I always try to work on both theoretical statistics (to develope useful
methods) and applied statistics

(to apply theoretical methods in real problems and to gain motivations
for enriching the statistical

theory). I am breaking my researchs according to the areas of my research
interests.

The numbers in the brackets refer to the numerical labels of my research
reports and

publications in the list attached to this summary.

**1.** **Goodness-of-Fit Statistics Based on Kernel Density Estimates**.

Bickel and Rosenblatt (1973, Ann. Statist.) introduced a goodness-

of-fit statistic based on kernel density estimate. They developed the

asymptotic distribution of the proposed statistic under both null

hypothesis and local function alternatives. The function local

alternatives are generalizations of Pitman alternatives. The problem

of choosing the "best" possible kernel was not studied by Bickel

and Rosenblatt. B.K. Ghosh and I studied this problem for a while

and we finally solved the problem and part of the results was

published in the Annals of Statistics [4]. The optimal choice of the

kernel is in terms of maximizing the local asymptotic power of the

test. The result is kind of surprising and unexpected. The

standard quadratic kernel ( the well-known Epanechnikov kernel in

density estimation) does not maximize the local asymptotic power.

In fact the simplest kernel, the uniform kernel, is the one which

maximizes the local asymptotic power of the Bickel-Rosenblatt test.

It implies that the moving histogram should be used in the kernel-

based goodness-of-statistic. The proof uses the Fourier

transformation of the target functional and then applies Parseval's

identity to the transformed functional. Standard variational

method using Gateaux differential and convexity argument then

used to obtain the optimal solution of the target functional. Beran

(1977, Ann. Statist.) proposed a goodness-of-fit test based on

Hellinger distance and the power of his test is also a decreasing

function of the target functional in our study. The resulting kernel

in [4] should also be a good choice for Beran's test as well. Using

Monte-Carlo method, Ghosh and I ([5]) also compared the power of

Bickel-Rosenblatt's test (for sample size n = 40) with those of

several well-known tests based on empirical distribution. In [10]

and [12] I studied nonparametric likelihood ratio and adaptive

tests and investigated the sampling distributions via Monte-Carlo

method. I found that the proposed likelihood ratio statistic is

closely related to the Bickel-Rosenblatt statictic. From the

simulation study I realized that the first order approximation (used

in Bickel and Rosenblatt (1973)) can be improved. The normal

approximation applied to the log-transformated statistic works

better than to the one without the log-transformation ([12] , [24]).

**2. Estimation of Discontinuous Density**.

Rosenblatt and Parzen proposed the well-known kernel estimator for
an

unknown continuous density function. Schuster (1985, Comm. in Statist.)

suggested a modification, the reflected version of the kernel

estimator, to rectify certain drawbacks of the kernel estimators

when the unknown density has some discontinuous points. The

optimal choice (in terms of minimizing the Integrated Mean Squares

Error (IMSE)) of the kernel for the Rosenblatt-Rosenblatt's

estimators is known to be the quadratic kernel obtained by

Epanchnikov (1969). But the kernel which minimizes the IMSE for

the folded kernel estimators was not before 1991. It is a rather

difficult problem. Ghosh and I worked on this problem for a period

of time. In fact, we didn't not know the reflected kernel estimator

proposed by Schuster when we submitted the work for publication.

We were looking at it totally from "folding" the ordinary kernel

estimator with respect to points of discontinuity. We found the

optimal kernel for this problem and it was included in [3] and [28]

along with some other related problems of rates of IMSE and

asymptotic relative efficiency. Some of the results in [3] were

treated in great details in [28]. Two defferent definitions of IMSE's

were used in the study and lead to some interesting results. Part of

the results also supplement earlier related results of van Eeden

(1985, Ann. Inst. Statist. Math.) and Cline and Hart (1990,

Statistics). The new optimal kernel has never been used in density

estimation and it appers somewhat unusual but the uniform (which

was described in the previous section in the Bickel-Rosenblatt's

goodness-of-fit test) and Epanechnikov (the optimal kernel in

estimating a continuous density) kernels can, in fact, be thought of

first and second order approximations to the new kernel. As an

application, the folded kernel estimator could be used in the Bickel-

Rosenblatt goodness-of-fit statistic since the uniform density, the

density under the null hypothesis, is discontinuous at the two

endpoints. In fact, it improves the normal approximation used by

Bickel and Rosenblatt (1973). Some of the results are included in

[24] and in a dissertation of one of my Ph.D. students at Lehigh.

