Decision-theoretic estimation of generalized variance and generalized precision

Consider a random data matrix X=(X₁,...,X_k):pXk with independent columns [sathik] and an independent p X p Wishart matrix [sathik]. Estimators dominating the best affine equivariant estimators of [sathik] are obtained under four types of loss functions. Improved estimators (Testimators) of generalized variance and generalized precision are also considered under convex entropy loss (CEL).     

Defective Galton–Watson processes

The Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k ? 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑_{k ? 0}p_k = 1 ? ? for some ? ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ?. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ? → 0.     

Probabilities and moments of generalized discrete distributfons using finite difference operators

The probabilities and factorial moments of the univar iate and multivariate generalized (or compound) discrete di st r-Lbut Lons with probability generating functions H(t)=F(G(t)) and H(t₁,…,t_k)=F(G(t₁,…,t_k))or H(t₁,…,t_k) = F(G₁(t₁),…, G_k( t_k)) are derived using finite difference operators.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

Optimal linear combinations using householder transformations

In pattern classification of sampled vector valued random variables it is often essential, due to computational and accuracy considerations, to consider certain measurable transformations of the random variable. These transformations are generally of a dimension-reducing nature. In this paper we consider the class of linear dimension reducing transformations, i.e., the k × n matrices of rank k where k < n and n is the dimension of the range of the sampled vector random variable.

In this connection, we use certain results (Decell and Quirein, 1973), that guarantee, relative to various class separability criteria, the existence of an extremal transformation. These results also guarantee that the extremal transformation can be expressed in the form (I_k∣ Z)U where I_k is the k × k identity matrix and U is an orthogonal n × n matrix. These results actually limit the search for the extremal linear transformation to a search over the obviously smaller class of k × n matrices of the form (I_k ∣Z)U. In this paper these results are refined in the sense that any extremal transformation can be expressed in the form (I_K∣Z)H_p … H₁ where p ≤ min{k, n?k} and H_i is a Householder transformation i=l,…, p, The latter result allows one to construct a sequence of transformations (L_K∣ Z)H₁, (I_K Z)H₂H₁ … such that the values of the class separability criterion evaluated at this sequence is a bounded, monotone sequence of real numbers. The construction of the i-th element of the sequence of transformations requires the solution of an n-dimensional optimization problem. The solution, for various class separability criteria, of the optimization problem will be the subject of later papers. We have conjectured (with supporting theorems and empirical results) that, since the bounded monotone sequence of real class separability values converges to its least upper bound, this least upper bound is an extremal value of the class separability criterion.

Several open questions are stated and the practical implications of the results are discussed.     

Combined Bayesian estimates for the equicorrelation coefficient

Combined Bayesian estimates for equicorrelation covariance matrices are considered. The case of a common equicorrelation p and possibly different standard deviations σ_l,σ_k among k experimental groups is examined first, and the Bayesian estimation of (σ, σ₁,σ_k) is discussed. Secondly, under the assumption of a common standard deviation and possibly different equicorrelations, the Bayesian estimation of (ρ₁,ρ_k,σ) is considered.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

Testing for and against a set of linear inequality constraints in a multinomial setting

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely p_ij = P_ji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing H_t,0 against H₁ - H₀ and H₁ against H₂ - H ₁ when H₀:k × i=1 p_ix_ji = 0, j = 1,…, s, H₁:k × i=1 p_ix_ji × 0, j = 1,…, s, and does not impose any restrictions on p. The x_ji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

Estimation of the location parameter and the average worth of the selected subset of two parameter exponential populations under LINEX loss function

Suppose a subset of populations is selected from k exponential populations with unknown location parameters θ₁, θ₂, …, θ_k and common known scale parameter σ. We consider the estimation of the location parameter of the selected population and the average worth of the selected subset under an asymmetric LINEX loss function. We show that the natural estimator of these parameters is biased and find the uniformly minimum risk-unbiased (UMRU) estimator of these parameters. In the case of k = 2, we find the minimax estimator of the location parameter of the smallest selected population. Furthermore, we compare numerically the risk of UMRU, minimax, and the natural estimators.     

