On concurrence of several linear regressions and applications to problems of enzyme kinetics

In comparing several regressions E(y_ij) =α_i + β_ix_ij i = 1, 2, ..., k, j = 1,2, ..., n_i, researchers are generally interested in the following five problems: whether they have (1) equal slope, (2) equal intercept, (3) coincidence, (4) common intersection on X-axis, and (5) common intersection on (X,Y) - plane. Problems (1) - (3) can be put into the framework of the general linear hypothesis and the F-test can be used. However, problems (4) and (5) cannot be put into the general linear hypothesis because they are ratios of parameters. Hence, in this paper we consider the generalized likelihood ratio test for hypothesis testing. An application to an enzyme kinetics problem in Aniline Metabolism is demonstrated     

TESTING THE EQUALITY OF PARAMETERS IN THE DISTRIBUTIONS WHEN THE RANGE DEPENDS UPON THE PARAMETER

The authors derive the null and non-null distributions of the test statistic v=y_min/y_max (where y_min= min x_ij, y_max= max x_ij, J=1,2, …, k) connected with testing the equality of scale parameters θ₁, θ₂, …θ_k in certain, class of density functions given by      

Calculating nonparametric confidence intervals for quantiles using fractional order statistics

In this paper, we provide an easy-to-program algorithm for constructing the preselected 100(1 - alpha)% nonparametric confidence interval for an arbitrary quantile, such as the median or quartile, by approximating the distribution of the linear interpolation estimator of the quantile function Q L ( u ) = (1 - epsilon) X \[ n u ] + epsilon X \[ n u ] + 1 with the distribution of the fractional order statistic Q I ( u ) = Xn u , as defined by Stigler, where n = n + 1 and \[ . ] denotes the floor function. A simulation study verifies the accuracy of the coverage probabilities. An application to the extreme-value problem in flood data analysis in hydrology is illustrated.     

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t_i = iΔ , i = 0, ..., n . We study a particular class of estimating functions of the form ∑ f (θ, X _{t i −1}) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.     

Test of independence between tow sets of variates

In this paper asymptotic expansions of the null as well as non-null distributions of the likelihood ratio criterion for testing independence between two sets of variates are obtained. These appear to be better than the ones available in the literature . In factin the null case for p = 1 and p = 2 , t h e expansion reduces to the exact di stribution. In the non-null case, the expansion is given i n terms of non-central beta distributions and for the case when the population canonical correlation coefficients are small.     

On a conjecture of Kakosyan,Klebanov and Melamed

Suppose X₁, X₂, ..., X_m is a random sample of size m from a population with probability density function f(x), x>0 and let X_1,m<...m,m be the corresponding order statistics. We assume m as an integer valued random variable with P(m=k)=p(1?p)^k?1, k=1, 2, ... and 0 and n X_1,n for fixed n characterizes the exponential distribution. In this paper we prove that under the assumption of monotone hazard rate the identical distribution of and (n?r+1) (X_r,n?X_r?1,n) for some fixed r and n with 1≤r≤n, n≥2, X_0,n=0, characterizes the exponential distribution. Under the assumption of monotone hazard rate the conjecture of Kakosyan, Klebanov and Melamed follows from the above result with r=1.     

Non-parametric Kernel Estimation of the Coefficient of a Diffusion

In this work we exhibit a non-parametric estimator of kernel type, for the diffusion coefficient when one observes a one-dimensional diffusion process at times i / n for i = , ..., n and study its asymptotics as n ←∞. When the diffusion coefficient has regularity r ≥ 1, we obtain a rate 1/ n ^{r /(1+2 r )}, both for pointwise estimation and for estimation on a compact subset of R: this is the same rate as for non-parametric estimation of a density with i.i.d. observations.     

