Monotone Likelihood Research Articles

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.

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.

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255–264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69–78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660–671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255–264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318–1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.

Admissible treatment rules for a risk-averse planner with experimental data on an innovation

Consider a planner choosing treatments for observationally identical persons who vary in their response to treatment. There are two treatments with binary outcomes. One is a status quo with known population success rate. The other is an innovation for which the data are the outcomes of an experiment. Karlin and Rubin [1956. The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27, 272–299] assumed that the objective is to maximize the population success rate and showed that the admissible rules are the KR-monotone rules. These assign everyone to the status quo if the number of experimental successes is below a specified threshold and everyone to the innovation if experimental success exceeds the threshold. We assume that the objective is to maximize a concave-monotone function f ( · ) of the success rate and show that admissibility depends on the curvature of f ( · ) . Let a fractional monotone rule be one where the fraction of persons assigned to the innovation weakly increases with the number of experimental successes. We show that the class of fractional monotone rules is complete if f ( · ) is concave and strictly monotone. Define an M-step monotone rule to be a fractional monotone rule with an interior fractional treatment assignment for no more than M consecutive values of the number of experimental successes. The M -step monotone rules form a complete class if f ( · ) is differentiable and has sufficiently weak curvature. Bayes rules and the minimax-regret rule depend on the curvature of the welfare function.

Channel-optimized quantizers for decentralized detection in sensor networks

Motivated by the delay and resource constraints omnipresent in most wireless sensor network applications, we design channel-optimized scalar quantizers for a canonical decentralized detection system. Aimed at minimizing the error probability of the fusion center output, we first establish the optimality of monotone likelihood ratio partition of the observation space for the local quantizer design. We then devise an iterative algorithm to construct distributed quantizers that are person-by-person optimal. The channel-optimized approach is shown to offer better performance compared with various alternatives. It also exhibits inherent adaptivity in resource (bit) allocation in response to varying channel conditions.

On the existence of an increasing symmetric equilibrium in (k + 1)-st price common value auctions

In a classical result, Milgrom (1981a) established that the Monotone Likelihood Ratio Property (MLRP) is a sufficient condition for the existence of an increasing symmetric equilibrium in (k + 1)-st price common value auctions. We show: (1) If MLRP is violated, then for any number of bidders and objects there exists a distribution of the common value such that no increasing symmetric equilibrium exists; (2) If MLRP is violated, then for any distribution of the common value there exist infinitely many pairs of the number of bidders and the number of objects such that an increasing symmetric equilibrium does not exist; (3) There are examples where an increasing symmetric equilibrium exists even when the signal distribution violates MLRP.

Increased Risk-Bearing with Background Risk

Analyses of risk-bearing often assume that agents face only one risk. Agents however usually face several risks and the interaction between them can affect the willingness to bear any one of them. We consider how the introduction of background risk affects the comparative statics predictions of distribution changes in the standard two asset portfolio model. We show that such predictions are fairly robust, no matter what the correlation between the two risks. We consider changes in the conditional distributions of the risky asset's return; and changes in the marginal distribution of the asset's return. For the first question, a version of Gollier's (1995) Central Riskiness order is sufficient and necessary to increase risk-bearing. For the second question, Monotone Likelihood Ratio improvements are sufficient and necessary. Many of our proofs illustrate the "basis" approach to comparative statics under uncertainty.

Fiducial inference in the pivotal family of distributions

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.

An optimal policy for partially observable Markov decision processes with non‐independent monitors

Purpose – This research investigated the optimal structure of a discrete‐time Markov deterioration system monitored by multiple non‐independent monitors. The purpose is to obtain a sufficient condition with which the optimal policy is given by a control limit policy.Design/methodology/approach – The model of this research is formulated as a partially observable Markov decision process. The problem is to obtain an optimal policy which can minimize the expected total discounted cost over an infinite horizon.Findings – The research found that the expected optimal cost function over an infinite horizon has a property of control limit policy given the conditions that a transition probability having a property of totally positive of order 2 and a conditional probability of the monitors having a property of weak multivariate monotone likelihood ratio. Furthermore, we showed that the optimal policy has only four action regions at most.Practical implications – If the optimum policy can be limited to a control limi...

