Monotone Likelihood Ratio Research Articles

Information space conditions for the first-order approach in agency problems

When the principal is risk-neutral, the optimal contract for the agent which is derived using the first-order approach depends on the signals of the agent's effort only through the information variable (i.e., the likelihood ratio of the signals). By analyzing the principal-agent problem based on the information variable rather than the signals, we derive three new sets of conditions under which the first-order approach is justified. We show not only that they are more general than any sets of conditions in the existing literature, including Conlon's conditions in the multi-signal case and Jewitt's conditions in the one-signal case but also that they do not require the monotone likelihood ratio property (MLRP) for the density function of the signals. We also derive a set of conditions which applies when the principal is risk-averse and show that those conditions are more general than Conlon's corresponding conditions.

Optimal Insurance with Heterogeneous Beliefs and Disagreement About Zero-Probability Events

Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub (2013) re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the monotone likelihood ratio condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub (2013) only gave a characterization of these optimal indemnity schedules in the special case of a monotone likelihood ratio. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result as well as one of the results of Ghossoub (2013) as corollaries. Finally, we formalize Marshall’s (1991) argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any” shape.

Variance and covariance inequalities for truncated joint normal distribution via monotone likelihood ratio and log-concavity

Let X Nv(0, {\Lambda}) be a normal vector in v dimensions, where {\Lambda} is diagonal. With reference to the truncated distribution of X on the interior of a v-dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by F. Palombi and S. Toti [J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure the convergence of an algorithm for the reconstruction of {\Lambda} only on the basis of the covariance matrix of X truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove and utilize the fact that the cumulative distribution function of any positive linear combination of independent chi-square variates is log-concave, even though the same may not be true for the corresponding density function.

Ants, Traders, and Fat Tails: An Application of the Kirman (1993) Model

If you put two mounds of sugar with identical conditions near an ants’ nest, which one will the ants congregate at? Kirman explained the process of ant social herding using a simple model, and he conducted an interesting simulation. The fat tail distribution in the security market is well known, but its causes have not been sufficiently clarified. As an extension of the research of Kirman, Nirei and Watanabe tried to connect the fat tail distribution to microeconomic foundations, but there are many strong assumptions that are indispensable to the model such as the monotone likelihood ratio property (MLRP), risk neutrality, a demand curve that bends to the upper right, and an automatic supply; furthermore, the theoretical structure of the model is extremely complex. In this paper, by establishing a simple security market model and by applying the model of Kirman, the fat tail observed for price fluctuations is reproduced. Identical to Kirman and Nirei and Watanabe, the beauty contest of Keynes is kept in mind, and it is shown that a cause of the fat tail is the balance between independence and interdependence of the economic agents.

Sequential Testing for Sparse Recovery

This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared with fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), where s is the level of sparsity and D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) is the Kullback-Leibler divergence between the underlying distributions. A series of sequential probability ratio tests, which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real-world experiments and recent work in adaptive sensing, we introduce a simple procedure termed sequential thresholding, which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), achieving the lower bound. For comparison, we show any nonsequential procedure fails provided the number of measurements grows at a rate less than log(n)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ), where n is the total dimension of the problem.

Optimal insurance design of ambiguous risks

We examine the characteristics of the optimal insurance contract under linear transaction costs and an ambiguous distribution of losses. Under the standard expected utility model, we know from Arrow (1965) that it contains a straight deductible. In this paper, we assume that the policyholder is ambiguity averse in the sense of Klibanoff et al. (Econometrica 73(6):1849–1892, 2005). The optimal contract depends upon the structure of the ambiguity. For example, if the set of possible priors can be ranked according to the monotone likelihood ratio order, the optimal contract contains a disappearing deductible. We also show that the policyholder’s ambiguity aversion may have the counterintuitive effect to reduce the optimal insurance coverage of an ambiguous risk.

Using Geometry to Select One Dimensional Exponential Families That Are Monotone Likelihood Ratio in the Sample Space, Are Weakly Unimodal and Can Be Parametrized by a Measure of Central Tendency

One dimensional exponential families on finite sample spaces are studied using the geometry of the simplex Δn°-1 and that of a transformation Vn-1 of its interior. This transformation is the natural parameter space associated with the family of multinomial distributions. The space Vn-1 is partitioned into cones that are used to find one dimensional families with desirable properties for modeling and inference. These properties include the availability of uniformly most powerful tests and estimators that exhibit optimal properties in terms of variability and unbiasedness.

Who's Afraid of Selection Bias? Robust Inference in the Presence of Competitive Selection

This paper considers competitive selection dominance: what conditions on the unconditional distribution of of a random prospect will ensure that the prospect stochastically dominates a rival random prospect conditioned on competitive selection, i.e., conditioned on the prospect's realized value exceeding its rivals? Because standard distributional orders, such as stochastic dominance and the monotone likelihood ratio property (MLRP), do not provide either necessary or sufficient restrictions on the unconditional distributions to ensure selection dominance, new distribution orders are required. We provide the requisite orders, which we term supermultiplicativity on average and geometric dominance. These orderings generate conditions, satisfied by many, but not all, scale shifts of standard textbook distributions, under which the selection-conditioned distribution is stochastically dominant if and only if the unconditional distribution is stochastically dominant. When these conditions are satisfied, robust qualitative inferences concerning the unconditional population distribution can be drawn from the selection-conditioned subsample distribution and vice versa.

The meta-Gaussian Bayesian Processor of forecasts and associated preliminary experiments

Public weather services are trending toward providing users with probabilistic weather forecasts, in place of traditional deterministic forecasts. Probabilistic forecasting techniques are continually being improved to optimize available forecasting information. The Bayesian Processor of Forecast (BPF), a new statistical method for probabilistic forecast, can transform a deterministic forecast into a probabilistic forecast according to the historical statistical relationship between observations and forecasts generated by that forecasting system. This technique accounts for the typical forecasting performance of a deterministic forecasting system in quantifying the forecast uncertainty. The meta-Gaussian likelihood model is suitable for a variety of stochastic dependence structures with monotone likelihood ratios. The meta-Gaussian BPF adopting this kind of likelihood model can therefore be applied across many fields, including meteorology and hydrology. The Bayes theorem with two continuous random variables and the normal-linear BPF are briefly introduced. The meta-Gaussian BPF for a continuous predictand using a single predictor is then presented and discussed. The performance of the meta-Gaussian BPF is tested in a preliminary experiment. Control forecasts of daily surface temperature at 0000 UTC at Changsha and Wuhan stations are used as the deterministic forecast data. These control forecasts are taken from ensemble predictions with a 96-h lead time generated by the National Meteorological Center of the China Meteorological Administration, the European Centre for Medium-Range Weather Forecasts, and the US National Centers for Environmental Prediction during January 2008. The results of the experiment show that the meta-Gaussian BPF can transform a deterministic control forecast of surface temperature from any one of the three ensemble predictions into a useful probabilistic forecast of surface temperature. These probabilistic forecasts quantify the uncertainty of the control forecast; accordingly, the performance of the probabilistic forecasts differs based on the source of the underlying deterministic control forecasts.

The optimal power puzzle: scrutiny of the monotone likelihood ratio assumption in multiple testing.

In single hypothesis testing, power is a non-decreasing function of type I error rate; hence it is desirable to test at the nominal level exactly to achieve optimal power. The puzzle lies in the fact that for multiple testing, under the false discovery rate paradigm, such a monotonic relationship may not hold. In particular, exact false discovery rate control may lead to a less powerful testing procedure if a test statistic fails to fulfil the monotone likelihood ratio condition. In this article, we identify different scenarios wherein the condition fails and give caveats for conducting multiple testing in practical settings.

Pessimistic Portfolio Choice with One Safe and One Risky Asset and Right Monotone Probability Difference Order

As is well known, a first-order dominant deterioration in risk does not necessarily cause a risk-averse investor to reduce his holdings of that deteriorated asset under the expected utility framework, even in the simplest portfolio setting with one safe asset and one risky asset. The purpose of this paper is to derive conditions on shifts in the distribution of the risky asset under which the counterintuitive conclusion above can be overthrown under the rank-dependent expected utility framework, a more general and prominent alternative of the expected utility. Two new criterions of changes in risk, named the monotone probability difference (MPD) and the right monotone probability difference (RMPD) order, are proposed, which is a particular case of the first stochastic dominance. The relationship among MPD, RMPD, and the other two important stochastic orders, monotone likelihood ratio (MLR) and monotone probability ratio (MPR), is examined. A desired comparative statics result is obtained when a shift in the distribution of the risky asset satisfies the RMPD criterion.

The optimal power puzzle: scrutiny of the monotone likelihood ratio assumption in multiple testing

The optimal power puzzle: scrutiny of the monotone likelihood ratio assumption in multiple testing

Multivariate Monotone Bayesian Updating

We consider a Bayesian decision problem in which an agent must draw inferences about multiple unobserved variables. This problem appears in dierent economic settings, for instance in the theory of agency. Assume that both observed and unobserved variables are ordered. We say that the inference process (Bayesian updating) is monotone if higher observations imply higher posterior probabilities. When the parameter space is totally ordered, monotone Bayesian updating is characterized by the monotone likelihood ratio property. We show here that this can be extended to the case in which the parameter space is partially ordered, by defining a generalized monotone likelihood ratio property. However, when the parameter space is multidimensional and endowed with its componentwise order, Bayesian updating is monotone only in trivial cases.

