Summary The problem of specifying and estimating the variance of estimated parameters based on complex sample designs from finite populations is considered. The results of this paper are particularly useful when the parameter estimators cannot be defined explicitly as a function of other statistics from the sample. It is shown how these results can be applied to linear regression, logistic regression and log linear contingency table models. An example of the application of the technique to the Canada Health Survey is given.

https://www.jstor.org/stable/info/1402588

On the variances of asymptotically normal estimators from complex surveys

The effect of stratification and clustering on the asymptotic distributions of standard Pearson chi-squared test statistics for goodness of fit (simple hypothesis) and independence in a two-way contingency table, denoted as X 2 and XI 2, respectively, is investigated It is shown that both X 2 and XI 2 are asymptotically distributed as weighted sums of independent χ1 2 random variables The weights are then related to the familiar design effects (deffs) used by survey samplers A simple correction to X 2, which requires only the knowledge of variance estimates (or deffs) for individual cells in the goodness-of-fit problem, is proposed and empirical results on the performance of corrected X 2 provided Empirical work on XI 2 indicated that the distortion of nominal significance level is substantially smaller with XI 2 than with X 2 Some results under simple models for clustering are also given

https://www.jstor.org/stable/pdfplus/2287815.pdf

The Analysis of Categorical Data from Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables

The construction of sample designs and estimators under a linear regression superpopulation model is considered. The anticipated variance, the variance of the predictor computed with respect to the sampling design and the superpopulation model, is used as a criterion for evaluating probability designs and model-unbiased predictors. Regression predictors that are model unbiased and design consistent are constructed.

Survey Design under the Regression Superpopulation Model

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

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Optimum bounds for the distributions of martingales in banach spaces

A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional ones. Moreover, the bounds on the Rosenthal-Burkholder type of moments seem to be to certain extent new even for the sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

Optimum bounds for the distributions of martingales in Banach spaces

To each of the items $1,2,\cdots, N$ in a finite population there is associated a variate value. The population is sampled by successive drawings without replacement in the following way. At each draw the probability of drawing item $s$ is proportional to a number $p_s > 0$ if item $s$ remains in the population and is 0 otherwise. Let $\Delta(s; n)$ be the probability that item $s$ is obtained in the first $n$ draws and let $Z_n$ be the sum of the variate values obtained in the first $n$ draws. Asymptotic formulas, valid under general conditions when $n$ and $N$ both are "large", are derived for $\Delta(s; n), EZ_n$ and $\operatorname{Cov}(Z_{n_1}, Z_{n_2})$. Furthermore it is shown that, still under general conditions, the joint distribution of $Z_{n_1}, Z_{n_2},\cdots, Z_{n_d}$ is asymptotically normal. The general results are then applied to obtain asymptotic results for a "quasi"-Horvitz-Thompson estimator of the population total.

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Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, II

