Conditional Density Function Research Articles

Quadratic errors for nonparametric estimates under dependence

We investigate nonparametric curve estimation (including density, distribution, hazard, conditional density, and regression functions estimation) by kernel methods when the observed data satisfy a strong mixing condition. In a first attempt we show asymptotic equivalence of average square errors, integrated square errors, and mean integrated square errors. These results are extensions to dependent data of several works, in particular of those by Marron and Härdle (1986, J. Multivariate Anal. 20 91–113). Then we give precise asymptotic evaluations of these errors.

Stand density management strategies under risk: effects of stochastic prices

A method is described for determining optimal economic strategies for density management in loblolly pine (Pinustaeda L.) stands in the southern United States. A stochastic dynamic programming model employs a price state transition matrix constructed using a first-order conditional cumulative density function for price based on time-series data for national forest pine stumpage in the South. The model also incorporates WTHIN, a pine growth and yield simulator widely used in the South to support analyses of alternative stand management strategies. Results indicate that at current average prices for pulpwood, on average sites, optimal planting density decisions recommended by the stochastic model are very similar to those obtained with a deterministic price equivalent. However, there are many instances where the thinning decisions recommended by the two models are different, especially when they are used to evaluate better sites or when the price of pulpwood is substantially increased above current region-wide averages.

Indicator principal component kriging

An alternative to multiple indicator kriging is proposed which approximates the full coindicator kriging system by kriging the principal components of the original indicator variables. This transformation is studied in detail for the biGaussian model. It is shown that the cross-correlations between principal components are either insignificant or exactly zero. This result allows derivation of the conditional cumulative density function (cdf) by kriging principal components and then applying a linear back transform. A performance comparison based on a real data set (Walker Lake) is presented which suggests that the proposed method achieves approximation of the conditional cdf equivalent to indicator cokriging but with substantially less variogram modeling effort and at smaller computational cost.

A Note on Bivariate Distributions That Are Conditionally Normal

Abstract It is possible for a nonnormal bivariate distribution to have conditional distribution functions that are normal in both directions. This article presents several examples, with graphs, including a counterintuitive bimodal joint density. The graphs simultaneously display the joint density and the conditional density functions, which appear as Gaussian curves in the three-dimensional plots.

A Note on Bivariate Distributions That are Conditionally Normal

Abstract It is possible for a nonnormal bivariate distribution to have conditional distribution functions that are normal in both directions. This article presents several examples, with graphs, including a counterintuitive bimodal joint density. The graphs simultaneously display the joint density and the conditional density functions, which appear as Gaussian curves in the three-dimensional plots.

Asymptotically efficient estimation of prior probabilities in multiclass finite mixtures

A prior probability estimator, a candidate for asymptotic efficiency, from within the class of recursive estimators proposed by the authors (1990) is synthesized. The authors prove asymptotic efficiency and convergence with probability one by involving a stochastic approximation theorem. The estimator can be implemented in practice for continuous, discrete, and mixed class conditional density functions, although continuous and mixed densities generally require repeated evaluation of expectations of certain functions through numerical techniques. Results of a simulation. experiment with discrete densities are included. Variations of the estimator, for computational simplicity, are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimation of mixing probabilities in multiclass finite mixtures

The problem of estimating prior probabilities in a mixture of M classes with known class conditional distributions is studied. The observation is a sequence of n independent, identically distributed mixture random variables. The first moments of appropriately formulated functions of observations are used to facilitate estimation. The complexity of these functions may vary from linear functions of the observations (in some cases) to complex functions of class conditional density functions of observations, depending on the desired balance between computational simplicity and theoretical properties. A closed-form, recursive, unbiased, convergent estimator using the density function is presented: the result is valid for any problem in which prior probabilities are identifiable. Discrete and mixed densities require a minor modification. Three application examples are described. The class conditional expectations of density functions, required for the initialization of the estimator algorithm, are analytically evaluated for Gaussian and exponential densities. >

On the log-normal diffusion process

The log-normal conditional density function is the delta function initial condition solution of a four-parameter Fokker–Planck equation. It defines a diffusion process over the open first quadrant of the (x,t) plane. This process reaches a nonzero steady state as t increases indefinitely if the drift parameter is positive. The process may be monotonic or by expansion and contraction (breathing). If the drift parameter is negative the process goes to zero by expansion and contraction towards x=0 as t increases indefinitely.

Analysis of marginal and conditional density functions for separate inference

This paper discusses, with measure-theoretical rigor, some basic aspects of the theory of separate inference. To analyze densities of marginal and conditional submodels, certain operators are introduced. First a general concept of decomposition of a model is proposed, and the corresponding factorization of densities of the model is established. Next it is shown that the property of smoothness of a family of densities is retained in the operation of conditioning, and therefore it yields the differentiability of the conditional expectation of a real-valued statistic in a certain sense. On the basis of this result, two measures of the effectiveness of a submodel in separate inference are investigated.