Information Bounds and Quickest Change Detection in Decentralized Decision Systems

The quickest change detection problem is studied in decentralized decision systems, where a set of sensors receive independent observations and send summary messages to the fusion center, which makes a final decision. In the system where the sensors do not have access to their past observations, the previously conjectured asymptotic optimality of a procedure with a monotone likelihood ratio quantizer (MLRQ) is proved. In the case of additive Gaussian sensor noise, if the signal-to-noise ratios (SNR) at some sensors are sufficiently high, this procedure can perform as well as the optimal centralized procedure that has access to all the sensor observations. Even if all SNRs are low, its detection delay will be at most pi/2-1ap 57% larger than that of the optimal centralized procedure. Next, in the system where the sensors have full access to their past observations, the first asymptotically optimal procedure in the literature is developed. Surprisingly, the procedure has the same asymptotic performance as the optimal centralized procedure, although it may perform poorly in some practical situations because of slow asymptotic convergence. Finally, it is shown that neither past message information nor the feedback from the fusion center improves the asymptotic performance in the simplest model

The adaptive coherence estimator: a uniformly most-powerful-invariant adaptive detection statistic

We show that the Adaptive Coherence Estimator (ACE) is a uniformly most powerful (UMP) invariant detection statistic. This statistic is relevant to a scenario appearing in adaptive array processing, in which there are auxiliary, signal-free, training-data vectors that can be used to form a sample covariance estimate for clutter and interference suppression. The result is based on earlier work by Bose and Steinhardt, who found a two-dimensional (2-D) maximal invariant when test and training data share the same noise covariance. Their 2-D maximal invariant is given by Kelly's Generalized Likelihood Ratio Test (GLRT) statistic and the Adaptive Matched Filter (AMF). We extend the maximal-invariant framework to the problem for which the ACE is a GLRT: The test data shares the same covariance structure as the training data, but the relative power level is not constrained. In this case, the maximal invariant statistic collapses to a one-dimensional (1-D) scalar, which is also the ACE statistic. Furthermore, we show that the probability density function for the ACE possesses the property of total positivity, which establishes that it has monotone likelihood ratio. Thus, a threshold test on the ACE is UMP-invariant. This means that it has a claim to optimality, having the largest detection probability out of the class of detectors that are also invariant to affine transformations on the data matrix that leave the hypotheses unchanged. This requires an additional invariance not imposed by Bose and Steinhardt: invariance to relative scaling of test and training data. The ACE is invariant and has a Constant False Alarm Rate (CFAR) with respect to such scaling, whereas Kelly's GLRT and the AMF are invariant, and CFAR, only with respect to common scaling.

On a multidimensional oil exploration problem

This paper is concerned with optimal strategies for drilling in an oil exploration model. An exploration area contains n1 large and n2 small oilfields, where n1 and n2 are unknown, and represented by a two‐dimensional prior distribution π. A single exploration well discovers at most one oilfield, and the discovery process is governed by some probabilistic law. Drilling a single well costs c, and the values of a large and small oilfield are v1 and v2 respectively, v1 > v2 > c > 0. At each time t = 1, 2, …, the operator is faced with the option of stopping drilling and retiring with no reward, or continuing drilling. In the event of drilling, the operator has to choose the number k, 0 ≤ k ≤ m (m fixed), of wells to be drilled. Rewards are additive and discounted geometrically. Based on the entire history of the process and potentially on future prospects, the operator seeks the optimal strategy for drilling that maximizes the total expected return over the infinite horizon. We show that when π≻π′ in monotone likelihood ratio, then the optimal expected return under prior π is greater than or equal to the optimal expected return under π′. Finally, special cases where explicit calculations can be done are presented.

Experimentation, full revelation, and the monotone likelihood ratio property

We consider a simple two-period experimentation model and explore some comparative statics implications of failures in the monotone likelihood ratio property (MLRP). A canonical example of a non-MLRP noise density is a U-shaped density on a bounded interval. In that case, the optimal first period action and information at the optimum are non-monotonic in the volatility of noise. Furthermore, there may be discontinuities in the optimal action with respect to changes in the volatility of noise and other parameters, in which the agent switches back and forth between a low, non-fully-revealing action and a high, fully-revealing action.

