This paper considers estimates of multidimensional density functions based on a bounded and bandlimited weight function. The asymptotic behavior of quadratic functions of density function estimates useful in setting up a test of goodness of fit of the density function is determined. A test of independent is also given. The methods use a Poissonization of sample size. The estimates considered are appropriate if interested in estimating density functions or determining local deviations from a given density function.

Let $W_p^{(m)}(M) = \{f: f^{(\nu)} \operatorname{abs}. \operatorname{cont}., \nu = 0, 1,\cdots, m - 1, f^{(m)} \in \mathscr{L}_p, \|f^{(m)}\|_p \leqq M\}$, where $\|\cdot\|_p$ is the norm in $\mathscr{L}_p, m$ is a positive integer and $p$ is a real number, $p \geqq 1$. Let $\{\hat{f}_n(x)\}, n = 1, 2,\cdots$ be any sequence of estimates of a density at the point $x$ where $\hat{f}_n(x)$ depends on $n$ independent observations from some density $f \in W_p^{(m)}(M)$. It is shown that if $\sup_{f\in W_p^{(m)} (M)} E_f(f(x) - \hat{f}_n(x))^2 = b_nn^{-\phi(m, p+\varepsilon)}$, where $\phi(m, p) = (2m - 2/p)/(2m + 1 - 2/p)$, and $\varepsilon > 0$, then there exists a $D_0 > 0$ such that $b_n \geqq D_0$ for infinitely many $n$. Thus the best possible mean square convergence rate for a density estimate, which is uniform over $W_p^{(m)}(M)$, is not better than $n^{-\phi(m,p+\varepsilon)}$ for arbitrarily small $\varepsilon$. The following types of density estimates are shown to have mean square error at a point bounded above by $Dn^{-\phi(m,p)}$, provided that a certain parameter, usually depending on $m, p$ and $M$, is chosen optimally: the polynomial algorithm, kernel-type estimates, certain orthogonal series estimates, and the ordinary histogram. $D$'s for each method are given.

Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for $x$ in a finite interval, $x\in\lbrack a, b\rbrack = \lbrack 0, 1\rbrack$ say, by letting $\hat{F}_n(x), x \epsilon\lbrack 0, 1\rbrack$ be the unique cubic spline of interpolation to the sample cumulative distribution function $F_n(x)$ at equi-spaced points $x = jh, j = 0, 1,\cdots, l + 1, (l + 1)h = 1$, which satisfies specified boundary conditions $\hat{F}_n'(0) = a, \hat{F}_n'(1) = b$. The density estimate $\hat{f}_n(x)$ is then $\hat{f}_n(x) = d/dx\hat{F}_n(x)$. It is shown how to estimate $a$ and $b$. A formula for the optimum $h$ is given. Suppose $f$ has its support on [0, 1] and $f^{(m)}\in\mathscr{L}_p\lbrack 0, 1\rbrack$. Then, for $m = 1,2,3$ and certain values of $p$, it is shown that $E(f_n(x) - f(x))^2 = O(n^{-(2m - 2/p)/(2m+1-2/p)}).$ Bounds for the constant covered by the "$O$" are given. An extension to the $\mathscr{L}_p$ case of known convergence properties of the derivative of an interpolating spline is found, as part of the proofs.

This paper deals with the asymptotic distribution theory of least squares estimators in regression models having different analytical forms in different regions of the domain of the independent variable. An important special case is that of broken line regression, in which each segment of the regression function is a different straight line. The residual sum of squares function has many corners, and so classical least squares techniques cannot be directly applied. It is shown, however, that the problem can be transformed into a new problem in which the sum of squares function is locally smooth enough to apply the classical techniques. Asymptotic distribution theory is discussed for the new problem and it is shown that the results are also valid for the original problem. Results related to the usual normal theory are derived.

This paper deals with the asymptotic distribution of the log likelihood ratio statistic in regression models which have different analytical forms in different regions of the domain of the independent variable. It is shown that under suitable identifiability conditions, the asymptotic chi square results of Wilks and Chernoff are applicable. It is shown by example that if there are actually fewer segments than the number assumed in the model, then the least squares estimates are not asymptotically normal and the log likelihood ratio statistic is not asymptotically $\chi^2$. The asymptotic behavior is then more complicated, and depends on the configuration of the observation points of the independent variable.

