Indirect likelihood inference

Given a sample from a fully specified parametric model, let Zn be a given finite-dimensional statistic - for example, an initial estimator or a set of sample moments. We propose to (re-)estimate the parameters of the model by maximizing the likelihood of Zn. We call this the maximum indirect likelihood (MIL) estimator. We also propose a computationally tractable Bayesian version of the estimator which we refer to as a Bayesian Indirect Likelihood (BIL) estimator. In most cases, the density of the statistic will be of unknown form, and we develop simulated versions of the MIL and BIL estimators. We show that the indirect likelihood estimators are consistent and asymptotically normally distributed, with the same asymptotic variance as that of the corresponding efficient two-step GMM estimator based on the same statistic. However, our likelihood-based estimators, by taking into account the full finite-sample distribution of the statistic, are higher order efficient relative to GMM-type estimators. Furthermore, in many cases they enjoy a bias reduction property similar to that of the indirect inference estimator. Monte Carlo results for a number of applications including dynamic and nonlinear panel data models, a structural auction model and two DSGE models show that the proposed estimators indeed have attractive finite sample properties.

Applications of stochastic analysis in finance and statistical inference

First: A continuous-time version of Kyle's model (Kyle 1985), known as the Back's model (Back 1992), of asset pricing with asymmetric information, is studied. A larger class of price processes and of noise traders' processes are studied. The price process, as in Kyle's model, is allowed to depend on the path of the market order. The process of the noise traders' is an inhomogeneous Lévy process. Solutions are found by the Hamilton-Jacobi-Bellman equations. With the insider being risk-neutral, the price pressure is constant, and there is no equilibirium in the presence of jumps. If the insider is risk-averse, there is no equilibirium in the presence of either jumps or drifts. Also, it is analised when the release time is unknown. A general relation is established between the problem of finding an equilibrium and of enlargement of filtrations. Random announcement time is random is also considered. In such a case the market is not fully efficient and there exists equilibrium if the sensitivity of prices with respect to the global demand is time decreasing according with the distribution of the random time. Second: Power variations. it is considered, the asymptotic behavior of the power variation of processes of the form _integral_0^t u(s-)dS(s), where S_ is an alpha-stable process with index of stability 0&alpha&2 and the integral is an Itô integral. Stable convergence of corresponding fluctuations is established. These results provide statistical tools to infer the process u from discrete observations. Third: A bond market is studied where short rates r(t) evolve as an integral of g(t-s)sigma(s) with respect to W(ds), where g and sigma are deterministic and W is the stochastic Wiener measure. Processes of this type are particular cases of ambit processes. These processes are in general not of the semimartingale kind.

Bringing game theory to hypothesis testing: Establishing finite sample bounds on inference

Small sample properties are of fundamental interest when only limited data is avail-able. Exact inference is limited by constraints imposed by speci.c nonrandomizedtests and of course also by lack of more data. These e¤ects can be separated as we propose to evaluate a test by comparing its type II error to the minimal type II error among all tests for the given sample. Game theory is used to establish this minimal type II error, the associated randomized test is characterized as part of a Nash equilibrium of a .ctitious game against nature.We use this method to investigate sequential tests for the di¤erence between twomeans when outcomes are constrained to belong to a given bounded set. Tests ofinequality and of noninferiority are included. We .nd that inference in terms oftype II error based on a balanced sample cannot be improved by sequential sampling or even by observing counter factual evidence providing there is a reasonable gap between the hypotheses.

Absolute type shaft encoding using LFSR

Maximal-length binary sequences have been known for a long time. They have many interesting properties, one of them is that when taken in blocks of n consecutive positions they form 2ⁿ-1 different codes in a closed circular sequence. This property can be used for measuring absolute angular positions as the circle can be divided in as many parts as different codes can be retrieved. This paper describes how can a closed binary sequence with arbitrary length be effectively designed with the minimal possible block-length, using linear feedback shift registers (LFSR). Such sequences can be used for measuring a specified exact number of angular positions, using the minimal possible number of sensors that linear methods allow.

On Boas-Type Problem

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α(0 & α & 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of modulus of continuity. The aim of this paper is to obtain Boas-type theorems for generalized Lipschitz classes. To define generalized Lipschitz classes we use the concept of modulus of smoothness of fractional order.

Data size sufficiency analyses of haplotype inference algortihms

We present experimental and theoretical analyses of data requirements for haplotype inference algorithms. Our experiments include a broad range of problem sizes under two standard models of tree distribution and were designed to yield statistically robust results despite the size of the sample space. Our results validate Gusfield's conjecture that a population size of n log n is required to give (with high probability) sufficient information to deduce the n haplotypes and their complete evolutionary history. The experimental results inspired our experimental finding with theoretical bounds on the population size. We also analyze the population size required to deduce some fixed fraction of the evolutionary history of a set of n haplotypes and establish linear bounds on the required sample size. These linear bounds are also shown theoretically.

The identity type weak factorisation system

We show that the classifying category C(T)of a dependent type theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids.

Moduli of flat conformal structures of hyperbolic type

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A Data mining approach to indirect inference

Consider a model with parameter phi, and an auxiliary model with parameter theta. Let phi be a randomly sampled from a given density over the known parameter space. Monte Carlo methods can be used to draw simulated data and compute the corresponding estimate of theta, say theta_tilde. A large set of tuples (phi, theta_tilde) can be generated in this manner. Nonparametric methods may be use to fit the function E(phi|theta_tilde=a), using these tuples. It is proposed to estimate phi using the fitted E(phi|theta_tilde=theta_hat), where theta_hat is the auxiliary estimate, using the real sample data. This is a consistent and asymptotically normally distributed estimator, under certain assumptions. Monte Carlo results for dynamic panel data and vector autoregressions show that this estimator can have very attractive small sample properties. Confidence intervals can be constructed using the quantiles of the phi for which theta_tilde is close to theta_hat. Such confidence intervals are found to have very accurate coverage.

Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

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Weak-type estimates for Haar shift operators: Sharp power on the A(p) characteristic

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The Brezis-Nirenberg type problem involving the square root of the Laplacian

We establish existence and non-existence results to the Brezis-Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

Wiener type theorems for Jacobi series with nonnegative coefficients

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Repeated Interaction and the Revelation of the Monitor's Type: A Principal-Monitor-Agent Problem

This paper studies a dynamic principal-monitor-agent relation where a strategic principal delegates the task of monitoring the effort of a strategic agent to a third party. The latter we call the monitor, whose type is initially unknown. Through repeated interaction the agent might learn his type. We show that this process damages the principal's payoffs. Compensation is assumed exogenous, limiting to a great extent the provision of incentives. We go around this difficulty by introducing costly replacement strategies, i.e. the principal replaces the monitor, thus disrupting the agent's learning. We found that even when replacement costs are null, if the revealed monitor is strictly preferred by both parties, there is a loss in efficiency due to the impossibility of bene…tting from it. Nonetheless, these strategies can partially recover the principal's losses. Additionally, we establish upper and lower bounds on the payoffs that the principal and the agent can achieve. Finally we characterize the equilibrium strategies under public and private monitoring (with communication) for different cost and impatience levels.

A type of perturbation of the harmonic oscillator

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