**3. Semiparametric Modeling**.

A semiparametric model is a statistical model consisting of both parametric

and nonparametric components. Many practical problems in statistics
can be

modeled in terms of a semiparametric model with suitable choice of
both

components. We could also set up a semiparametric model such

that it includes a proposed parametric model with the freedom that

we don't have too specific about some of assumptions in the

proposed parametric model. In a sense we are hoping to get a more

"robust" model. But one of the consequences of "enlarging" the

parameter space is that it is rather difficult to evaluate the

efficiencies of statistical procedures. The problem of evaluating the

efficiency of an estimator in a purely parametric model or a purely

nonparametric model has been well-developed. But the efficiency

problem for a mixed parametric-nonparametric model is relatively

new. Under the supervision of Professor W.J. Hall and Professor

Jon A. Wellner , I worked on semiparametric problems for my Ph.D.

dissertation. Begun, Hall, Wellner and I ([16]) developed some

results in evaluating the efficiency of estimators in general

semiparametric models. The work was motivated by the pioneer

works of Stein (1956, Proc. Third Berkeley Symp.), Bickel (1982,

Ann. Statist), LeCam (1972, Proc. Sixth Berkeley Symp.) and others.

The work leads to information-type of bounds for estimation of the

parameters of semiparametric models and it also leads to methods

of constructing effective estimators based on "effective scores".

Some methods (motivated by works of Bickel (1982) and the general

score function method by C.R. Rao) were proposed in [33] and [34]

and have applied to some specific problems in [14] and [31]. The

problem of nonparametric estimation of the cumulative distribution

function under random truncation was treated as a probelm in

semiparametric model in [7]. Tsai and I established a convolution-

type representation theorem and a local asymptotic minimax

theorem to show the asymptotic efficiency of the nonparametric

estimator proposed by Lyndell-Bell (1971). The results are similar

to those results for the complete data case due to Beran (1977,

Ann. Statist.) and for the censored data case due to Wellner (1982,

Ann. Statist.). Both likelihood and functional (which was also used

in [14]) approaches are considered in the study. An attempt to

generalize the semiparametric theory to non-iid case was

considered for autoregressive models. Using the "projected score",
I

showed ([13]) that it is possible to construct adaptive estimators
of

the autoregressive coefficients. In a different direction, Hall and
I

([8]) developed bounds on the exponential rates of consistent

estimates in semiparametric models in the sense of large deviations.

The bounds can be treated as lower bounds for the asymptotic

effective standard deviation of such estimates. A directional

method was used in the study. The work is an extention of

Bahadur's results of parametric models. Semiparametric modeling

is very useful in statistics since many practical problems in

statistics are natually of semiparametric type. Most of the

statistical problems in semiparametric models are by now solved

and known. The book by Bickel, Klassen, Ritov and Wellner (1993)

includes general theory of semiparametric models and detailed

studies of many interesting models. The Fifth Lukacs Symposium

will be held in Browling Green, Ohio on March 24 and 25 of 1995.

The theme will be Statistical Inference in Semiparametric Models. I

was invited to give a talk in the Symposium.

**4. Tests for Lack-of-Fit in Linear Models**.

In linear models, we often want to verify the proposed model. When

replication is not available, the classical pure-error test no longer
works.

Such situations arise routinely under experimentalists who are simply

unaware of the need for replications. They may also arise due to

the very nature of an experiment where replications are impossible.

Many test procedures have been suggested under such

circumstances (Chow (1960), Shillington (1979), Utts (1982), and

Neill and Johnson (1985). The drawback is that the notion of

"cluster" or "near replicates" is sometimes highly subjective and,

more seriously, the exact procedures sacrifice certain information

or the approximate procedures do not leave any clue to the true p-

value and power of the tests. Ghosh and I ([19], [26]) propose a

simple class of exact tests which are based on subdivisions of the

observed points. We provide justifications for such a class from

different viewpoints. In particular, we (a) show that Utts' test and,

when replications are available, the classical test can be treated
as

special cases of our test, (b) demostrate by formal calculations that

our test can be more powerful than others under suitable choices of

subdivision, and (c) argue that our test should effectively dispense

with the need for the existing tests based on "near replicates".

Power comparisons and extensive numerical studies are also

included in our work. The paper is almost finished and is ready to

be submitted for publication. Some related generalizations will be

studied in near future.