Generating random vectors from a bivariate random walk density function

Two simple algorithms are presented for obtaining random vectors from a bivariate random walk density function. Both algorithms are based on the fact that the bivariate density function can be written as a product of a marginal density function and a conditional density function. An acceptance/rejection technique was used for generating random variates from the conditional density function. The efficiency of the two algorithms was studied.

The Application of Conditional Beta Density Functions to Model Time- Delay Measurement Errors

A conditional beta probability density function is employed to model time-delay measurement errors associated with sensors used in triangulation ranging applications. This type of density function is less restrictive than the beta density used in previous analyses in that end points can be extended out beyond the imposed range constraints and data editing is allowed before range calculations. The resulting conditional range density function is computed along with the mean and variance. Comparisons are made of the pertinent range statistics associated with the beta and conditional beta density functions. Finally, the conditional beta density function range statistics are compared with those obtained from a gated Gaussian density function for equivalent values of time-delay measurement error standard deviation. The Gaussian statistics are determined via Monte-Carlo simulation.

Strong consistency of the kernel estimators of conditional density function

Let (X, Y) be a ~ • RS-valued vector. Assume ~hat when X =~ ~ is given, ~her~ exists a conditional density of Y ~o be denoted by f ( y [ z ) , which is a Borel-measurable function of (~, y). Note that we do not assume the existence of a density function of (X, Y). Let (X1, Y~), --., (X., Y.) be i.i.d, samples of (X, Y). Our purpose is to estimate/(~/l~) based on these samples. This is an interesting problem in view of either pure theory or practical applications. Motivated by the idea suggests1 in kernel and NI~ estimations in the theory of nonparametric regression and density estimates, the first author proposes the following two classes of estimators of f (g[~) : 1) Let JE'I(~), K2(v) be density functions on ~Y and R' respectively. Take h ~ h . ~ 0 , and write

The non‐Gaussian joint probability density function of slope and elevation for a nonlinear gravity wave field

On the basis of the mapping method developed by Huang et al. (1983), an analytic expression for the non‐Gaussian joint probability density function of slope and elevation for nonlinear gravity waves is derived. Various conditional and marginal density functions are also obtained through the joint density function. The analytic results are compared with a series of carefully controlled laboratory observations, and good agreement is noted. Furthermore, the laboratory wind wave field observations indicate that the capillary or capillary‐gravity waves may not be the dominant components in determining the total roughness of the wave field. Thus, the analytic results, though derived specifically for the gravity waves, may have more general applications.

On a general storage problem and its approximating solution

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn+1(y) = ∫ l(y, w)gn(w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

On a general storage problem and its approximating solution

A GI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rate r(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the form gn +1(y) = ∫ l(y, w)gn (w) dw with l(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.

Maximum-likelihood estimation of the angular position and extent of a target

The maximum-likelihood estimates of the angular position and angular extent of a target relative to the boresight of a monopulse radar system is derived, and the statistics of the estimates are determined. It is shown that the moments of the estimates do not exist when the thermal noise power in the receiver is nonzero. However, conditional moments are determined which do exist, as well as conditional density and/or distribution functions.

Estimation of demand and supply functions from univariate sample with known separation

It is shown that the joint unconditional density function of demand and supply is a computable function of parameters of the conditional density function of the observed quantity. Propositions of formulating unconditional and conditional disequilibrium measures are suggested.

Approximating Conditional Distributions by the Mixed Edgeworth-Saddlepoint Expansion

Conditional density and distribution functions can be approximated by the mixed Edgeworth-saddlepoint expansion. This technique is briefly described, and its use and character are illustrated by three examples concerning miscellaneous exponential families of order two.

Exponential Fourier densities and optimal estimation for axial processes

Several models are proposed which are believed to be generic for the estimation of discrete-time axial processes. By introducing axial exponential Fourier densities and axial exponential trigonometric densities, finite-dimensional recursive schemes are obtained for updating the conditional density functions. The underlying idea is the closure properties under the Bayes rule of the various combinations of these exponential densities. An estimation error criterion is introduced which is compatible with a Riemannian metric. The corresponding Optimal axial estimates be easily computed from the conditional probability distributions.

Approximating conditional distributions by the mixed Edgeworth-saddlepoint expansion

SUMMARY Conditional density and distribution functions can be approximated by the mixed Edgeworth-saddlepoint expansion. This technique is briefly described, and its use and character are illustrated by three examples concerning miscellaneous exponential families of order two.