Maximum likelihood estimation in the presence of outiliers

This paper deals with the maximum likelihood estimation of parameters when the sample (x₁…x_n ) may heve k spuriously generated observations from another distribution, say G≠F, where F is the distribution of the target population. If G is stochastically larger than F, then these k observations may give rise to k extreme observations or ‘outliers’. This situation is often described by a so-called ‘k-outlier model’ in which in addition to the parameters involved in F and G, the set ν={ν₁,…,ν_k} of indices, for which x_νj , j=1,…,k, come from G, is also unknow.     

A NEW EXPANSION FOR THE MULTIVARIATE NORMAL DISTRIBUTION1

A new expansion is given for the integral of the multivariate normal distribution over a region of the form (V₁, m) ×.× (u_k, ∞). The regions of convergence and divergence are partially identified and shown to be different from those of the tetrachoric expansion.     

General scores statistics on ranks in the analysis of unbalanced designs

The authors consider the situation of incomplete rankings in which n judges independently rank k_i ∈ {2, …, t} objects. They wish to test the null hypothesis that each judge picks the ranking at random from the space of k_i! permutations of the integers 1, …, k_i. The statistic considered is a generalization of the Friedman test in which the ranks assigned by each judge are replaced by real‐valued functions a(j, k_i), 1 ≤ j ≤ k_i ≤ t of the ranks. The authors define a measure of pairwise similarity between complete rankings based on such functions, and use averages of such similarities to construct measures of the level of concordance of the judges' rankings. In the complete ranking case, the resulting statistics coincide with those defined by Hájek & ?idák (1967, p. 118), and Sen (1968). These measures of similarity are extended to the situation of incomplete rankings. A statistic is derived in this more general situation and its properties are investigated.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

A series of group divisible designs

Bose and Shrikhande C19763 proved that if D(m, k, ?) is a Baer subdesign of another SBIBD D₁ (v₁, k₁ ?), k₁>k, then it also contains a complementary subdesign D^* which is symmetric GDD, D^* (v^*, k^*; ?-1, ?; m, n). Utilising this, we give a necessary condition for a SBIBD D to be a Baer subdesign of D₁ and also give the parameters. Some GD designs are constructed.     

Estimation and comparison of lognormal parameters in the presence of censored data

It is assumed that k(k?>?2) independent samples of sizes n _i (i?=?1, …, k) are available from k lognormal distributions. Four hypothesis cases (H ₁ – H ₄) are defined. Under H ₁, all k median parameters as well as all k skewness parameters are equal; under H ₂, all k skewness parameters are equal but not all k median parameters are equal; under H ₃, all k median parameters are equal but not all k skewness parameters are equal; under H ₄, neither the k median parameters nor the k skewness parameters are equal. The Expectation Maximization (EM) algorithm is used to obtain the maximum likelihood (ML) estimates of the lognormal parameters in each of these four hypothesis cases. A (2k???1) degree polynomial is solved at each step of the EM algorithm for the H ₃ case. A two-stage procedure for testing the equality of the medians either under skewness homogeneity or under skewness heterogeneity is also proposed and discussed. A simulation study was performed for the case k?=?3.     

Maximum Likelihood Estimation in the Bivariate Binomial (0,1) Distribution:Application TO 2×2 Tables

We consider a 2×2 contingency table, with dichotomized qualitative characters (A,A) and (B,B), as a sample of size n drawn from a bivariate binomial (0,1) distribution. Maximum likelihood estimates p?₁p?₂ and p? are derived for the parameters of the two marginals p₁p₂ and the coefficient of correlation. It is found that p? is identical to Pearson's (1904)?=(χ²/n)½, where ?² is Pearson's usual chi-square for the 2×2 table. The asymptotic variance-covariance matrix of p?_lp?₂and p is also derived.