A one-sided test for testino homogeneity of scale parameters against simple ordered alternative

In an earlier paper the authors (1997) extended the results of Hayter (1990) to the two parameter exponential probability model. This paper addressee the extention to the scale parameter case under location-scale probability model. Consider k (k≧3) treatments or competing firms such that an observation from with treatment or firm follows a distribution with cumulative distribution function (cdf) F_i(x)=F[(x-μ_i)/Q_i], where F(·) is any absolutely continuous cdf, i=1,…,k. We propose a test to test the null hypothesis H₀:θ₁=…=θ_k against the simple ordered alternative H₁:θ₁≦…≦θ_k, with at least one strict inequality, using the data X_i,j, i=1,…k; j=1,…,n₁. Two methods to compute the critical points of the proposed test have been demonstrated by talking k two parameter exponential distributions. The test procedure also allows us to construct simultaneous one sided confidence intervals (SOCIs) for the ordered pairwise ratios θ_j/θ_i, 1≦i<j≦k. Statistical simulation revealed that: 9i) actual sizes of the critical points are almost conservative and (ii) power of the proposed test relative to some existing tests is higher.     

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.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

Characterization of generalized markov-polya and generalized polya-eggenberger distributions

A discrete model is considered where the original observation is subjected to partial destruction according to the Generalized Markov-Polya (GMP) damage model. A characterization of the Generalized Polya-Eggenberger distribution (GPED) is given in the context of the Rao-Rubin condition. More specifically, if the probability that an observation n of a non-negative integer valued r.v.X is reduced to an integer k during a damage, process is given by the GMPD, and if the resulting r.v.Y is such thatrit satisfies the RR-conditlon, then X has a GPED. Secondly, if N = A + B, where B is the missing part and A is the recorded part such that the conditional distribution P(A= x|N=n) is the GMPD, then the r.v.'s A and B are independent if, and only if, N has a GPED. Several other characterizations are also given for these two distributions. The results of Rao-Rubin ‘1964’, Patil-Ratnaparkhi (1977) and Consul (1975) follow as special cases.     

Characterizations of Competing Risks in Terms of Independent-Risks Proxy Models

In competing risks a failure time T and a cause C , one of p possible, are observed. A traditional representation is via a vector ( T ₁, ..., T_p ) of latent failure times such that T = min( T ₁, ..., T_p ); C is defined by T = T_C in the basic situation of failure from a single cause. There are several results in the literature to the effect that a joint distribution for ( T ₁, ..., T_p ), in which the T_j are independent, can always be constructed to yield any given bivariate distribution for ( C , T ). For this reason the prevailing wisdom is that independence cannot be assessed from competing risks data, not even with arbitrarily large sample sizes (e.g. Prentice et al. , 1978). A result was given by Crowder (1996) which shows that, under certain circumstances, independence can be assessed. The various results will be drawn together and a complete characterization can now be given in terms of independent-risks proxy models.     

Extended tables of the distribution of friedman’ s s-statistic in the two-way layout

Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.     

Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models

We use Owen's (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z ₁, ..., z _n which is observed directly from a distribution F ( z ). The other one is x ₁, ..., x _m which is observed indirectly from F ( z ), where the x _i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H ₀: p ( x | z ) = p ( x | z ; θ ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi-squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F ( z ) when H ₀ is true. The connection between upgraded mixture models and general estimating equations is pointed out.     

NON-EXISTENCE OF AN EXACT β-CONFIDENCE INTERVAL FOR A SCALE PARAMETER*

Abstract There are given k (≥22) independent distributions with c.d.f.'s F(x;θ_j) indexed by a scale parameter θ_j, j = 1,…, k. Let θ_[i] (i = 1,…, k) denote the ith smallest one of θ₁,…, θ_k. In this paper we wish to show that, under some regularity conditions, there does not exist an exact β-level (0≤β1) confidence interval for the ith smallest scale parameter θ_i based on k independent samples. Since the log transformation method may not yield the desired results for the scale parameter problem, we will treat the scale parameter case directly without transformation. Application is considered for normal variances. Two conservative one-sided confidence intervals for the ith smallest normal variance and the percentage points needed to actually apply the intervals are provided.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

Ordering of Sequentially Sampled Exponential Experiments

Let X ₁, X ₂, ... be a sequence of i.i.d. random variables, X _i∼ F _θ, θ∈Θ. Let N ₁ and N ₂ be two stopping rules. For a class of exponential families { F _θ: θ∈Θ} we show that the experiment Y ₁ = ( X ₁, ..., X _N1) carries more statistical information than Y ₂ = ( X ₁, ..., x _N2) only if N ₁ is stochastically larger then N ₂     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .