There has been many studies in the area of the ‘improved’ estimation follow- ing the seminal work of Bancroft (1944) and later Han and Bancroft (1968). They developed the **preliminary** **test** estimator that uses uncertain non-sample prior in- formation (not in term of prior distribution), in addition to the sample information. Stein (1956) introduced the Stein-rule (**shrinkage**) estimator for multivariate nor- mal population that dominates the usual maximum likelihood estimator under the squared error loss criterion. In a series of papers Saleh and Sen (1978, 1985) ex- plored the **preliminary** **test** approach to Stein-rule estimation. Many authors have contributed to this area, notably Sclove et al. (1972), Judge and Bock (1978), Stein (1981), Maatta and Casella (1990), and Khan (1998), to mention a few. Ahmed and Saleh (1989) provided comparison of several improved **estimators** for two mul- tivariate normal populations with a common covariance matrix. Later Khan and Saleh (1995, 1997) investigated the problem for a family of Student-t populations. However, the relative performance of the **preliminary** **test** and **shrinkage** **estimators** of the slope parameter of linear regression equation has not been investigated.

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There has been many studies in the area of the ‘improved’ estimation follow- ing the seminal work of Bancroft (1944) and later Han and Bancroft (1968). They developed the **preliminary** **test** estimator that uses uncertain non-sample prior in- formation (not in the form of prior distributions), in addition to the sample infor- mation. Stein (1956) introduced the Stein-rule (shinkage) estimator for multivariate normal population that dominates the usual maximum likelihood **estimators** under the square error loss function. In a series of papers Saleh and Sen (1978, 1985) explored the **preliminary** **test** approach to Stein-rule estimation. Many authors have contributed to this area, notably Sclove et al. (1972), Judge and Bock (1978), Stein (1981), Maatta and Casella (1990), and Khan (1998), to mention a few. Ahmed and Saleh (1989) provided comparison of several improved **estimators** for two mul- tivariate normal populations with a common covariance matrix. Later Khan and Saleh (1995, 1997) investigated the problem for a family of Student-t populations. However, the relative performance of the **preliminary** **test** and **shrinkage** **estimators** of the univariate normal mean has not been investigated.

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This paper considers alternative **estimators** of the intercept parameter of the linear regression model with normal error when uncertain non-sample prior in- formation about the value of the slope parameter is available. The maximum likelihood, restricted, **preliminary** **test** and **shrinkage** **estimators** are consid- ered. Based on their quadratic biases and mean square errors the relative performances of the **estimators** are investigated. Both analytical and graph- ical methods are explored. None of the **estimators** is found to be uniformly dominating the others. However, if the non-sample prior information regard- ing the value of the slope is not too far from its true value, the **shrinkage** estimator of the intercept parameter dominates the rest of the **estimators**.

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very close to the true value of the parameter α (i.e.; 𝜁 is approximate close to one), the proposed **estimators** perform better than the Bayes estimator. If one has no confidence in the guessed value then proposed **preliminary** **test** shrunken **estimators** can be suggested. We can safely use the proposed **estimators** for small sample size at usual level of significance Δ and moderate value of shrunken weight factor Ψ(. ) .

for the inclusion of non-sample information in conjunction with the sample information is to improve the statistical properties of the **estimators**. Recently, Khan and Saleh (2001) have used the coefficient of distrust 0 ≤ d ≤ 1, a measure of degree of lack of trust on the null hypothesis, in the estimation of parameters. This coefficient of distrust reflects on the reliability of the prior information. In particular, d = 0 implies no distrust on the null hypothesis, d = 0.5 implies equal distrust and trust in the null hypothesis, and d = 1 implies total distrust in the null hypothesis. The selection of an appropriate value of d is subjective, and individual researcher would determine a specific value of d based on expert knowledge and, or, practical experiences. Combining the sample and non-sample information as well as the coefficient of distrust we propose the **shrinkage** restricted esti- mator (SRE) and **shrinkage** **preliminary** **test** estimator (SPTE), as a generalization of the restricted and **preliminary** **test** **estimators**, for the slope parameters of two suspected parallel linear regression models. Khan (2003) discussed different **estimators** of the slope under the suspected parallelism problem. However, it does not deal with the **shrinkage** **preliminary** **test** estimator.

