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Register a new userApproximation by normal distribution for a sample sum in sampling without replacement from a finite population
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Added by
Sherzod Mirakhmedov
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
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A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.
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Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders alpha some further series are provided, including a convex combination of gamma distributions for alpha = 1 and nearly of this type for alpha > 1. Furthermore, some representations of the distribution of the angle Phi between (X1,...,Xn) and (1,...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Phi).
Let $X,X_1,dots, X_n$ be i.i.d. Gaussian random variables in a separable Hilbert space ${mathbb H}$ with zero mean and covariance operator $Sigma={mathbb E}(Xotimes X),$ and let $hat Sigma:=n^{-1}sum_{j=1}^n (X_jotimes X_j)$ be the sample (empirical) covariance operator based on $(X_1,dots, X_n).$ Denote by $P_r$ the spectral projector of $Sigma$ corresponding to its $r$-th eigenvalue $mu_r$ and by $hat P_r$ the empirical counterpart of $P_r.$ The main goal of the paper is to obtain tight bounds on $$ sup_{xin {mathbb R}} left|{mathbb P}left{frac{|hat P_r-P_r|_2^2-{mathbb E}|hat P_r-P_r|_2^2}{{rm Var}^{1/2}(|hat P_r-P_r|_2^2)}leq xright}-Phi(x)right|, $$ where $|cdot|_2$ denotes the Hilbert--Schmidt norm and $Phi$ is the standard normal distribution function. Such accuracy of normal approximation of the distribution of squared Hilbert--Schmidt error is characterized in terms of so called effective rank of $Sigma$ defined as ${bf r}(Sigma)=frac{{rm tr}(Sigma)}{|Sigma|_{infty}},$ where ${rm tr}(Sigma)$ is the trace of $Sigma$ and $|Sigma|_{infty}$ is its operator norm, as well as another parameter characterizing the size of ${rm Var}(|hat P_r-P_r|_2^2).$ Other results include non-asymptotic bounds and asymptotic representations for the mean squared Hilbert--Schmidt norm error ${mathbb E}|hat P_r-P_r|_2^2$ and the variance ${rm Var}(|hat P_r-P_r|_2^2),$ and concentration inequalities for $|hat P_r-P_r|_2^2$ around its expectation.
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development complements recent results by Wei and Dudley (2011) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving corresponding results for one-sided statistics with emphasis on adjusted inequalities of the type proved originally by Dvoretzky, Kiefer, and Wolfowitz (1956) and by Massart (1990) for one-samp
Let $pi_1$ and $pi_2$ be two independent populations, where the population $pi_i$ follows a bivariate normal distribution with unknown mean vector $boldsymbol{theta}^{(i)}$ and common known variance-covariance matrix $Sigma$, $i=1,2$. The present paper is focused on estimating a characteristic $theta_{textnormal{y}}^S$ of the selected bivariate normal population, using a LINEX loss function. A natural selection rule is used for achieving the aim of selecting the best bivariate normal population. Some natural-type estimators and Bayes estimator (using a conjugate prior) of $theta_{textnormal{y}}^S$ are presented. An admissible subclass of equivariant estimators, using the LINEX loss function, is obtained. Further, a sufficient condition for improving the competing estimators of $theta_{textnormal{y}}^S$ is derived. Using this sufficient condition, several estimators improving upon the proposed natural estimators are obtained. Further, a real data example is provided for illustration purpose. Finally, a comparative study on the competing estimators of $theta_{text{y}}^S$ is carried-out using simulation.
Program synthesis has emerged as a successful approach to the image parsing task. Most prior works rely on a two-step scheme involving supervised pretraining of a Seq2Seq model with synthetic programs followed by reinforcement learning (RL) for fine-tuning with real reference images. Fully unsupervised approaches promise to train the model directly on the target images without requiring curated pretraining datasets. However, they struggle with the inherent sparsity of meaningful programs in the search space. In this paper, we present the first unsupervised algorithm capable of parsing constructive solid geometry (CSG) images into context-free grammar (CFG) without pretraining via non-differentiable renderer. To tackle the emph{non-Markovian} sparse reward problem, we combine three key ingredients -- (i) a grammar-encoded tree LSTM ensuring program validity (ii) entropy regularization and (iii) sampling without replacement from the CFG syntax tree. Empirically, our algorithm recovers meaningful programs in large search spaces (up to $3.8 times 10^{28}$). Further, even though our approach is fully unsupervised, it generalizes better than supervised methods on the synthetic 2D CSG dataset. On the 2D computer aided design (CAD) dataset, our approach significantly outperforms the supervised pretrained model and is competitive to the refined model.
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