Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic R. The standard normal approximation to the conditional distribution of R typically has error of order O(n^{-1/2}), where n is the sample size. There are several modifications for R, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations.
We investigate saddlepoint approximations applied to the score test statistic in genome-wide association studies with binary phenotypes. The inaccuracy in the normal approximation of the score test statistic increases with increasing sample imbalance and with decreasing minor allele count. Applying saddlepoint approximations to the score test statistic distribution greatly improve the accuracy, even far out in the tail of the distribution. By using exact results for an intercept model and binary covariate model, as well as simulations for models with nuisance parameters, we emphasize the need …
We consider the problem of comparing two Poisson parameters from the Bayesian perspective. Kawasaki and Miyaoka (2012b) proposed the Bayesian index $P(λ_1 < λ_2 | X_1,X_2)$ and expressed it using the hypergeometric series. In this paper, under some conditions, we give four other expressions of the Bayesian index in terms of the cumulative distribution functions of beta, $F$, binomial, and negative binomial distribution. Next, we investigate the relationship between the Bayesian index and the $p$-value of the conditional test with the null hypothesis $H_0: λ_1 \geq λ_2 $ versus an alternative h…
We propose a novel Bayesian model framework for discrete ordinal and count data based on conditional transformations of the responses. The conditional transformation function is estimated from the data in conjunction with an a priori chosen reference distribution. For count responses, the resulting transformation model is novel in the sense that it is a Bayesian fully parametric yet distribution-free approach that can additionally account for excess zeros with additive transformation function specifications. For ordinal categoric responses, our cumulative link transformation model allows the i…
In this paper we study the cumulative conditional expectation function (CCEF) in the copula context. It is shown how to compute CCEF in terms of the cumulative copula function, this natural representation allows to deduce some useful properties, for instance with applications to convex combination of copulas. We introduce approximations of CCEF based on Bernstein polynomial copulas. We introduce estimators for CCEF, which were constructed through Bernstein polynomial estimators for copulas. The estimators are asymptotically normal and biased for CCEF.
Conventional multiclass conditional probability estimation methods, such as Fisher's discriminate analysis and logistic regression, often require restrictive distributional model assumption. In this paper, a model-free estimation method is proposed to estimate multiclass conditional probability through a series of conditional quantile regression functions. Specifically, the conditional class probability is formulated as difference of corresponding cumulative distribution functions, where the cumulative distribution functions can be converted from the estimated conditional quantile regression f…
An individualized risk prediction model that dynamically updates the probability of a clinical event from a specific cause is valuable for physicians to be able to optimize personalized treatment strategies in real-time by incorporating all available information collected over the follow-up. However, this is more complex and challenging when competing risks are present, because it requires simultaneously updating the overall survival and the cumulative incidence functions (CIFs) while adjusting for the time-dependent covariates and time-varying covariate effects. In this study, we developed a …
Using cumulative residual processes, we propose joint goodness-of-fit tests for conditional means and variances functions in the context of nonlinear time series with martingale difference innovations. The main challenge comes from the fact the cumulative residual process no longer admits, under the null hypothesis, a distribution-free limit. To obtain a practical solution one either transforms the process in order to achieve a distribution-free limit or approximates the non-distribution free limit using a numerical or a re-sampling technique. Here the three solutions will be considered.It is …
Nonparametric or distribution-free charts can be useful in statistical process control problems when there is limited or lack of knowledge about the underlying process distribution. In this paper, a phase II Shewhart-type chart is considered for location, based on reference data from phase I analysis and the well-known Mann-Whitney statistic. Control limits are computed using Lugannani-Rice-saddlepoint, Edgeworth, and other approximations along with Monte Carlo estimation. The derivations take account of estimation and the dependence from the use of a reference sample. An illustrative numerica…
In this paper, we introduce a new class of bivariate distributions called the bivariate exponentiated extended Weibull distributions. The model introduced here is of Marshall-Olkin type. This new class of bivariate distributions contains several bivariate lifetime models. Some mathematical properties of the new class of distributions are studied. We provide the joint and conditional density functions, the joint cumulative distribution function and the joint survival function. Special bivariate distributions are investigated in some detail. The maximum likelihood estimators are obtained using t…
We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asy…
We consider estimating a low-dimensional parameter in an estimating equation involving high-dimensional nuisances that depend on the parameter. A central example is the efficient estimating equation for the (local) quantile treatment effect ((L)QTE) in causal inference, which involves as a nuisance the covariate-conditional cumulative distribution function evaluated at the quantile to be estimated. Debiased machine learning (DML) is a data-splitting approach to estimating high-dimensional nuisances using flexible machine learning methods, but applying it to problems with parameter-dependent nu…
In this paper, we are interested in nonparametric kernel estimation of a generalized regression function, including conditional cumulative distribution and conditional quantile functions, based on an incomplete sample $(X_t, Y_t, ζ_t)_{t\in \mathbb{ R}^+}$ copies of a continuous-time stationary ergodic process $(X, Y, ζ)$. The predictor $X$ is valued in some infinite-dimensional space, whereas the real-valued process $Y$ is observed when $ζ= 1$ and missing whenever $ζ= 0$. Pointwise and uniform consistency (with rates) of these estimators as well as a central limit theorem are established. Con…
A random variable $Y_1$ is said to be smaller than $Y_2$ in the increasing concave stochastic order if $\mathbb{E}[ϕ(Y_1)] \leq \mathbb{E}[ϕ(Y_2)]$ for all increasing concave functions $ϕ$ for which the expected values exist, and smaller than $Y_2$ in the increasing convex order if $\mathbb{E}[ψ(Y_1)] \leq \mathbb{E}[ψ(Y_2)]$ for all increasing convex $ψ$. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y) = \mathbb{P}(Y \leq y \mid X = x)$ of a response variable $Y$ given a covariate $X$, solely under the assumption that the conditiona…