Procedures are considered for testing whether the means of each variable in a sequence of independent random variables can be taken to be the same, against alternatives that a shift might have occurred after some point $r$. Bayesian test statistics as well as some statistics depending on estimates of $r$ are presented and their powers compared. Exact and asymptotic distribution functions are derived for some of the Bayesian statistics.

The elementary constructive methods for BBD's and GYD's which have been used in optimality considerations for a number of years, are listed and illustrated. These are not detailed algebraic or geometric prescriptions for listing each entry, but rather methods for combining LS's and known BIBD's to yield the desired products. Nevertheless, there results a large class of useful and previously unpublished designs.

A method is presented for making statistical inferences about the upper tail of a distribution function. It is useful for estimating the probabilities of future extremely large observations. The method is applicable if the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.

Many hypotheses in the multidimensional normal distribution are given or can be given by symmetries or, in other words, invariance. This means that the variances are invariant under a given subgroup of the general linear group in the vector space of observations. In this paper we define a class of hypotheses, the Invariant Normal Models, including all symmetry hypotheses. We derive the maximum likelihood estimator of the mean and variance and its distribution under the hypothesis. The value of the paper lies in the mathematical formulation of the theory and in the general results about hypotheses given by symmetries. Especially the formulation gives an easy simultaneous derivation of the real, complex and quaternion version of the Wishart distribution. Furthermore, we show that every invariant normal model with mean-value zero can be obtained by a symmetry.

In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.

Let $(N_1, \cdots, N_k)$ be a multinomial vector with $n = \sum N_i$ and with parameter $(p_1, \cdots, p_k), \sum p_i = 1$. Let $f_1, \cdots, f_k$ be real-valued functions defined on the integers $\{0, 1, \cdots, n\}$, and let $S_k = \sum^k_{i=1} f_i(N_i)$. Suppose $k \rightarrow \infty$, and as $k \rightarrow \infty$, that $n \rightarrow \infty$ and $\max_{1\leqq i \leqq k}p_i \rightarrow 0$. Conditions on the $\{f_i\}$ are given which guarantee that $S_k$, suitably centered and scaled, has a normal limit in law. An application shows that if $\min_{1\leqq i \leqq k} (np_i)$ is bounded away from zero and the $\{f_i\}$ are polynomials of bounded degree as $k \rightarrow \infty$, that $S_k$ is asymptotically normal provided only that a "uniformly asymptotically negligible" (uan) condition on the $\{f_i\}$ holds. For testing the specified simple hypothesis $p_i = p_i^0$ for all $1 \leqq i \leqq k$, Pearson's "chi-square" statistic and the likelihood ratio statistic can be written in the form of $S_k$. It is shown that these two statistics are asymptotically normal as $k \rightarrow \infty$ provided they satisfy simple conditions which are equivalent to their respective uan conditions.

Two independent Bernoulli processes (arms) have unknown success probabilities $\rho$ and $\lambda$. The initial (a priori) information about $\rho$ and $\lambda$ is expressed by probability distributions $dR(\rho) = C_R \rho{^r_0}(1 - \rho)^{r_0'} d\mu(\rho) \text{for the right arm},$ and $dL(\lambda) = C_L \lambda^{l_0}(1 - \lambda)^{l_0'} d\mu(\lambda) \text{for the left arm},$ where $\mu$ is any arbitrary measure on the unit interval. A specified number $n$ of observations is made sequentially, the arm selected at each stage depending on the previous observations and the initial information. A conjecture of Berry states that if the initial information present about the right arm (given by $r_0 + r_0'$) is not greater than that present for the left arm $(l_0 + l_0')$ and the initial expected value of $\rho$ is not less than that of $\lambda$, then for any $n$ the advantage (in terms of expected number of successes) of taking the first observation on the right arm is never less than that for the left arm. A proof of this conjecture is given in this paper.

This paper considers the discrete time finite state Markovian decision problem with the average return criterion. A modified form of the iterative method of dynamic programming is studied. Under the assumption that the maximal average return is independent of the initial state the asymptotic behaviour of the sequence of functions generated by this modified method is found. It is shown that the modified iterative method supplies both upper and lower bounds on the maximal average return and $\varepsilon$-optimal policies. Moreover, a convergence result is proved for the policies produced by the modified iterative method.

An extension of Strawderman's results yields minimax admissible estimates for the mean of a $p$-variate normal distribution where the known nonsingular covariance matrix is not necessarily the identity and $p > 2$. Minimax estimators for the case where the covariance matrix is unknown are also given.