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The most important challenge of using these inks is that they must be flexible enough to withstand the **shrinkage** process. If not, the ink could crack or break when the substrate shrinks (Genuario, 2004). In addition to flexibility, inks need to dry quickly under low heat from the dryers, have improved adhesion, and be heat resistant. Because heat is involved in the process, these inks need to be formulated with pigments that won't change color or fade when heat is applied (Sharon, 2004). The printer also has to consider other important factors such as the container material, bottle-blocking resistance, scuff resistance and a high level of opacity of the white ink (George, 2004).

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Objective: Eulophia ochreata Lindl (Orchidaceae) is an important traditional medicinal plant,but unknown to most of the people and widely used in treatment of variety of disease by tribes of taluka maval. Plant is unexplored scientifically yet for their identification and use. Therefore, the current study was carried out to perform detailed pharmacognostical and phytochemical analysis of E.ochreata Lindl Method: Systematic pharmacognostic evaluation of tubers of E.ochreata Lindl has been carried out with respect to macroscopy, microscopy, and followed by **preliminary** phytochemical investigation and estimation of various chemical standards.Result: Tubers are fibrous, woody and perennial with numerous rootlets. Microscopic study shows the presence of cork layer with alternate lignified cells pink in colour, cortex, layer circular, parenchymatus ground tissue, fibro vascular bundles, muciligenous cells and crystals. Qualitative phytochemical **test** revealed the presence of most important that is alkaloids then saponin glycosides, mucilage, tannins, flavonoids, steroids and triterpenoid. Conclusions: Morphological, and phyto-chemical parameters studied in this paper may be proposed to establish the authenticity of plant Eulophia ochreata, and most important differentiating characteristic is yellow flowers which can probably, helps to differentiate the crude drug from its other species with respect to quality, purity and identification and revealing its important constituents present in plant.

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Recently, in several seminal empirical papers Professor Badi Baltagi and associates have focused on investigating which estimator is the “best” when the specified model has to be used for forecast purposes. Baltagi and Grif- fin (1997), Baltagi, Griﬃn and Xiong (2000), Baltagi, Bresson, Griﬃn and Pirotte (2003) and Baltagi, Bresson and Pirotte (2002) apply dynamic panel specifications to industrial level data and find that the predictive ability of homogeneous **estimators** outperforms the predictive ability of heterogeneous and Bayesian **estimators** over any forecast horizon. Amongst the homoge- neous **estimators**, GLS and within-2SLS emerge as the best **estimators** for forecasting purposes, especially when we forecast over a long time span. The superiority of the homogeneous **estimators** can sound quite reasonable when the panel is short, and when the degree of heterogeneity across units is lim- ited, but it is rather puzzling when the time length T of the panel is large or when the degree of heterogeneity is high. This genuine empirical finding is particularly interesting because the model where we impose homogene- ity is in general rejected by the data. A first interpretation of this apparent counter-intuitive empirical regularity is that a model that is “simple and par- simonious” oﬀers a better forecasting performance. However, using a diﬀerent dataset, Baltagi, Bresson and Pirotte (2004) find that Bayesian **estimators** provide the best forecasting performance.

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The SAR model was first introduced by Whittle (1954) who showed that the least squares estimator of β is inconsistent while Ord (1975) showed that the MLEs for such parameters are consistent **estimators**. Statistical inference of the SAR model appears mostly in economics literature. For example, Bell and Bockstael (2000) used the generalized-moments estimation technique for the SAR model in the context of micro level spatially correlated data. Lee and Yu (2010) established the asymptotic properties of the quasi-maximum likelihood estimator in economic panel data with fixed effects and SAR errors. Su (2012) proposed generalized method of moments (GMM) **estimators** for a semiparametric SAR model and derived their limiting distri- butions. Su and Jin (2010) proposed a profile quasi-maximum likelihood estimation of a partially linear SAR model and showed that such **estimators** are consistent at the usual √ n rate of convergence. An overview of the statistical inference for the SAR model and its variants can be found in Anselin (1988), Cressie (1993), Wall (2004) and Kazar and Celik (2012)

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The paper is outlined as follows: In Sect. 2, some **preliminary** results are addressed. Section 3 includes the main result, where we give the conditions under which the proposed class of **shrinkage** **estimators** dominates the natural estimator under balance loss function, while the numerical performance analysis is investigated by a simulation study in Sect. 4. In Sect. 5, we use the air pollution dataset of USA cities to further demonstrate the superior performance of the **shrinkage** estimation. The paper is concluded in Sect. 6.

