This paper contains an Edgeworth-type expansion for the distribution of a minimum contrast estimator, and expansions suitable for the computation of critical regions of prescribed error (type one) as well as confidence intervals of prescribed confidence coefficient. Furthermore, it is shown that, for one-sided alternatives, the test based on the maximum likelihood estimator as well as the test based on the derivative of the log-likelihood function is uniformly most powerful up to a term of order $O(n^{-1})$. Finally, an estimator is proposed which is median unbiased up to an error of order $O(n^{-1})$ and which is--within the class of all estimators with this property--maximally concentrated about the true parameter up to a term of order $O(n^{-1})$. The results of this paper refer to real parameters and to families of probability measures which are "continuous" in some appropriate sense (which excludes the common discrete distributions).

Suppose $n$ tea-tasting ladies are asked to compare two brands of brandy. Each lady drinks two cups, one of each brand. The order is determined by coin-tossing. The lady is forced to state whether she prefers the first or the second cup. Thus each lady provides one out of four possible outcomes: which brand did she try first, did she prefer the first or the last cup. One usually summarizes this information by telling for each lady which of the brands she preferred. This paper shows that by doing so some relevant information is lost. One obtains a theory for a $2 \times 2$-table where no UMP unbiased test exists. Several modifications and generalizations are considered.

The rank statistics $S_{\Delta N} = N^{-1} \sum^N_{i=1} c_{Ni} R^\Delta_{Ni}$, with $R^Delta_{Ni}$ being the rank of $X_{Ni} + \Delta d_{Ni}, i = 1, 2, \cdots, N$ and $X_{N1}, \cdots, X_{NN}$ being the random sample from the basic distribution with density function $f$, are considered as a random process with $\Delta$ in the role of parameter. Under some assumptions on $C_{Ni}$'s, $d_{Ni}$'s and on the underlying distribution, it is proved that the process $\{S_{\Delta N} - S_{0N} - ES_{\Delta N}; 0 \leqq \Delta \leqq 1\}$, being properly standardized, converges weakly to the Gaussian process with covariances proportional to the product of parameter values. Under additional assumptions, $\Delta b_N$ can be written instead of $ES_{\Delta N}$, where $b_N = \sum^N_{i=1} C_{Ni}d_{Ni}\int f^2(x) dx$. As an application, this result yields the asymptotic normality of the standardized form of the length of a confidence interval for regression coefficient based on statistic $S_{\Delta N}$.

In this paper, the $c$-sample location problem with ordered or restricted alternatives is considered. Linear combinations of Chernoff-Savage type two-sample statistics computed among the $c(c - 1)/2$ pairs of samples are proposed as test statistics. It is shown that for each linear combination of two-sample statistics there is another linear combination, using only the $c - 1$ two-sample statistics based on adjacent samples as determined by the alternative, which has the same Pitman efficacy. If the ordered alternative is restricted further by specifying the relative spacings in the alternative, then the weighting coefficients can be chosen to maximize the Pitman efficacy over the class of linear combinations. It is also shown that the statistics proposed by Jonkheere [4] and Puri [9] have maximum Pitman efficacy when the alternative specifies equal spacings.

We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

Let $F_n(x)$ be the empirical distribution function based on a random sample of size $n$ from a continuous symmetric distribution with center $\theta$. As a nonparametric estimator of $\theta$, we propose $a^\ast$ where $a^\ast$ is chosen so as to minimize the function $h$ where $h(a) = \max_x |F_n(x) + F_n((2a - x)^-) - 1|$. In this paper we present an algorithm for constructing the interval of all $a$ which minimize $h$. We show that if $a^\ast$ is chosen as the center of this interval then $a^\ast$ is an unbiased estimator of $\theta$ which converges to $\theta$ with probability one at a rate of $n^{1/2-\delta}$ for $\delta > 0$. We then use the large or small sample distribution of $h(\theta)$ given by Butler (1969) to construct confidence intervals for $\theta$ and show how one can test for symmetry when the center is not specified under the null hypothesis.

The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is developed with no assumptions concerning the continuous or discrete nature of the underlying distribution function. Conditional rank tests, given the vector of ties, are shown to be similar, and the locally most powerful conditional rank test is given. The asymptotic distribution of linear rank statistics is given under the null hypotheses of randomness and symmetry (which includes the one-sample problem), and under contiguous alternatives. Three methods of assigning scores, the average scores, midranks, and randomized ranks methods, are discussed and briefly compared.

The asymptotic behavior is given for the error rates and ASN of the Wald SPRT and of invariant sequential tests. An asymptotic justification of Bhate's conjecture is also provided for invariant sequential tests. Expressions are obtained for the asymptotic relative efficiency of the Wald SPRT as compared with the corresponding best non-sequential test.

