Monotone Likelihood Ratio Research Articles

Simultaneous directional inference

Abstract We consider the problem of inference on the signs of n>1 parameters. We aim to provide 1−α post hoc confidence bounds on the number of positive and negative (or non-positive) parameters, with a simultaneous guarantee, for all subsets of parameters. We suggest to start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the p-values of the one-sided hypotheses for the selection, and use the adjusted p-values for simultaneous inference on the selected n one-sided hypotheses. The adjustment is straightforward assuming the p-values of one-sided hypotheses have densities with monotone likelihood ratio, and are mutually independent. We show the bounds we provide are tighter (often by a great margin) than existing alternatives, and that they can be obtained by at most a polynomial time. We demonstrate their usefulness in the evaluation of treatment effects across studies or subgroups. Specifically, we provide a tight lower bound on the number of studies which are beneficial, as well as on the number of studies which are harmful (or non-beneficial), and in addition conclude on the effect direction of individual studies, while guaranteeing that the probability of at least one wrong inference is at most 0.05.

Stochastic ordering results on the duration of the gambler’s ruin game

AbstractIn the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0<z\leq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} \leq z<a$ , the family of distributions of the duration indexed by $p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p \in [0,1]$ by $p={\frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.

Comparing information in general monotone decision problems

I study the value of information in monotone decision problems with potentially multidimensional action spaces. As a criterion for comparing information structures, I develop a condition called monotone quasi-garbling, which involves adding reversely monotone noise to an existing information structure. Specifically, this noise is more likely to return a higher signal in a lower state and a lower signal in a higher state. I show that monotone quasi-garbling is a necessary and sufficient condition for decision makers to obtain a higher ex-ante expected payoff. This new criterion refines the garbling condition by Blackwell (1951, 1953) and is equivalent to the accuracy condition by Lehmann (1988) under the monotone likelihood ratio property. To illustrate, I apply the result to problems in nonlinear monopoly pricing and optimal insurance.

Risk Classification in Insurance Markets with Risk and Preference Heterogeneity

Abstract This paper studies a competitive model of insurance markets in which consumers are privately informed about their risk and risk preferences. We provide a characterization of the equilibria, which depend non-trivially on consumers’ type distribution, a desirable feature for policy analysis. The use of consumer characteristics for risk classification is modeled as the disclosure of a public informative signal. A novel property of signals, monotonicity, is shown to be necessary and sufficient for their release to be welfare improving for almost all consumer types. We also study the effect of changes to the risk distribution in the population as the result of demographic changes or policy interventions. We show that an increase in the risk distribution, according to the monotone likelihood ratio ordering of distribution, leads to lower utility for almost all consumer types. In contrast, the effect is ambiguous when considering the first-order stochastic dominance ordering.

Variable selection, monotone likelihood ratio and group sparsity

In the pivotal variable selection problem, we derive the exact nonasymptotic minimax selector over the class of all s-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2, and is also exact minimax with respect to the probability of wrong recovery. As a consequence, we establish explicit lower bounds under the monotone likelihood ratio property and we obtain a tight characterization of the minimax risk in terms of the best separable selector risk. We apply these general results to derive necessary and sufficient conditions of exact and almost full recovery in the location model with light tail distributions and in the problem of group variable selection under Gaussian noise and under more general anisotropic sub-Gaussian noise. Numerical results illustrate our theoretical findings.

Sequential Detection of an Arbitrary Transient Change Profile by the FMA Test

This article addresses the sequential detection of transient changes by using the finite moving average (FMA) test. It is assumed that a change occurs at an unknown (but nonrandom) change point and the duration of postchange period is finite and known. We relax the assumption that the profile of a transient change is chosen so that the log-likelihood ratios of the observations are associated random variables (r.v.s) in the prechange mode. Hence, the profile of the transient change is arbitrary and it is not necessarily of constant sign for a distribution with monotone likelihood ratio. A new upper bound for the worst-case probability of false alarm is proposed. It is shown that the optimization of the window-limited cumulative sum (CUSUM) test again leads to the FMA test. Three particular transient changes are considered: in the Gaussian mean, in the Gaussian variance, and in the parameter of exponential distribution. In the first case, a comparison between the bounds for the FMA test operating characteristics and the exact operating characteristics calculated by numerical integration is used to estimate the sharpness of the bounds. In the second and third cases, special attention is paid to the calculation of the FMA distribution in the case of arbitrary profile. The method of convolution is used to solve the problem.

A Statistical Test for the Detection of Item Compromise Combining Responses and Response Times

AbstractA test of item compromise is presented which combines the test takers' responses and response times (RTs) into a statistic defined as the number of correct responses on the item for test takers with RTs flagged as suspicious. The test has null and alternative distributions belonging to the well‐known family of compound binomial distributions, is simple to calculate, and has results that are easy to interpret. It also demonstrated nearly perfect power for the detection of compromise with no more than 10 test takers with preknowledge of the more difficult and discriminating items in a set of empirical examples. For the easier and less discriminating items, the presence of some 20 test takers with preknowledge still sufficed. A test based on the reverse statistic of the total time by test takers with responses flagged as suspicious may seem a natural alternative but misses the property of a monotone likelihood ratio necessary to decide between a test that should be left or right sided.