The incentive effect of a handicap

The incentive effect of a handicap in a tournament competition is studied. A handicap may decrease the effort levels of the advantaged group or the disadvantaged group. However, the average effort level will always increase as long as the performance measure is informative of effort in a specific sense: the monotone likelihood ratio property.

Belief Heterogeneity in the Arrow-Borch-Raviv Insurance Model

In Arrow’s classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer - or decision maker (DM) - is a deductible contract, when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU-maximizer. In Arrow’s framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This paper re-examines Arrow’s problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer’s and the DM’s subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical result is then obtained as a special case.

Excessive Perks to Restrain the Hidden Saving Problem

We oer an explanation to why perks are over-provided to high prole CEOs. Hidden saving of an agent makes it dicult for a principal to control the agent’s moral hazard problem, e.g. Rogerson (1985). However, an agent typically cannot save perks, for example, a CEO who owns the right to use a private jet for personal use cannot bank the unused airplane hours. Thus, the principal may over-supply the agent perks to avoid the hidden saving problem. When the agent can both exert lower eort and save wage income, i.e., in the presence of double deviation problem, we show that the principal over-supplies perks more than the agent would have purchased on their own under two fairly standard environments: (i) when there are two states of outcome, or (ii) when the probability distribution satises the monotone likelihood ratio property, and the agent’s utility function exhibits constant absolute risk aversion. More generally, we show that excessive perks are a part of the second-best compensation if the optimal wage increases in the outcome, or if the condition of increasing wage is imposed.

Convolution and composition of totally positive random variables in economics

This paper studies a class of multidimensional screening models where different type dimensions can be aggregated into a single-dimensional sufficient statistic. The paper applies results of totally positive functions to show that some critical properties of distributions of asymmetric information parameters, such as increasing hazard rate, monotone likelihood ratio, and single-peakedness are preserved under convolution or composition. Under some general conditions, these invariance results also provide a natural ordering of alternative screening mechanisms. I illustrate how these preservation results provide a unifying framework to interpret several contributions in economic models of adverse selection, moral hazard, and voting.

Portfolio Choices and Asset Prices: The Comparative Statics of Ambiguity Aversion

This paper investigates the comparative statics of “more ambiguity aversion” as defined by Klibanoff, Marinacci and Mukerji (2005, “A Smooth Model of Decision Making under Ambiguity”, Econometrica, 73 (6), 1849–1892). The analysis uses the static two-asset portfolio problem with one safe asset and one uncertain one. While it is intuitive that more ambiguity aversion would reduce demand for the uncertain asset, this is not necessarily the case. We derive sufficient conditions for a reduction in the demand for the uncertain asset and for an increase in the equity premium. An example that meets the sufficient conditions is when the set of plausible distributions for returns on the uncertain asset can be ranked according to their monotone likelihood ratio. It is also shown how ambiguity aversion distorts the price kernel in the alternative portfolio problem with complete markets for contingent claims.

Reversals of signal-posterior monotonicity for any bounded prior

Paul Milgrom [Milgrom, P.R., 1981. Goods news and bad news: representation theorems and applications. The Bell Journal of Economics 12, 380–391] showed that if the strict monotone likelihood ratio property (MLRP) does not hold for a conditional distribution then there exists some non-degenerate prior and pair of signals where the higher-signal posterior does not stochastically dominate the lower-signal posterior. We show that for any non-degenerate prior with bounded support there exists a conditional distribution (satisfying several natural properties) and pair of signals such that the lower signal’s posterior stochastically dominates that of the higher signal. Thus, for every bounded prior, higher signals may represent strictly “worse” news.

On the first-order approach in principal–agent models with hidden borrowing and lending

We provide sufficient conditions for the validity of the first-order approach for two-period dynamic moral hazard problems where the agent can save and borrow secretly. The first-order approach is valid if the following conditions hold: (i) the agent has non-increasing absolute risk aversion utility (NIARA), (ii) the output technology has monotone likelihood ratios (MLR), and (iii) the distribution function of output is log-convex in effort (LCDF). Moreover, under these three conditions, the optimal contract is monotone in output. We also investigate a few possibilities of relaxing these requirements.