlations.

Capture-Recapture Models Allowing for Age-Dependent Survival and Capture Rates

A K-sample capture-recapture model for an open population of animals, which allows for different identifiable age categories to have different survival and capture probabilities, is developed. Explicit maximum likelihood estimators of population size and survival rates, together with their asymptotic variances and covariances, are given. A test of whether survival and capture rates are independent of age is shown to be of simple hypergeometric form. An illustrative example based on resighting data of neck-collared giant Canada geese (Branta canadergsis maxima) is presented. The relationship of this model to other capture-recapture and band-recovery models is discussed.

Explicit Maximum Likelihood Estimates from Balanced Data in the Mixed Model of the Analysis of Variance

A result of Szatrowski, giving necessary and sufficient conditions for the existence of explicit maximum likelihood estimates for multivariate normal means and covariances with linear structure, can be applied to the problem of obtaining explicit maximum likelihood estimates in the mixed model of the analysis of variance. This application yields a simple procedure for checking whether or not explicit maximum likelihood estimates exist for the parameters in the balanced mixed model of the analysis of variance. Examples of this procedure as well as a discussion on finding the explicit maximum likelihood estimates are given.

Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances

The problem of finding maximum likelihood estimates for patterned means and covariance matrices in multivariate analysis is considered. Necessary and sufficient conditions are presented for the existence of explicit solutions and the obtaining of these explicit solutions in one iteration of the scoring equations from any positive definite starting point. Cases in which averaging yields the explicit maximum likelihood estimates are discussed. These results can be applied to the problems of finding maximum likelihood estimates for the parameters in the complete, compound and circular symmetry patterns; mixed models in the analysis of variance; and for finding asymptotic distributions of likelihood ratio statistics when the parameters under the null hypothesis have explicit maximum likelihood estimates.

A large population approach to estimation of parameters in Markov population models

SUMMARY It is known that in the limit of large population sizes certain classes of Markov population processes can be approximated by a deterministic process plus a noise term. In the usual approximation procedures the noise term is proportional to a Wiener process. This allows us to derive an estimation procedure for Markov population processes that are observed periodically at times A, 2A,.... 1. INTRODIUCTION In this paper we are concerned with the general problem of estimating parameters in a time homogeneous Markov population process. If the trajectory of the process can be observed in continuous time, then parameter estimation is straightforward in principle, for it is possible to write down the likelihood explicitly, and even if explicit maximum likelihood estimators for the required parameters are not obtainable they can be obtained numerically. We refer to Billingsley (1975) for a general approach to the problem, and to Moran (1951, 1953), Bartlett, Brennan & Pollock (1971), Bailey & Thomas (1971) and Keiding (1974) for detailed analyses in particular cases. In many situations, it is not possible to observe the process continuously but rather, observations are sampled at regular or irregular epochs. In this case the likelihood involves a product of factors, where a typical factor is the conditional distribution of the process at time t + A, say, given its value at time t. In most applications these distributions are intractable, and some approximation is required. Fortunately, there is a straightforward Gaussian approximation which is valid whenever the number of jumps between observation epochs is large. There have been a number of recent investigations into the asymptotic behaviour of Markov population processes under these conditions (Barbour, 1972, 1974; Kurtz, 1971; McNeil, 1972; McNeil & Schach, 1973; van Kampen, 1961, 1973). In ?? 2 and 3, below, we specify the class of processes to which our results will be usefully applied, and outline the general estimation procedure. In the remaining sections the procedure is illustrated using the example of the linear growth birth and death process, and an example is given in which a problem of identifiability of parameters arises because of the Gaussian approximation used.

Explicit maximum likelihood estimators for certain patterned covariance matrices

Given a random sample of size N from a p-variate normal distribution with mean vector μ and covariance matrix Σ, max-imum likelihood estimation of Σ is considered when both Σ and Σ−1 have linear structure, that is, when and where G1,…,Gm and H1,…,Hn are sets of known p×p symmetric linearly independent matrices. Theorems are given relating m and n and the sets G1,…,Gm and H1,…,Hn in the general case and when Σ is totally reducible. Explicit maximum likelihood estimators of σ1,…,σ m are found when m = n, and several examples, are given.

Estimating age-specific survival rates from bird-band returns when the reporting rate is constant

SUMMARY A probability model consisting of a product of multinomial distributions is derived for a bird-banding experiment in which a new batch of nestlings is banded and released at the beginning of each year and the bands from the dead birds are returned by observers in subsequent years. The annual reporting probability is assumed to be the same for all birds and to remain constant from year to year, while the probability of survival is assumed to depend on the age of the bird rather than on the calendar year. By considering the probability distribution of band returns conditional on the total numbers of returns from each release, explicit maximum likelihood estimates of the survival probabilities are obtained together with their asymptotic variance-covariance matrix. A goodness-of-fit test of the model is outlined and the theory is demonstrated by a worked example.

Explicit estimates from capture-recapture data with both death and immigration-stochastic model

A brief review, with references, of the literature on capture-recapture theory is given in Jolly (1963). More recently, Cormack (1964) gives a solution, including asymptotic variances, for a specific situation involving the marking and release of a non-random sample of fulmar petrels. His model is stochastic and will be referred to in ? 4. Seber (1962, 1965) has also produced some interesting solutions, and these together with Darroch (1958, 1959) are most directly relevant to our present problem. Darroch (1958, 1959) shows that in a fully stochastic model with either immigration (often called dilution) or death (or emigration), the population parameters can be easily estimated by maximum likelihood. For the more general case when death and immigration are operating simultaneously he derives estimation equations by equating certain observations to their expectations, but does not give variances or covariances for the estimates. In a later paper, (Darroch, 1961), he considers estimation for a closed population consisting of different strata. Seber (1962) establishes a stochastic model for what he calls the multi-sample single recapture census in which an individual cannot be recaptured more than once. This situation arises, for example, when the recaptures are made in the course of hunting or fishing. He allows for both death and immigration in the population, provides explicit maximumlikelihood estimates of the parameters with variances, and suggests tests for certain of the assumptions. In a second paper, (Seber, 1965), he considers a multiple-recapture model differing only slightly from that of Darroch (1959), with both death and immigration. Again, he provides explicit maximum-likelihood estimates of the parameters with variances. A test is also given for equi-catchability in a closed population of individuals with different capture histories. The first purpose of the present paper is to derive a general probability distribution designed to fit the majority of capture-recapture problems involving a 'single' population. The word 'single' here denotes a population covering an area within whose boundaries the animals (or, in general, individuals or members) are free to move and to mix with others of their kind, but which is regarded as a single area in respect of which parameters are to be estimated. The type of situation which is thus excluded by this definition is one where the population is split into a number of defined areas, and separate population estimates are required for each area as well as for numbers of animals moving from one area to another. The single population, however, need not be homogeneous but may consist of different classes of animals behaving in different ways. The other assumptions underlying the model are stated with the notation in ? 2, and the generalized probability distribution is derived in ?3.