On monotone likelihood ratio of stationary probabilities in bonus-malus systems

Abstract Bonus-malus system is an often used risk management tool in the insurance industry, and it is usually modeled with Markov chains. Under mild conditions it can be stated that the bonus-malus system converges to a unique stationary distribution in the long run. The maximum likelihood ratio property is a well-known statistical concept and we define it for the stationary distribution of a bonus-malus system. For two special cases we could justify it algebraically. For other cases we describe a numerical method with which we can test this property in any case. With the help of the described method, we checked this property for cases that appear in actuarial practice.

Simple contracts with adverse selection and moral hazard

We study a principal–agent model with moral hazard and adverse selection. Risk‐neutral agents with limited liability have arbitrary private information about the distribution of outputs and the cost of effort. We show that under a multiplicative separability condition, the optimal mechanism offers a single contract. This condition holds, for example, when output is binary. If the principal's payoff must also satisfy free disposal and the distribution of outputs has the monotone likelihood ratio property, the mechanism offers a single debt contract. Our results generalize if the output distribution is “close” to multiplicatively separable. Our model suggests that offering a single contract may be optimal in environments with adverse selection and moral hazard when agents are risk‐neutral and have limited liability.

Fair Cake Division Under Monotone Likelihood Ratios

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

Rational inattention and the monotone likelihood ratio property

This paper explores the connection between rational inattention and the monotone likelihood ratio property (MLRP). In many economic problems, agents prefer to take a higher action when there is a higher underlying state. It is often assumed that these agents have noisy signals that are strongly correlated with the underlying state, i.e. ordered by the MLRP. Separately, a large literature has recently explored the behavior of rationally inattentive agents. Under some common separability and smoothness assumptions on the class of attention costs, I show that agents will always acquire signals that are ordered by the MLRP within such economic problems if and only if they have attention costs proportional to entropy reduction. With any other attention costs, agents err regarding the relative benefit of actions at different states. Extending to games, I provide conditions under which players first acquire MLRP-ordered signals, and then choose higher actions given higher signals. I then apply these results to independent private-value auctions.

A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical Development and Applications

In this work, a three-parameter Weibull Inverse Rayleigh (WIR) distribution is proposed. The new WIR distribution is an extension of a one-parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log-logistic as quantile function. The statistical properties such as quantile function, order statistic, monotone likelihood ratio property, hazard, reverse hazard functions, moments, skewness, kurtosis, and linear representation of the new proposed distribution were studied theoretically. The maximum likelihood estimators cannot be derived in an explicit form. So we employed the iterative procedure called Newton Raphson method to obtain the maximum likelihood estimators. The Bayes estimators for the scale and shape parameters for the WIR distribution under squared error, Linex, and Entropy loss functions are provided. The Bayes estimators cannot be obtained explicitly. Hence we adopted a numerical approximation method known as Lindley's approximation in other to obtain the Bayes estimators. Simulation procedures were adopted to see the effectiveness of different estimators. The applications of the new WIR distribution were demonstrated on three real-life data sets. Further results showed that the new WIR distribution performed credibly well when compared with five of the related existing skewed distributions. It was observed that the Bayesian estimates derived performs better than the classical method.

Online Appendix for Security Design for Private Acquisitions

Online Appendix A to Security Design for Asset Acquisitions shows that seller debt securities can be optimal security designs when informed sellers, rather than uninformed buyers, make acquisition offers, and when acquirers are cash constrained. In contrast, acquirer overconfidence and seller private information are essential components of our analysis of seller debt financing. Online Appendix B to Security Design for Asset Acquisition' shows that the ratio condition, Assumption 4, is satisfied by all standard textbook distributions (supported by the non-negative real line) whenever the standard monotone likelihood ratio conditions (Assumptions 1 and 2) of security design are satisfied. The distributions families considered are Chi-Squared, Exponential, Gamma, Gompertz, Half Normal, Lognormal, and Weibull. General sufficient conditions for the satisfaction of Assumption 5 are also developed. The paper is available at https://ssrn.com/abstract=3731086.

Generalized FWER control procedures for testing multiple hypotheses

Most methods for testing multiple hypotheses assume the monotone likelihood ratio (MLR) condition holds. In fact, in some cases, MLR does not hold, and the widely used Bonferroni procedure may be inefficient. In this paper, we aim to improve the Bonferroni procedure without assuming the MLR condition holds. By combining the advantages of the test method constructed according to the Neyman–Pearson lemma and the Bonferroni procedure, we propose two generalized Bonferroni procedures, denoted as GB1 and GB2. Further, we propose to plug the proportion of true null hypotheses in the GB2 procedure to improve power. We numerically compare the proposed methods with existing methods. Simulation results show that the proposed methods perform very close to or significantly better than existing ones; the proposed plug-in procedure performs best among all in terms of power performance. Two real data sets are analyzed with the proposed procedures.