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function, and enriched for risk of schizophrenia, a neuro- developmental disorder with its highest incidence in young adults. We used quantitative multiparameter mapping (MPM) (31) to **test** these hypotheses on MRI data acquired from a sample of 297 healthy young people sampled from primary healthcare registers, stratified by age, and balanced for sex in the adolescent age range 14–24 y old, with ∼60 participants in each of five age- defined strata: 14–15 y old inclusive, 16–17 y old, 18–19 y old, 20–21 y old, and 22–24 y old (Methods and SI Appendix, Table S1). From the MRI data, we measured CT (millimeters) and MT [percentage units (PU)] at each of 308 cortical regions for all participants. We used linear models to estimate baseline CT and MT at 14 y old and age-related rates of change in the period from 14 to 24 y old (ΔCT and ΔMT) from data on participants of all ages at each regional node. We explored the relationships between these local cortical MRI markers and a few key metrics of complex network topology that have been widely used in prior neuroimaging and other neuroscience studies (review in ref. 32). We focused on the degree and closeness of each node—as measures of nodal “hubness”—and the community structure of the network—defined as a set of sparsely interconnected mod- ules; details of topological connectome analysis are in Methods and SI Appendix. We investigated the relationships between gene transcriptional profiles and colocalized MRI (CT and MT) and network topological phenotypes by multivariate analysis of MRI data on 297 adolescents and whole-genome gene expression maps of six adult human brains (postmortem) provided by the Allen Institute for Brain Science (AIBS) (33); details are in Methods and SI Appendix.

A major problem implementing portfolio optimization using sample estimations for 2 nd , 3 rd and 4 th co-moments is that of estimation error. Due to the curse of high dimensionality, the estimation error problem becomes far worse for the 3 rd and 4 th co-moments. Ledoit and Wolf appropriately related the use of the sample covariance matrix to be like “error maximization” because of the high estimation error compounded by the tendency of optimizers to hone in on stocks with the highest estimation errors in practice [13]. They suggest a **shrinkage** estimator for the covariance matrix using a constant correlation utility function (CRRA) approach and they demonstrate how their **shrinkage** estimator reduces tracking error in real data. Martellini and Ziemann extended Ledoit and Wolf’s methodology to the 3 rd and 4 th co-moments using the constant correlation utility function with a single statistical factor model approach [13, 14, 16]. They demonstrate that the portfolio optimization with improved **estimators** outperforms portfolios that use sample **estimators** for optimization. They also show that the single- factor approach out-performs the constant correlation without a statistical factor model in their sample data [16]. Boudt, Lu, and Peeters investigated a multifactor approach to the 3 rd and 4 th co-moments in portfolio optimization [4]. They revealed that portfolio optimization using higher co-moments with a multifactor approach outperformed their benchmark portfolio and

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Model averaging or model mixing **estimators** have received increased interest in recent years; see, e.g., Yang (2000, 2003, 2004), Magnus (2002), Leung and Barron (2006), and the references therein. [For a discussion of model averaging from a Bayesian perspective see Hoeting et al. (1999).] The main idea behind this class of **estimators** is that averaging **estimators** obtained from di¤erent models should have the potential to achieve better overall risk performance when compared to a strategy that only uses the estimator obtained from one model. As a consequence, the above mentioned literature concentrates on studying the risk properties of model averaging **estimators** and on associated oracle inequalities. In this paper we derive the …nite-sample as well as the asymptotic distribution (under …xed as well as under moving parameters) of the model averaging estimator studied in Leung and Barron (2006); for the sake of simplicity we concentrate on the special case when only two candidate models are considered. Not too surprisingly, it turns out that the …nite- sample distribution (after centering and scaling) depends on unknown parameters, and thus cannot be directly used for inferential purposes. As a consequence, one may be interested in **estimators** of this distribution, e.g., for purposes of conducting inference. We establish an impossibility result by showing that any estimator of the …nite-sample distribution of the model averaging estimator is necessarily “bad” in a sense made precise in Section 4. While we concentrate on Leung and Barron’s (2006) estimator (in the context of only two candidate models) as a prototypical example of a model averaging estimator in this paper, similar results will typically hold for other model averaging **estimators** (and more than two candidate models) as well.

