Finite population (FP :ℱ ) total prediction using a survey sample (SS :s^* ) taken from the ℱ using a suitable sampling design (D_s^*) has been an important topic of research interest in survey sampling over the last five decades. In general a model based approach is followed for such a prediction where the ℱ based hypothetical data involving the responses and covariates are assumed to follow a super-population (SP :𝒮 ) regression model, and the non-sampled (ns) response total is predicted by its 𝒮 model based expectation. Next, as the expected total contains unknown model parameters such as regression effects, an optimal prediction estimate is supposed to be obtained by optimally estimating the 𝒮 regression parameters using the SS (s^* ⊂ℱ) taken from the FP. For the purpose, the decades long existing studies, have been using the sample ( s^* ) based conditional (using conditioning on the sample) OLS/GLS (ordinary/generalized least square) estimates for the 𝒮 regression parameters in the independence and correlation setups, respectively. However as we demonstrate in this paper in the context of a binary population total (equivalent to proportion) prediction, in contrary to the claims made in the existing studies, the aforementioned OLS/GLS estimates fail to be model unbiased for the 𝒮 regression parameters, and hence they fail to produce valid MU prediction for the total. More specifically, it is shown that the OLS estimates are not MU, rather they are design cum model unbiased (DCMU); whereas the GLS estimates are neither MU nor DCMU. As a remedy, the sampling weighted OLS (SWOLS) estimates are used (as they are DCMU) for optimal prediction for independent binary data, whereas a new survey sample based doubly weighted (SSDW) regression estimates are derived which are DCMU under the correlation setup and the produce DCMU prediction. This DCM based principle based on the sampling sequence (s^* ⊂ℱ⊂𝒮) is further implemented to obtain variance of the regression effects estimators and subsequently the variance of the DCMU predictor for the total. The asymptotic properties such as consistency of the DCMU predictor is also established.
We study nonlinear processes for which quadratic predictors provide superior forecasts to using linear predictors alone. For this class, designated as second order forecastable, we develop theoretical results for quadratic prediction, offering three main contributions: (1) a new theory for infinite arrays and their factorization is developed, generalizing multivariate spectral factorization theory; (2) two formulas for the quadratic h-step ahead forecast filter (based on an infinite past) are derived; (3) a necessary and sufficient condition involving the bispectrum is derived, for discerning whether quadratic prediction offers a benefit over linear prediction. For the two quadratic forecast filters, the first is an exact expression whereas the second involves some implicitly defined quantities, and is more practical for computation. The new results are numerically illustrated on several nonlinear processes. Finally, the quadratic forecasting filter is applied to survey time series data of the U.S. Census Bureau’s Annual Retail Trade Survey, and the forecasts are used to detect survey discontinuity potentially arising from reclassification.
For clinical trials that address questions of non-inferiority or superiority, the null hypothesis specifies a non-zero difference between response probabilities p_0 ( p_1 ) under control(treatment) regimens. A confidence interval for p_0 under this hypothesis provides a summary of our knowledge of the baseline probability p_0 , which can easily be converted into an interval for the treatment probability p_1 . When the hypothesised difference between p_0 and p_1 is zero, there is an exact interval for p_0 based on the total number of responses known as the Clopper-Pearson interval. Unfortunately, this construction does not extend to the hypothesis of non-zero difference. There is no quickly computed accurate method currently available. In this paper, we propose and investigate the properties of a conceptually simple and computationally feasible interval and demonstrate that it has reasonable one-sided and excellent two-sided coverage properties. This is not only useful as a summary of post-experimental knowledge, but is a key input into the testing procedure of Berger and Boos (1994).
We investigate the problem of nonparametric density estimation on ℝ_+ in the presence of incomplete responses under a Missing At Random (MAR) mechanism. Since standard symmetric kernel estimators are known to suffer from severe boundary effects on positive supports, we develop an inverse-probability-weighted gamma kernel methodology specifically designed to accommodate simultaneously the support constraint and the missingness structure. The proposed procedure is built from an ideal reweighted estimator involving the unknown observation probabilities and from a feasible estimator obtained by replacing these propensity scores with a nonparametric Nadaraya–Watson-type estimate based on fully observed auxiliary covariates. Under suitable regularity and smoothing assumptions, we derive a detailed asymptotic theory for both estimators. In particular, we establish explicit pointwise expansions for the bias and variance, obtain the corresponding mean squared error and mean integrated squared error, and prove asymptotic normality in the interior region. These results extend the classical gamma-kernel framework for complete data to the substantially more delicate setting of MAR incompleteness, and they quantify the additional stochastic cost induced by propensity-score estimation. The finite-sample behavior of the method is assessed through an extensive Monte Carlo study covering several positive-supported models, varying dependence levels between the study variable and the auxiliary covariates, and multiple missingness rates. The numerical evidence shows that the proposed estimator retains the boundary-adaptive advantages of gamma kernels while providing stable and accurate recovery of the target density under incomplete observation. A real-data analysis further illustrates its practical relevance for distributional inference with positive-valued variables subject to missingness.
Let N=∏ _i=1^kp_i^a_i be a fixed positive integer with known prime factorization. We investigate bootstrap resampling from the prime-factor multiset of N and study two associated statistics, namely the log-product of the resampled integer and the number of distinct observed primes appearing in the resample. It is shown that the log-product has explicit mean and variance determined by the empirical law of the logarithms of the observed primes, while bootstrap- ω is an occupancy sum with explicit mean, variance, and negative pairwise covariance structure. In the square-free case, exact formulas are obtained and asymptotic constants are identified, yielding a benchmark on the Erdős–Kac scale in random-like regimes. Numerical experiments on a synthetic corpus of factorizations illustrate the formulas, the square-free benchmark, and the behavior of the diagnostics across smooth, square-free, spiky, and mixed factorization regimes.