**5. Fixed-Width Confidence Interval for Bernoulli Probability**.

Almost everyday we see statistical reports on newspapers,

magazines, journals and many books. Very often reports are

developed on the basis of statistical samplings. The yes-no type

questions are typically asked in a survey and inferences can then

be made from these Bernoulli variables. To reduce the cost of

sampling and to save time, we often want to conduct the

experiment in terms of sequential method. Ghosh and I ([20])

developed a sequential rule for fixed-width confidence intervals of

the Bernoulli probability. We are following the pioneer work by

Chow and Robins (1965, Ann. Math. Statist.) and especially making

uses of the special structures of binomial distribution in the

derivation of the confidence interval. Comparing to some known

methods, we found that the proposed fixed-width confidence

interval seems to have more reliable control on the coverage

probability and smaller expected sample size. Some interesting

links (in terms of first order approximations) to Chow and Robbin's

results are also developed. We intent (a) to investigate 2nd order

asymptotic properties of the proposed rule and (b) to use bootstrap

method to obtain useful approximations of the related distributions

and momemts. Part of the results was presented at the IMS

Sequential Workshop at UNC on June 18-19 of 1994. A draft of our

work is available.

**6. Approximate Entropy and Applications**.

For a while I was involved in analyzing medical data from hospitals
in

Lehigh Valley Area. One of the problems was to try to use the given
medial

data to detect the onset of an apprent life threatening event. The

analysis is to prevent sudden infant death syndrome. The data is

typically collected in terms of ECG, EEG and respiratory data.

Pincus and I ([2]) propose to use entropy as a chaos-related

patternness measure and to estimate it by a quantity called

approximate entropy. We use approximate entropy to quantify the

regularity (complexity) in data and to provide an information-

theoretic quantity for time series data. One of the difficult problems

is to obtain the the sampling distribution of the proposed

approximate entropy, especially in two-sample setup. Some results

were developed in the paper and we especially introdued the idea of

randomized approximate entropy in a spirit of sample reusing

method. The randomized approximate entropy derives an empirical

significant probability that two processes differ on the basis of one

data set from each process. As an application, it also provides a

test for the hypothesis that an underlying time-series is generated

by iid variables. It is always exciting to study real cases but it
is

also true that most of the real cases are not easy to handle. A

project at St. Lukes Hospital was proposed several months ago to

use the approximate entropy in a real case study and we hope to

see if it really works.

**7. Bounds for Measures of Associations with Given Marginals &**
**Related Distributional Problems**.

W.C. Shih of Merck Sharp Research Labs asked me a question if it is
possible

to have better bounds for the correlation coefficient when the marginal

distributions are specified. He raised the question because of the

needs in medical related statistics. Shih and I ([1], [27]) worked
out

the problem and provided a answer to the question. In [27], we

derived the lower and upper bounds using an argument on the

basis of Neyman-Pearson Lemma whcih is different from the

method used in Cambanis, Simon and Stout (1976, Zeit. Wahr.). In

fact, we came up with useful representations of the lower and upper

bounds in terms of quantile functions. Numerial examples are

included in [1]. During the Fall semester of 1993 I was visiting

Rutgers University, I had a great chance to discuss this problem

with Professor Kemperman. Kemperman gave me some useful

comments and I was able to redrive the bounds using completely

different argument. Hoeffding's formula for correlation and

Frechet's dirtributions were used in the new derivation and the

work is included in [25] and [21]. Part of it was presented at the

International Joint Statistical Conference in the Winter of 1993 in

Taiwan. In practice, the theoretical marginal distributions are

unknown, I proposed to use the empirical versions to estimate the

theoretical bounds. This empirical substitution leads to an

interesting link to the Hardy-Littlewood inequalities and

probabilistic version of Hardy-Littlewood inequalities. Related

distributional problems are treated in [22]. Paper [21] is pretty

much done and will be submitted for publication. Some interesting

distributional results are already obtained in a draft of [22]. Some

rather difficult and deep questions are involved in the distribuional

studies of the empirical lower and upper bounds.

**8. Others**. Other miscellaneous works I produced are permutation

tests for comparing two regressions ([6]), martingales induced by

Markov chains ([9]}, asymptotic efficient estimation of the interval

failure rates ([15]}. I also have some on going works in different

directions which I will not mention here.