We shall introduce a set of random variables and give interpretations of them in terms of coupon collection. A person collects coupons with different colors. Let there be in all $N$ different colors, which we label $1,2, \cdots, N$. The different colors may occur with different frequencies. The colors of successive coupons are independent. Let $J_\nu$ be the color of the $\nu$th coupon. Our formal assumptions are: $J_1, J_2, \cdots$ are independent random variables, all with the following distribution \begin{equation*}\tag{1.1} P(J = s) = p_s,\quad s = 1,2, \cdots, N\end{equation*} where \begin{equation*}\tag{1.2} p_s \geqq 0,\quad p_1 + p_2 + \cdots + p_N = 1.\end{equation*} Thus, $p_s$ is the probability that a coupon has color $s$. Let \begin{equation*}\tag{1.3} M_n = {\tt\#} \text{different elements among} (J_1, J_2, \cdots, J_n),\quad n = 1,2, \cdots.\end{equation*} Thus, $M_n$ is the number of different colors in the collection after $n$ coupons. Let \begin{equation*}\tag{1.4} T_n = \min \{\nu: M_\nu = n\},\quad n = 1,2, \cdots, N.\end{equation*} $T_n$ is the number of coupons needed in order to get a collection with $n$ different colors in. Define \begin{align*} \tag{1.5} D_\nu &= 1\quad\text{if} J_\nu \notin (J_1, J_2, \cdots, J_{\nu-1}),\quad \nu = 1,2, \cdots \\ &= 0\quad \text{otherwise}.\end{align*} Thus, $D_\nu$ tells if the $\nu$th coupon adds a new color to the collection or not. We shall assume that the coupons also carry a bonus value, which is a real number. All coupons with the same color have the same bonus value, while the bonus value may differ from color to color. Let $a_s$ be the bonus value of coupons with color $s, s = 1,2, \cdots, N$. The bonus sum of a collection of coupons is obtained by adding the bonus values of the different colors which are represented in the collection. Thus, duplicates do not count. Formally we define the bonus sum as follows. \begin{equation*}\tag{1.6} Q_n = a_{J_1}D_1 + a_{J_2}D_2 + \cdots + a_{J_n}D_n,\quad n = 1,2, \cdots.\end{equation*} The random variable $Q_n$ will be referred to as the bonus sum after $n$ coupons for a collector in the situation $\Omega = ((p_1, a_1), (p_2, a_2), \cdots, (p_N, a_N))$. We define for $B > 0$ \begin{equation*}\tag{1.7} W(B) = \min \{n: Q_n \geqq B\}.\end{equation*} $W(B)$ will be referred to as the waiting time to obtain bonus sum $B$ for a coupon collector in the situation $\Omega$. The following lemma, which is obvious, states that we have introduced a slight abundance of terminology and notation. LEMMA 1.1. The random variables $M_n$ and $T_n$ in (1.3) and (1.4) are respectively the bonus sum after $n$ coupons and the waiting time to obtain bonus sum $n$ for a coupon collector in the situation $((p_1), (p_2, 1), \cdots, (p_n, 1))$. Our main concern will be to study the random variable $W(B)$ and its particular case $T_n$. We confine ourselves to the case when all bonus values, $a_s$, are positive. The main result is that $W(B)$, under general conditions, is asymptotically (as $n$ and $N$ increase simultaneously) normally distributed. We give a brief sketch of the idea of proof, which is well known. When all $a$'s are positive, the distributions of the random variables $W(B)$ and $Q_n$ are related according to the formula \begin{equation*}\tag{1.8} P(W(B) > x) = P(Q_{\lbrack x \rbrack} 0.\end{equation*} With the aid of formula (1.8) one can "invert" results concerning either of the random variables $Q$ or $W$ to yield results concerning the other variable. In [5] we showed that $Q_n$, under general conditions, is asymptotically normally distributed. The asymptotic normality of $W$ will be derived by inversion of the results in [5]. The asymptotic behavior of the collector's waiting time has, to the best of our knowledge, earlier only been considered in the classical case, i.e. $p_s = 1/N$ and $a_s = 1, s = 1,2, \cdots, N$. In [4] Section 3, Renyi derives results about $M$ by first deriving results about $T$ and then "inverting." His basic tool is the representation \begin{equation*}\tag{1.9} T_n = U_1 + U_2 + \cdots + U_n\end{equation*} where $U_v$ is the waiting time from bonus sum $\nu - 1$ to bonus sum $\nu$. In the classical case $U_1, U_2, \cdots$ are independent random variables and $P(U_\nu = k) = ((\nu - 1)/N)^{k-1}(N - \nu + 1)/N, k = 1,2, \cdots$. Thus, results concerning the asymptotic behavior of sums of independent random variables can be applied. A complete investigation along these lines is given by Baum and Billingsley in [1]. A generalized version of the problem is considered by Ivchenko and Medvedev in [2]. In their problem, as in our problem here, a representation of the type (1.9) no longer holds. They proceed along the path we shall follow here, i.e. to obtain results about the waiting time by "inverting" results concerning the bonus sum. The following notation will be used throughout the paper. $E$ and $\sigma^2$ stand for expectation and variance. $^c$ denotes centering at expectation, i.e. $X^c = X - EX. X = _\mathscr{L} Y$ means that the random variables $X$ and $Y$ have the same distribution. $\Rightarrow$ denotes convergence in law. The normal distribution with mean $\mu$ and variance $\sigma^2$ is denoted by $N(\mu, \sigma^2)$. The integral part of a real number is denoted by \lbrack \rbrack.

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On the Coupon Collector's Waiting Time

/pdf/on-an-inequality-of-hoeffding-3qdrzqwuv0.pdf

On an Inequality of Hoeffding

/pdf/on-bounds-on-the-central-moments-of-even-order-of-a-sum-of-25drswzrap.pdf

Bengt Rosen

Papers

Asymptotic Theory for Successive Sampling with Varying Probabilities Without Replacement, II

On the Coupon Collector's Waiting Time

On an Inequality of Hoeffding

On Bounds on the Central Moments of Even Order of a Sum of Independent Random Variables