Monotone Likelihood Ratio Research Articles

Optimal Brokerage Commissions for Fair Insurance: A First Order Approach

This paper studies a principal-agent insurance brokerage problem with a risk-averse principal (an insured) and a risk-neutral agent (a broker). The concept of “mean-preserving, spread-reducing” (MPSR) effort is introduced to model the broker's activities. Using the first-order approach, it is shown that under some common conditions, the insured may “concavify” the reward function to induce the risk-neutral agent to exert MPSR brokering effort. These conditions, together with an additional condition, guarantee the validity of the first-order approach even when the monotone likelihood ratio condition (used exclusively to justify the first-order approach) is violated.

Optimism and Communication

I examine how the communication incentive of an agent (sender) changes when the prior of the principal (receiver) about the agent's private information becomes more optimistic (in the sense of monotone likelihood ratio dominance). I use the canonical model of strategic communication (Crawford and Sobel, 1982) under the assumption that the agent has an upward bias. The main result is that under a more optimistic prior, more information can be transmitted in equilibrium in the sense that the principal's expected payoff is higher. Applying the result to three models, I find: (1) If the principal can choose among intermediaries to make decisions, she may benefit from choosing someone more optimistic than herself, but never benefits from choosing someone more pessimistic. (2) A more optimistic principal is more likely to gain from centralization relative to delegation. (3) In a game of two-way communication in which a privately informed principal reports to an agent first, the principal cannot credibly reveal her private signal unless it is informative enough.

Locally Maximin Test for the Mixture Proportion with Type-I Censoring

Mixtures of distributions arise frequently in the context of life testing experiments, where one also encounters the problem of censored data. In this article, we derive the locally most powerful (LMP) test for testing the mixing proportion in a general mixture model based on type-I censored data. We also prove the additional properties of unbiasedness and locally maximin using a novel approach. To this end, we prove an extension of a standard lemma in the testing literature (Lehmann, 1986) relating to families with monotone likelihood ratio.

Stochastic ordering of classical discrete distributions

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤st Q can be characterized by their extreme tail ordering equivalent to P({k *})/Q({k *}) ≥ 1 ≥ lim k→k * P({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k *})/Q({k *}) for finite k *. This includes in particular all pairs where P and Q are both binomial (b n 1,p 1 ≤st b n 2,p 2 if and only if n 1 ≤ n 2 and (1 - p 1) n 1 ≥ (1 - p 2) n 2 , or p 1 = 0), both negative binomial (b − r 1,p 1 ≤st b − r 2,p 2 if and only if p 1 ≥ p 2 and p 1 r 1 ≥ p 2 r 2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

Stochastic ordering of classical discrete distributions

For several pairs (P,Q) of classical distributions on ℕ0, we show that their stochastic orderingP≤stQcan be characterized by their extreme tail ordering equivalent toP({k*})/Q({k*}) ≥ 1 ≥ limk→k*P({k})/Q({k}), withk*andk*denoting the minimum and the supremum of the support ofP+Q, and with the limit to be read asP({k*})/Q({k*}) for finitek*. This includes in particular all pairs wherePandQare both binomial (bn1,p1≤stbn2,p2if and only ifn1≤n2and (1 -p1)n1≥ (1 -p2)n2, orp1= 0), both negative binomial (b−r1,p1≤stb−r2,p2if and only ifp1≥p2andp1r1≥p2r2), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

Asymmetric Information in Financial Markets: Anything Goes

I study a standard Grossman and Stiglitz (1980) noisy rational expectations economy, but relax the usual assumption of the normality of fundamental and supply. My solution approach dispenses with the typical \conjecture and verify method and enables me to analytically solve an entire class of previously intractable nonlinear models that nests the standard model. I show how: (1) price jumps and crashes may arise endogenously, purely due to learning eects, (2) observation of the net trading volume of informed and noise traders may be valuable for investors in the economy as it can provide a renement of the information conveyed by price, (3) the value of acquiring information may be non-monotonic in the number of informed traders, leading to multiple equilibria in the information market, and (4) the relation between disagreement and future returns is ambiguous. In short, many of the results from noisy rational expectations models are not robust. Finally, I introduce monotone likelihood ratio conditions that determine the signs of the various comparative statics, which represents the rst demonstration of the importance of the MLRP for comparative statics