The comparative advantage of cities

What determines the distributions of skills, occupations, and industries across cities? We develop a theory to jointly address these fundamental questions about the spatial organization of economies. Our model incorporates a system of cities, their internal urban structures, and a high-dimensional theory of factor-driven comparative advantage. It predicts that larger cities will be skill-abundant and specialize in skill-intensive activities according to the monotone likelihood ratio property. We test the model using data on 270 US metropolitan areas, 3 to 9 educational categories, 22 occupations, and 19 industries. The results provide support for our theory's predictions.

Optimal incentive contract with endogenous monitoring technology

Recent technology advances have enabled firms to flexibly process and analyze sophisticated employee performance data at a reduced and yet significant cost. We develop a theory of optimal incentive contracting where the monitoring technology that governs the above procedure is part of the designer's strategic planning. In otherwise standard principal–agent models with moral hazard, we allow the principal to partition agents' performance data into any finite categories, and to pay for the amount of information the output signal carries. Through analysis of the trade‐off between giving incentives to agents and saving the monitoring cost, we obtain characterizations of optimal monitoring technologies such as information aggregation, strict monotone likelihood ratio property, likelihood ratio–convex performance classification, group evaluation in response to rising monitoring costs, and assessing multiple task performances according to agents' endogenous tendencies to shirk. We examine implications of these results for workforce management and firms' internal organizations.

Contracting with moral hazard, adverse selection and risk neutrality: when does one size fit all?

This paper studies a principal-agent relationship when both are risk-neutral and in the presence of adverse selection and moral hazard. Contracts must satisfy the limited-liability and monotonicity conditions. We provide sufficient conditions under which the optimal contract is simple, in the sense that each type is offered the same contract. These are: the action and the agent’s type are complements, and the output’s cumulative distribution function is such that the marginal rate of substitution between the action and the agent’s type is the same for each possible output realization. Furthermore, under the average monotone likelihood ratio property, the optimal contract is a call-option contract as in Innes (J Econ Theory 52(1):45–67, 1990). The results shed light on the fact that sometimes contracts are not highly dependent on individual characteristics as predicted in most pure moral hazard and pure adverse selection settings.

Monotonicity Properties for Two-Action Partially Observable Markov Decision Processes on Partially Ordered Spaces

This paper investigates monotonicity properties of optimal policies for two-action partially observable Markov decision processes when the underlying (core) state and observation spaces are partially ordered. Motivated by the desirable properties of the monotone likelihood ratio order in imperfect information settings, namely the preservation of belief ordering under conditioning on any new information, we propose a new stochastic order (a generalization of the monotone likelihood ratio order) that is appropriate for when the underlying space is partially ordered. The generalization is non-trivial, requiring one to impose additional conditions on the elements of the beliefs corresponding to incomparable pairs of states. The stricter conditions in the proposed stochastic order reflect a conservation of structure in the problem – the loss of structure from relaxing the total ordering of the state space to a partial order requires stronger conditions with respect to the ordering of beliefs. In addition to the proposed stochastic order, we introduce a class of matrices, termed generalized totally positive of order 2, that are sufficient for preserving the order. Our main result is a set of sufficient conditions that ensures existence of an optimal policy that is monotone on the belief space with respect to the proposed stochastic order.

Monotonicity properties for two-action partially observable Markov decision processes on partially ordered spaces

This paper investigates monotonicity properties of optimal policies for two-action partially observable Markov decision processes when the underlying (core) state and observation spaces are partially ordered. Motivated by the desirable properties of the monotone likelihood ratio order in imperfect information settings, namely the preservation of belief ordering under conditioning on any new information, we propose a new stochastic order (a generalization of the monotone likelihood ratio order) that is appropriate for when the underlying space is partially ordered. The generalization is non-trivial, requiring one to impose additional conditions on the elements of the beliefs corresponding to incomparable pairs of states. The stricter conditions in the proposed stochastic order reflect a conservation of structure in the problem – the loss of structure from relaxing the total ordering of the state space to a partial order requires stronger conditions with respect to the ordering of beliefs. In addition to the proposed stochastic order, we introduce a class of matrices, termed generalized totally positive of order 2, that are sufficient for preserving the order. Our main result is a set of sufficient conditions that ensures existence of an optimal policy that is monotone on the belief space with respect to the proposed stochastic order.

Budget-constrained optimal insurance with belief heterogeneity

We re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow (1971), but allow for divergence in beliefs between the insurer and the insured; and in particular for singularity between these beliefs, that is, disagreement about zero-probability events. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under a mild consistency requirement between these beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterize the optimal indemnity and show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event, which depends on the state of the world through a likelihood ratio. The latter is obtained from a Lebesgue decomposition of the insured’s belief with respect to the insurer’s belief.