In a full rank normal regression model, the existence of an unbiased estimate of the distribution associated to any "future" observation is studied. A necessary and sufficient condition in terms of the point at which the "future" observation will be taken is given for such an estimate to exist. This criterion is introduced to consider the validity of extrapolation.

Let $f(x)$ be a $\operatorname{pdf}$ of exponential form with respect to the measure $\mu$. Suppose a prior $\operatorname{pdf}$ $\pi$ has been placed on the natural parameter space $\Omega$, where $\pi$ is a density (with respect to $m$-dimensional Lebesgue measure) which is both positive and continuous at $\tau^\ast$, the true but unknown parameter value. Using basic properties of exponential families and certain associated convex functions, it is shown that the posterior pdf tends to the multivariate normal.

We consider the problem of estimating an unknown parameter $\theta$ on the basis of independent identically distributed observations with a common density $f(x,\theta)$ and give some lower bounds for the accuracy of estimates of $\theta$ expressed in terms of the Hellinger distance $\rho(\theta; \theta') = \int_\mathscr{X} (f^{\frac{1}{2}}(x; \theta) - f^{\frac{1}{2}}(x; \theta'))^2 d\nu.$

The rate at which the probability $P_\theta\{|t_n - \theta| \geqq \varepsilon\}$ of consistent estimator $t_n$ tends to zero is of great importance in large sample theory of point estimation. The main tools available at present for finding the rate are Bernstein-Chernoff-Bahadur's theorem and Sanov's theorem. In this paper, we give two new techniques for finding the rate of convergence of certain consistent estimators. By using these techniques, we have obtained an upper bound for the rate of convergence of consistent estimators based on sample quantities and proved that the sample median is an asymptotically efficient estimator in Bahadur's sense if and only if the underlying distribution is double-exponential. Furthermore, we have proved that the Bahadur asymptotic relative efficiency of sample mean and sample median coincides with the classical Pitman asymptotic relative efficiency.

In this paper locally most powerful rank tests for independence with censored data for a one-parameter family are derived. The statistic derived has discrete score functions and its asymptotic normality follows from a theorem essentially given by Ruymgaart [6].

A sample $(n_1, n_2,\cdots, n_t)$ is taken from a $t$-category multinomial population. The hypothesis of equiprobability, that the $t$ physical probabilities associated with the cells are all equal to $1/t$, is called the null hypothesis. Conditional on the non-null hypothesis, a symmetric Dirichlet prior of parameter $k$ is assumed $(k \geqq 0)$ and the Bayes factor against the null hypothesis, with this assumption, is denoted by $F(k)$. A conjecture made in 1965 is almost proved, namely that $F(k)$ has a unique local maximum and that this occurs for a finite value of $k$ if and only if Pearson's $X^2$ exceeds its number of degrees of freedom. The result is required for the calculation of $\max F(k)$, which provides a non-Bayesian significance criterion whose simple asymptotic distribution is good even in the extreme tail, and even for sample sizes less than $t$. This criterion arose from an attitude involving a Bayes/non-Bayes compromise.

Let $t$ be the translation parameter of a process $X(t), -\infty < t < \infty$. The likelihood ratio of the process $X(t)$ at $t$ against $t = 0$ can be written as $\exp\lbrack W(t) - \frac{1}{2}|t|\rbrack, -\infty < t < \infty$, where $W(t)$ is a standard Wiener process. For the absolute error-loss function the best invariant estimator of the translation parameter is the median of the posterior distribution. The distribution of the median for the posterior distribution is obtained, when the prior distribution for $t$ is the Lebesgue measure on the real line.

Consider the model equation $y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk} (i = 1,2,\cdots, r; j = 1,2,\cdots, s; k = 1,2,\cdots, n_{ij})$, where $\mu$ is a constant and $\alpha_i, \beta_j, \gamma_{ij}, e_{ijk}$ are distributed independently and normally with zero means and variances $\sigma_A^2, \sigma_B^2, \sigma^2_{AB}, \sigma^2$, re spectively. In this paper procedures for testing hypotheses on $\sigma_A^2/\sigma_B^2, \sigma^2/\sigma^2$, and $\sigma^2_{AB}/\sigma^2$ are given. The test procedure for $\sigma^2_{AB}/\sigma^2$ is compared with the corresponding test procedures when $\alpha_i, \beta_j$, and $\gamma_{ij}$ are fixed effects instead of being random.