The First-Order Approach to Moral Hazard Problems with Hidden Saving

Moral hazard models with hidden saving decisions are useful to study such diverse problems as unemployment insurance, income taxation, executive compensation, or human capital policies. How can we solve such models? In general, this is very difficult. Under the conditions derived in this paper, however, we can replace the incentive constraint with the associated first-order condition. This allows the application of simple Lagrangian methods and yields a precise characterization of optimal contracts. To obtain tractable conditions for the validity of this approach, the paper draws on the concept of log-convexity. Since logconvexity, unlike convexity, is preserved under multiplication, the paper is able to separate the assumptions on the output distribution from the assumptions on the agent’s preferences in a sense, even though the interaction between these two is important for the agent’s incentives. The first-order approach is valid if the following conditions hold: a) the agent has nonincreasing absolute risk aversion (NIARA) utility, b) the output technology has monotone likelihood ratios (MLR), and c) the distribution function of output is log-convex in effort (LCDF). Finally, the paper shows how the curvature of optimal wage schemes can be used to relax the above conditions.

Is the Selected Population the Best?—Location and Scale Parameter Cases

We observe X 1,…,X k , where X i has density f(x,θ i ) possessing monotone likelihood ratio. The best population corresponds to the largest θ i . We select the population corresponding to the largest X i . The goal is to attach the best possible p-value to the inference: the selected population has the uniquely largest θ i . Gutmann and Maymin (1987) considered the location parameter case and derived the supremum of the error probability by conditioning on S, the index of the largest X i . Using this conditioning approach, Kannan and Panchapakesan (2009) considered the problem for the gamma family. We consider here a unified approach to both the location and scale parameter cases, and obtain the supremum of the error probability without using conditioning.

Partially Observed Markov Decision Process Multiarmed Bandits—Structural Results

This paper considers multiarmed bandit problems involving partially observed Markov decision processes (POMDPs). We show how the Gittins index for the optimal scheduling policy can be computed by a value iteration algorithm on each process, thereby considerably simplifying the computational cost. A suboptimal value iteration algorithm based on Lovejoy's approximation is presented. We then show that for the case of totally positive of order 2 (TP2) transition probability matrices and monotone likelihood ratio (MLR) ordered observation probabilities, the Gittins index is MLR increasing in the information state. Algorithms that exploit this structure are then presented.

On the monotone likelihood ratio property for the convolution of independent binomial random variables

Given that r and s are natural numbers and X ∼ Binomial ( r , q ) and Y ∼ Binomial ( s , p ) are independent random variables where q , p ∈ ( 0 , 1 ) , we prove that the likelihood ratio of the convolution Z = X + Y is decreasing, increasing, or constant when q < p , q > p or q = p , respectively.

Optimality of k-out-of-n Structures for Condition Monitoring Maintenance Using Dependent Information

A maintenance monitoring system is considered that is subject to two types of contradictory failures, “false alarm” and “failure to alarm”. It is shown that, for a monitoring system with dependent observations, the optimal decision procedure is given by a k-out-of-n structure under a weaker condition than that used in previous research. The condition is that the ratios of the same size elements of the conditional probabilities for the observational matrix of the monitors given the true state of the observed system are the same besides the probability matrix having a weak-MLR (weak-multivariate monotone likelihood ratio) property. If the optimal procedure is given by a k-out-of-n structure, the amount of calculation time needed to identify an optimal decision procedure can be greatly reduced. This enables an optimal action to be found quickly.