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Skin grafts are widely used and any information regarding their characteristics is valuable. Our **preliminary** report reveals an expected increased **shrinkage** of FTSGs compared to STSGs and in a limited number of specimens, the shape of the skin graft seems to affect primary contraction of the STSGs. Although it is difficult to dramatically change the shape of skin grafts, if this feature is ultimately found to alter primary contraction, the results could possibly be applied in clinical practice.

In line with this, a numerical approach is developed in this study to simulate the **shrinkage** behavior of different thicknesses concrete in restrained ring specimens drying from the outer circumference. A fictitious temperature field was applied on concrete ring specimens in numerical analyses to simulate the mechanical effect of concrete **shrinkage** on rings under restrained condition. The fictitious temperature field is derived based on free **shrinkage** **test** of concrete prisms by considering that the free **shrinkage** strain of the prisms is caused by the fictitious temperature field applied on them which leads to the same value of contraction as **shrinkage** does. The stress developmentis analyzed to predict cracking age and position in both thin and thick concrete ring specimens subject to restrained **shrinkage**. It is expected that the experimental and numerical investigations presented here will help to better understand how the ring **test** works, stress development and crack initiation/propagation in restrained concrete rings under outer circumferential drying.

3.1 Mechanical sieve analysis and hydrometer **test** The sieve analysis and hydrometer **test** were conducted on the clay soil sample and the results are shown in Figure 1. Figure 1 shows that the soil is fine-grained as more than 30% of the soil fraction passes sieve No 200 (0.0075mm). The high value of the fine content indicates the soil might be clay or silt. Adopting AASHTO classification system, the soil falls within the range of A4 to A7 soils. Further classification reveals that it is an A-7–5 soil as the liquid limit and plasticity index are greater than 41 and 11, respectively. The soils in this group are classified as clay and with group index of zero and they are rated as poor subgrade material.

Interactive teaching materials that have been developed further validated by experts and tested in limited scale to a number of seventh grade junior high school students in Serang City. Based on the validation results, it is known that the ITMSA obtained a percentage of total score 85,30% from mathematics experts, 87,80% from educational experts, and 83,60% from multimedia experts. The results of limited trials to 30 junior high school students are obtained a percentage of total score 89,40%. The results of validity **test** and limited **test** can be seen more clearly in Figure 3 below.

The **shrinkage** estimator of the shape parameter has been considered by several authors (Singh and Bhatkulikar 1977, Pandey 1983, Pandey, et. al. 1989, Pandey and Singh 1993, and Singh and Shukla 2000). Estimator (5) is also studied for the shape parameter but in different contexts (Singh et. al. 2002). It may be noted here that other authors (e.g., Kambo et. al. 1990, 1992, Parkash et. al. 2008, and Al-Hemyari et. al. 2009, 2011) have tried to develop new **shrinkage** **estimators** of the form (5) for special populations by choosing different weight functions.

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It is advantageous to utilize a linear combination of the two sample means ponderated by the sample size of them if E X coincides with E Y , i.e., to use the restricted estimator. In many situations it is not clear if E X = E Y . In order to tackle this uncertainty we can perform a **preliminary** **test** and then choose between a restricted and an unrestricted estimator. This line of thought was first proposed by Bancroft (1944). For a wide study of **preliminary** **test** **estimators** in different statistical problems see Saleh (2006) and references therein.

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**Shrinkage** estimation is a method to improve a raw estimator in some sense, by combining it with other information. Although the **shrinkage** estimator is biased, it is well known that it has minimum quadratic risk compared to natural **estimators** (mostly the maximum likelihood estimator) ( Karamikabir, Afshari and Arashi, 2018). The **shrinkage** estimator, have evolved