Simultaneous estimation of the multivariate normal mean under balanced loss function

This paper considers simultaneous estimation of multivariate normal mean vector using Zellner's(1994) balanced loss function which is defined as follows:

where 0 < w < 1 and for i = 1,…,p and j = 1,…,n, X_ij is distributed as normal with mean θi and variance 1. It is shown that the sample mean, X, is admissible when p <3. For p ≥3, we obtain that James-Stein type estimator which has uniformly smaller risk than that of sample mean X.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Nonparametric statistical procedures for the changepoint problem

Let X₁,…,X_r?1,X_r,X_r+1,…,X_n be independent, continuous random variables such that X_i, i = 1,…,r, has distribution function F(x), and X_i, i = r+1,…,n, has distribution function F(x?Δ), with -∞ <Δ< ∞. When the integer r is unknown, this is refered to as a change point problem with at most one change. The unknown parameter Δ represents the magnitude of the change and r is called the changepoint. In this paper we present a general review discussion of several nonparametric approaches for making inferences about r and Δ.     

Upper tolerance limit for the largest observation from censored samples

This paper deals with obtaining an upper tolerance limit for a largest observation X_(n) in an ordered sample of size n from a continuous distribution where the first m observations X₍₁₎ < X₍₂₎ < … < X_(m), l ≤ m < n, have been observed. A criterion of “goodness” of tolerance limit is developed, and a method is given to obtain the best tolerance limit. This method is applied to exponential and Pareto distributions.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Dependence Structure of Spacings of Order Statistics

Let X_1:n ≤ X_2:n ≤···≤ X_n:n denote the order statistics of a sample of n independent random variables X₁, X₂,…, X_n, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (X_k1:n, X_k2:n ? X_k1:n,…, X_kr:n ? X_kr?1:n) is MTP₂ [S-MRR₂] whenever 1 ≤ k₁ < k₂ <···< k_r ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.     

Locally best invariant and locally minimax test of independence

Let X₁ X₂ … X_N be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sG_i) [(1–p) I + pE] Diag ($sG_i), $sG_i> 0, i = 1… p, 1>p>=1/p–1. Write r_ij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H₀ : p = 0 against the alternative H₁ : p>0 where $$ and $sG₁,…, $sG_p are unknown, the test which rejects H₀ for large value of $$ r_ij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

On the asymptotic distribution of an estimate of the change point in a failure rate

The failure rate r(t) is assumed to have the shape of the"first"part of the"bathtub"model, i.e.r(t) is non-increasing for t<r and is constant for t> r. Asymptotic distribution of one of the estimates proposed earlier has been investigated in this paper. This leads to a test for the hypothesis HQ r<r ₀ vs H :r>r (where TQ > 0). Asymptotic expression for the power of this test under Pitman alternatives is derived. Some simulations are reported.     

On ar(1) processes with exponential white noise

We consider the autoregressive model X_t= bX_t-1= Y_twhere 0 ≤ b < 1 and Y_tare independent random variables with an exponential distribution. The moments of the stationary distribution of X_tare calculated and the distribution of an approximation to the maximum likelihood estimator for b is derived. The result is used for a construction of a confidence interval for b.     

Exponential models,brownian motion,and independence

Some examples of steep, reproductive exponential models are considered. These models are shown to possess a τ-parallel foliation in the terminology of Barndorff-Nielsen and Blaesild. The independence of certain functions follows directly from the foliation. Suppose X(t) is a Wiener process with drift where X(t) = W(t) + ct, 0 < t < T. Furthermore let Y = max [X(s), 0 < s < T]. The joint density of Y and X = X(T), the end value, is studied within the framework of an exponential model, and it is shown that Y(Y – X) is independent of X. It is further shown that Y(Y – X) suitably scaled has an exponential distribution. Further examples are considered by randomizing on T.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

Correlation estimation in Downton's bivariate exponential distribution using incomplete samples

Suppose (X, Y) has a Downton's bivariate exponential distribution with correlation ρ. For a random sample of size n from (X, Y), let X _r:n be the rth X-order statistic and Y _[r:n] be its concomitant. We investigate estimators of ρ when all the parameters are unknown and the available data is an incomplete bivariate sample made up of (i) all the Y-values and the ranks of associated X-values, i.e. (i, Y _[i:n]), 1≤i≤n, and (ii) a Type II right-censored bivariate sample consisting of (X _i:n , Y _[i:n]), 1≤i≤r<n. In both setups, we use simulation to examine the bias and mean square errors of several estimators of ρ and obtain their estimated relative efficiencies. The preferred estimator under (i) is a function of the sample correlation of (Y _i:n , Y _[i:n]) values, and under (ii), a method of moments estimator involving the regression function is preferred.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

Variants of the graph dependent model in extreme value theory

With a set X₁, X₂, .... X_n n random variables, a graph is associated whose vertices are the integers 1,2,..., n and whose edges represent those pairs i and j for which the events {X_i>X} and {X_j>X} do not become “almost independent” for “large X”. With a variety of assumption on the edge set of the graph, the asymptotic distribution of the extremes of the X_j, when properly normalized, is determined. This refines the earlier result of the present author on this kind of dependence, and extends and unifies several known dependent extreme value models.