Transaction Size and Effective Spread: An Informational Relationship

The relationship between quantity traded and transaction costs has been one of the main focuses among financial scholars and practitioners. The purpose of this thesis is to investigate the informational relationship between these variables. Following insights and results of Milgrom (1981), Feldman (2004), and Feldman and Winer (2004), we use New York Stock Exchange (NYSE) data and kernel estimation methods to construct the distribution of one variable conditional on the other. Then, we study the information in these conditional distributions: the extent to which they are ordered by first order stochastic dominance (FOSD) and by the monotone likelihood ratio property (MLRP). We find that transaction size and effective spread are significantly correlated. FOSD, a necessary condition for a signaling equilibrium, holds under certain conditions. We start from 2-subsample case. We choose a cut-off point in transaction size and categorize the observations with transaction sizes smaller than the cut-off point into group low. The remaining data is classified as high. We repeat this procedure for all possible transaction size cut-off points. It turns out that FOSD holds nowhere. However, once we eliminate transactions at the quote midpoint, the crossings between exchange members not specialists, FOSD holds for all the cut-off points fewer than 15800 shares. The MLRP, a necessary and sufficient condition for the separating equilibrium to hold point by point of the conditional density functions, does not hold but might not be ruled out considering the error in the estimates. We also find that large trades are not necessarily associated with large spread. Instead, it is more likely that larger trades are transacted at the quote midpoint (again, the non-specialist crossings) than smaller trades. Our results confirm the findings of Barclay and Warner (1993) regarding the informativeness of medium-size transactions: we identify informational relationships between mid-size transactions and spreads but not for zero effective spreads and large transactions. That is, we identify two regimes, an informational one and a non-informational/liquidity one.

A note on monotone likelihood ratio of the total score variable in unidimensional item response theory

This note provides a direct, elementary proof of the fundamental result on monotone likelihood ratio of the total score variable in unidimensional item response theory (IRT). This result is very important for practical measurement in IRT, because it justifies the use of the total score variable to order participants on the latent trait. The proof relies on a basic inequality for elementary symmetric functions which is proved by means of few purely algebraic, straightforward transformations. In particular, flaws in a proof of this result by Huynh [(1994). A new proof for monotone likelihood ratio for the sum of independent Bernoulli random variables. Psychometrika, 59, 77-79] are pointed out and corrected, and a natural generalization of the fundamental result to non-linear (quasi-ordered) latent trait spaces is presented. This may be useful for multidimensional IRT or knowledge space theory, in which the latent 'ability' spaces are partially ordered with respect to, for instance, coordinate-wise vector-ordering or set-inclusion, respectively.

On Abel's Concept of Doubt and Pessimism

In this paper, we characterize subjective probability beliefs leading to a higher equilibrium market price of risk. We establish that Abel's result on the impact of doubt on the risk premium is not correct (see Abel, A., 2002. An exploration of the effects of pessimism and doubt on asset returns. Journal of Economic Dynamics and Control, 26, 1075-1092). We introduce, on the set of subjective probability beliefs, market price of risk dominance concepts and we relate them to well known dominance concepts used for comparative statics in portfolio choice analysis. In particular, the necessary first order conditions on subjective probability beliefs in order to increase the market price of risk for all nondecreasing utility functions appear as equivalent to the monotone likelihood ratio property.

On Abel's concept of doubt and pessimism

In this paper, we characterize subjective probability beliefs leading to a higher equilibrium market price of risk. We establish that Abel's result on the impact of doubt on the risk premium is not correct in general; see Abel [2002. An exploration of the effects of pessimism and doubt on asset returns. Journal of Economic Dynamics and Control 26, 1075–1092]. We introduce, on the set of subjective probability beliefs, market-price-of-risk dominance concepts and we relate them to well-known dominance concepts used for comparative statics in portfolio choice analysis. In particular, the necessary first-order conditions on subjective probability beliefs in order to increase the market price of risk for all nondecreasing utility functions appear as equivalent to the monotone likelihood ratio property.

