In this paper we provide an introduction to statistical inference for the classical linear birth‒death process, focusing on computational aspects of the problem in the setting of discretely observed processes. The basic probabilistic properties are given in Section 2, focusing on computation of the transition functions. This is followed by a brief discussion of simulation methods in Section 3, and of frequentist methods in Section 4. Section 5 is devoted to Bayesian methods, from rejection sampling to Markov chain Monte Carlo and approximate Bayesian computation. In Section 6 we consider the time-inhomogeneous case. The paper ends with a brief discussion in Section 7.

In this paper we study the Robbins–Monro procedure Xn+1 = Xn - an-1Yn with some fixed number a > 0 and establish the moderate deviation principle of the process {Xn}.

In this paper we present results on the concentration properties of the smoothing and filtering distributions of some partially observed chaotic dynamical systems. We show that, rather surprisingly, for the geometric model of the Lorenz equations, as well as some other chaotic dynamical systems, the smoothing and filtering distributions do not concentrate around the true position of the signal, as the number of observations tends to ∞. Instead, under various assumptions on the observation noise, we show that the expected value of the diameter of the support of the smoothing and filtering distributions remains lower bounded by a constant multiplied by the standard deviation of the noise, independently of the number of observations. Conversely, under rather general conditions, the diameter of the support of the smoothing and filtering distributions are upper bounded by a constant multiplied by the standard deviation of the noise. To some extent, applications to the three-dimensional Lorenz 63 model and to the Lorenz 96 model of arbitrarily large dimension are considered.

A prevalent problem in general state-space models is the approximation of the smoothing distribution of a state conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous analysis of such approximations of smoothed distributions provided by the two-filter algorithms. We extend the results available for the approximation of smoothing distributions to these two-filter approaches which combine a forward filter approximating the filtering distributions with a backward information filter approximating a quantity proportional to the posterior distribution of the state, given future observations.

Multiplicative noise removal is a challenging problem in image restoration. In this paper, by applying Box-Cox transformation, we convert the multiplicative noise removal problem into the additive noise removal problem and the block matching three dimensional (BM3D) method is applied to get the final recovered image. Indeed, BM3D is an effective method to remove additive Gaussian white noise in images. A maximum likelihood method is designed to determine the parameter in the Box-Cox transformation. We also present the unbiased inverse transform for the Box-Cox transformation which is important. Both theoretical analysis and experimental results illustrate clearly that the proposed method can remove multiplicative noise very well especially when multiplicative noise is heavy. The proposed method is superior to the existing methods for multiplicative noise removal in the literature.

Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.

We derive some limit theorems associated with the Ewens sampling formula when its parameter is increasing together with a sample size. Moreover, the limit results are applied in order to investigate asymptotic properties of the maximum likelihood estimator.

In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.

An urn contains black and red balls. Let Z n be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball b n is drawn. If b n is black and Z n-1<U, then b n is replaced together with a random number B n of black balls. If b n is red and Z n-1>L, then b n is replaced together with a random number R n of red balls. Otherwise, no additional balls are added, and b n alone is replaced. In this paper we assume that R n =B n . Then, under mild conditions, it is shown that Z n →a.s. Z for some random variable Z, and D n ≔√n(Z n -Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(D n ∈⋅|𝒢 n )→w 𝒩(0,σ2) a.s., where ℙ(D n ∈⋅|𝒢 n ) is a regular version of the conditional distribution of D n given the past 𝒢 n . Thus, in particular, one obtains D n →𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.

This paper introduces a new estimation technique for discretely observed diffusion processes. Transform functions are applied to transform the data to obtain good and easily calculated estimators of both the drift and diffusion coefficients. Consistency and asymptotic normality of the resulting estimators is investigated. Power transforms are used to estimate the parameters of affine diffusions, for which explicit estimators are obtained.

This paper applies the theory of the quasi-likelihood method to model-based inference for sample surveys. Currently, much of the theory related to sample surveys is based on the theory of maximum likelihood. The maximum likelihood approach is available only when the full probability structure of the survey data is known. However, this knowledge is rarely available in practice. Based on central limit theory, statisticians are often willing to accept the assumption that data have, say, a normal probability structure. However, such an assumption may not be reasonable in many situations in which sample surveys are used. We establish a framework for sample surveys which is less dependent on the exact underlying probability structure using the quasi-likelihood method.