Let $p$ be the probability of any event in repeated independent trials. Sequential tests of the composite hypothesis $p \leqq p_0$ against the composite hypothesis $p > p_0$ are proposed, which asymptotically minimize the maximum risk when the cost of experimentation tends to zero, if the loss depends only on $p$ and satisfies some natural regularity conditions. Asymptotic power, expected sample size and risk of the tests are also given.

In this paper the minimax estimators of a cumulative distribution function $F$ is obtained for four types of loss functions. The result is quite general in that no restrictions are imposed on the unknown $F$. Moreover, the estimates do not depend upon the weight function used in the definition of the loss functions. It is also shown that the sample distribution function is minimax under one of these types of loss functions.

The general stochastic approximation procedure: $X_{n+1} = X_n - a_n c_n^{-1}h(Y_n),\quad n = 1,2, \cdots$ is considered, where $h$ is a Borel measurable transformation on the random observations $Y_n$. Under some mild requirements on $h$ and on the error random variables, the asymptotic properties, the a.s. convergence and the asymptotic normality are studied. The analysis is confined to the case where the error random variables are (conditionally) distributed according to a distribution function $G$ which is symmetric around 0 and admits a density $g$. The optimal choices of the design sequences $a_n$ and $c_n$ as well as the transformation $h$ are studied. The optimal transformation turned out to be equal to $-C(g'/g)$ (a.e. with respect to $G$) for a $C > 0$ and it is the transformation which minimizes the second moment of the asymptotic distribution of $n^\beta(X_n - \theta)$. The Robbins-Monro and the Kiefer-Wolfowitz situations are emphasized as special cases. With the optimal transformation, the new proposed generalized procedure is shown to yield asymptotically efficient estimators.

The problem of estimating the zero of a regression function by means of Robbins Monro type of stochastic approximation procedures is discussed. Optimality of the procedures is defined in terms of asymptotic variance. The discussion is restricted to the case of identically distributed errors. In that case we suggest transforming the observed random variables in order to minimize the asymptotic variance of the estimators. The optimal transformation turns out to depend on the underlying distribution of the errors and on the slope of the regression function at the zero.

The complete asymptotic expansion in powers of $N^{-\frac{1}{2}}$ for the distribution of the two-sample statistic $V_{NN}$ is derived under the null hypothesis. The proof is based on methods developed by Kemperman (1959).

We investigate the problem of estimating the mean vector $\mathbf{\theta}$ of a multivariate normal distribution with covariance matrix equal to $\sigma^2\mathbf{I}_p, \sigma^2$ unknown, and loss $\|\delta - \mathbf{\theta}\|^2/\sigma^2$. We first find a class of minimax estimators for this problem which enlarges a class given by Baranchik. This result is then used to show that for sufficiently large sample sizes (which never need exceed 4) proper Bayes minimax estimators exist for $\mathbf{\theta}$ if $p \geqq 5$.

A sequential test of a statistical hypothesis $H_0$ versus $H_1$ is said to be a test of Robbins type if there is a positive probability that the test will not stop if $H_0$ is true. Tests of this nature were introduced for testing the Bernoulli case by Darling and Robbins [1]; an earlier paper of Farrell [2] deals implicitly with the asymptotic expected sample size of such tests for testing the hypothesis $\theta = 0$ in the parametrized family of generalized density functions $h(\theta)e^{\theta x} d\mu$.

For the location parameter family of distributions $F(x - \theta)$ under some regularity conditions, a confidence interval for $\theta$ of fixed width $2d$ and given confidence coefficient $1 - \alpha$ in the limit as $d$ tends to zero is obtained using Hodges-Lehmann estimates based on Wilcoxon statistics. An upper bound on the average sample size is also given.

Two problems of sequential estimation, viz. the estimation of the mean of a normal distribution with unknown variance and the estimation of a binomial proportion are studied as the cost per observation tends to 0. For the first problem the asymptotic distribution of the stopping time of a procedure due to Robbins (1959) is shown to be normal. For the second problem the stopping time of a modification of Wald's (1951) procedure is asymptotically normal when the parameter is different from $\frac{1}{2}$. When the parameter is $\frac{1}{2}$, this stopping time does not enjoy asymptotic normality. The method employed is to first prove the convergence in probability of the stopping time which is then converted to convergence in distribution by using a theorem of Wittenberg (1964). This method also yields a new proof of a theorem of Siegmund (1968).

Godambe (1955) and Godambe and Joshi (1965) established the joint optimality of the Horvitz-Thompson estimates and the corresponding sampling designs with respect to a class of prior distributions which were "product measures." In the present paper the optimality is established replacing the "product measures" by prior distributions which are exchangeable in appropriately transformed variates. Furthermore, the optimality criterion is justified in terms of Chebychev's inequality.

The unbiasedness and the monotonicity property of the power functions of a class of tests for the equality of covariance matrices of two $p$-variate normal distributions have been studied. For testing $\Sigma = I_p$ in a $p$-variate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$, a class of tests is proposed and their power functions and admissibility are studied.