MONOTONE PROPERTIES OF A GENERAL DIAGNOSTIC MODEL

ABSTRACTMonotonicity properties of a general diagnostic model (GDM) are considered in this paper. Simple data summaries are identified to inform about the ordered categories of latent traits. The findings are very much in accordance with the statements made about the GPCM (Hemker, Sijtsma, Molenaar, &amp; Junker, 1996, 1997). On the one hand, by fitting a GDM with equal slopes across items, the observed total score X+ = Σj:qjk=1 xj demonstrates the monotone likelihood ratio (MLR) property in individual coordinate at the cost of losing model fit. On the other hand, fitting a GDM without slope restriction increases the model fit but by sacrificing MLR property. Trade‐offs between these two situations should be considered in practice.

Structured Threshold Policies for Dynamic Sensor Scheduling—A Partially Observed Markov Decision Process Approach

We consider the optimal sensor scheduling problem formulated as a partially observed Markov decision process (POMDP). Due to operational constraints, at each time instant, the scheduler can dynamically select one out of a finite number of sensors and record a noisy measurement of an underlying Markov chain. The aim is to compute the optimal measurement scheduling policy, so as to minimize a cost function comprising of estimation errors and measurement costs. The formulation results in a nonstandard POMDP that is nonlinear in the information state. We give sufficient conditions on the cost function, dynamics of the Markov chain and observation probabilities so that the optimal scheduling policy has a threshold structure with respect to a monotone likelihood ratio (MLR) ordering. As a result, the computational complexity of implementing the optimal scheduling policy is inexpensive. We then present stochastic approximation algorithms for estimating the best linear MLR order threshold policy.

Nonparametric item response theory axioms and properties under nonlinearity and their exemplification with knowledge space theory

This paper investigates the dichotomous Mokken nonparametric item response theory (IRT) axioms and properties under incomparabilities among latent trait values and items. Generalized equivalents of the unidimensional nonparametric IRT axioms and properties are formulated for nonlinear (quasi-ordered) person and indicator spaces. It is shown that monotone likelihood ratio (MLR) for the total score variable and nonlinear latent trait implies stochastic ordering (SO) of the total score variable, but may fail to imply SO of the nonlinear latent trait. The reason for this and conditions under which the implication holds are specified, based on a new, simpler proof of the fact that in the unidimensional case MLR implies SO. The approach is applied in knowledge space theory (KST), a combinatorial test theory. This leads to a (tentative) Mokken-type nonparametric axiomatization in the currently parametric theory of knowledge spaces. The nonparametric axiomatization is compared with the assumptions of the parametric basic local independence model which is fundamental in KST. It is concluded that this paper may provide a first step toward a basis for a possible fusion of the two split directions of psychological test theories IRT and KST.

Better May be Worse: Some Monotonicity Results and Paradoxes in Discrete Choice Under Uncertainty

It is not unusual in real-life that one has to choose among finitely many alternatives when the merit of each alternative is not perfectly known. Instead of observing the actual utilities of the alternatives at hand, one typically observes more or less precise signals that are positively correlated with these utilities. In addition, the decision-maker may, at some cost or disutility of effort, choose to increase the precision of these signals, for example by way of a careful study or the hiring of expertise. We here develop a model of such decision problems. We begin by showing that a version of the monotone likelihood-ratio property is sufficient, and also essentially necessary, for the optimality of the heuristic decision rule to always choose the alternative with the highest signal. Second, we show that it is not always advantageous to face alternatives with higher utilities, a non-monotonicity result that holds even if the decision-maker optimally chooses the signal precision. We finally establish an operational first-order condition for the optimal precision level in a canonical class of decision-problems, and we show that the optimal precision level may be discontinuous in the precision cost.

When Do Ordered Prior Distributions Induce Ordered Posterior Distributions?

It has been shown that if two probability distributions satisfy the monotone likelihood ratio property (MLRP), and are independently updated using common public information and traditional Bayesian updating, then the resulting two posterior distributions will also satisfy the MLRP. I discuss this result and extend it by characterizing the full set of updating operations which preserve the MLRP in this manner, of which Bayesian updating is an element. I also find the set of updating operations which preserve first order stochastic dominance (FOSD). The only operator which preserves both the MLRP and FOSD is that which ignores all new information.