Testing for the Monotone Likelihood Ratio Assumption

Although monotonicity of the likelihood ratio for conditioned densities is a common technical assumption in economic models, empirical tests for its plausibility do not appear to have been conducted. This article develops such a test based on the theory of order-restricted inference that is robust with respect to the correlation structure of the distributions being compared. The test is applied to study the technology revealed by agricultural production experiments. For the data under scrutiny, the results support the monotone likelihood ratio assumption. In a second application, we find some support for the assumption of affiliation among bids cast in a multiple-round Vickrey auction for a consumption good.

Monotone Forecasts

In this paper we provide necessary and sufficient conditions for the distribution of demand in the future to be stochastically increasing in the demand that has been observed in the past. We base our analysis on the multiperiod inventory model examined by Eppen and Iyer (1997). In the process of establishing the necessary and sufficient conditions we develop a new property called the sequential monotone likelihood ratio property.

Separating signaling equilibria under random relations between costs and attributes: discrete attributes

We identify conditions for separating signaling equilibria where discrete attributes are randomly related to a continuum of costs. A necessary condition is the ordering of the cost distributions conditional on attributes by first order stochastic dominance. A necessary and sufficient condition is the ordering of the cost distributions conditional on attributes by the monotone likelihood ratio property. An equivalent necessary and sufficient condition is the monotone ordering of the cost elasticities of these distributions.

Separating signaling equilibria under random relations between costs and attributes: continuum of attributes

We identify conditions for separating signaling equilibria where costs and attributes are randomly related and where both take a continuum of values. A necessary and sufficient condition is the ordering by the cost elasticities of the cost density functions with respect to the original probability measure and with respect to a probability measure modified by the “attribute payoff function”. This condition is the equivalent, under the continuum of attributes, to the condition, under discrete attributes, of ordering by the Monotone Likelihood Ratio Property (MLRP) that Feldman (Mathematical Social Sciences, 2004, in press) found in a companion paper. We, thus, introduce the concept of generalized MLRP (GMLRP). While the original MLRP ranks only posterior distributions induced by particular realizations, the GMLRP ranks posterior distributions induced by distributions as well.

Separating Signaling Equilibria under Random Relations Between Costs and Attributes: Continuum of Attributes

We identify conditions for separating signaling equilibria where costs and attributes are randomly related and where both take a continuum of values. A necessary and sufficient condition is the ordering by the cost elasticities of the cost density functions with respect to the original probability measure and with respect to a probability measure modified by the attribute payoff function. This condition is the equivalent, under the continuum of attributes, to the condition, under discrete attributes, of ordering by the Monotone Likelihood Ratio Property (MLRP) that Feldman (2003) found in a companion paper. We, thus, introduce the concept of Generalized MLRP (GMLRP). While the original MLRP ranks only posterior distributions induced by particular realizations, the GMLRP ranks posterior distributions induced by distributions as well.

SPRT Fixed Length Confidence Intervals

In this paper we will present a method for constructing fixed length confidence intervals using Sequential Probability Ratio Tests (SPRT). We will see that under the conditions of monotone likelihood ratio and that the log likelihood function is concave as a function of the parameter, the set of parameters corresponding to null hypotheses that has not been rejected is an interval. We apply the SPRT fixed length confidence interval to a sequence of observations on a Bernoulli distributed random variable where we get a SPRT fixed length interval for the parameter p.

Separating Signaling Equilibria Under Random Relations Between Costs and Attributes: Discrete Attributes

We identify conditions for separating signaling equilibria where discrete attributes are randomly related to a continuum of costs. A necessary condition is the ordering of the cost distributions conditional on attributes by first order stochastic dominance. A necessary and sufficient condition is the ordering of the cost distributions conditional on attributes by the monotone likelihood ratio property. An equivalent necessary and sufficient condition is the monotone ordering of the cost elasticities of these distributions.