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Open Access
MODIFIED RATIO AND PRODUCT TYPE ESTIMATORS UNDER ADAPTIVE CLUSTER SAMPLING
(Sher-e-Kashmir University of Agricultural Sciences & Technology of Jammu, 2023-10-27) Bhat, Arshid Ahmad; Sharma, Manish Kr.
In sample surveys there are the cases when the population is rare and clustered. The sampling method used to obtain the sample in these cases is known as Adaptive Cluster Sampling. Adaptive cluster sampling (ACS) refers to the design in which an initial unit is selected by probability sampling procedure namely Simple Random Sampling without replacement and whenever the variable of interest of a selected unit satisfies a given pre-defined condition, additional units in the neighborhood of that unit are added to form the sample. The present investigation entitled “Modified Ratio and Product Type Estimators under Adaptive Cluster Sampling” has been undertaken with the objectives to develop some generalized ratio and product type estimators for estimating the population mean under ACS. The additional information used in order to improve the efficiency of the ratio and product type estimators is known as auxiliary information. The large sample property of mean square error (MSE) for the proposed estimators have been derived up to first order of approximation and are compared with the conventional and existing estimators both theoretically as well as empirically. The generalized class of ratio type estimators (t_(p,q)^ri) under ACS has been developed on the basis of linear combinations of mean, variance, skewness, kurtosis and correlation both at unit and network level at different values of i, p,q where i = 1, 2, 3, 4, p and q are the constants to be determined in order to improve the efficiency of the proposed estimators. All the cases developed with the different combinations of i, p and q are found efficient than the existing estimators taken in the literature. Among the developed cases the different values of p and q have been determined and the estimators namely 〖 t〗_2,2^r1,t_5,10^r2,t_8,9^r3 and t_2,2^r4 are found most efficient theoretically and empirically than the existing estimators as proposed by Cochran (1940), Sisodia and Dwivedi (1981), Upadhyay and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2003), Dryver and Chao (2007), Yan and Tian (2010), Chutiman (2013), Yadav et al. (2016). The generalized class of product type estimators (t_(k,k')^pi), (t^p4) and (t_(k,k')^p5) under ACS have also been developed on the basis of linear combinations of mean, variance, skewness, kurtosis and correlation both at unit and network level at different values of i, k, k ' where i = 1, 2, 3,k and k ' are the constants to be determined in order to improve the efficiency of the proposed estimators. All the cases developed with the different combinatioIn sample surveys there are the cases when the population is rare and clustered. The sampling method used to obtain the sample in these cases is known as Adaptive Cluster Sampling. Adaptive cluster sampling (ACS) refers to the design in which an initial unit is selected by probability sampling procedure namely Simple Random Sampling without replacement and whenever the variable of interest of a selected unit satisfies a given pre-defined condition, additional units in the neighborhood of that unit are added to form the sample. The present investigation entitled “Modified Ratio and Product Type Estimators under Adaptive Cluster Sampling” has been undertaken with the objectives to develop some generalized ratio and product type estimators for estimating the population mean under ACS. The additional information used in order to improve the efficiency of the ratio and product type estimators is known as auxiliary information. The large sample property of mean square error (MSE) for the proposed estimators have been derived up to first order of approximation and are compared with the conventional and existing estimators both theoretically as well as empirically. The generalized class of ratio type estimators (t_(p,q)^ri) under ACS has been developed on the basis of linear combinations of mean, variance, skewness, kurtosis and correlation both at unit and network level at different values of i, p,q where i = 1, 2, 3, 4, p and q are the constants to be determined in order to improve the efficiency of the proposed estimators. All the cases developed with the different combinations of i, p and q are found efficient than the existing estimators taken in the literature. Among the developed cases the different values of p and q have been determined and the estimators namely 〖 t〗_2,2^r1,t_5,10^r2,t_8,9^r3 and t_2,2^r4 are found most efficient theoretically and empirically than the existing estimators as proposed by Cochran (1940), Sisodia and Dwivedi (1981), Upadhyay and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2003), Dryver and Chao (2007), Yan and Tian (2010), Chutiman (2013), Yadav et al. (2016). The generalized class of product type estimators (t_(k,k')^pi), (t^p4) and (t_(k,k')^p5) under ACS have also been developed on the basis of linear combinations of mean, variance, skewness, kurtosis and correlation both at unit and network level at different values of i, k, k ' where i = 1, 2, 3,k and k ' are the constants to be determined in order to improve the efficiency of the proposed estimators. All the cases developed with the different combinations of i, k and k' are found efficient than the existing estimators. Among the developed cases the proposed values of k and k ' have been determined from those estimators and the estimators namely〖 t〗_4,8^p1,〖 t〗_(-1,-2)^p2,t_(-1,-2)^p3, t^p4 , t_(1,-1)^p5 and are found the most efficient theoretically and empirically than the estimators proposed by Robson (1957), Bahl and Tuteja (1991), Shahzad and Hanif (2016), Panda and Samantary (2018) and Hussain et al. (2021). Thus the proposed generalized ratio and product type estimators under ACS are superior over existing estimators theoretically as well as empirically. ns of i, k and k' are found efficient than the existing estimators. Among the developed cases the proposed values of k and k ' have been determined from those estimators and the estimators namely〖 t〗_4,8^p1,〖 t〗_(-1,-2)^p2,t_(-1,-2)^p3, t^p4 , t_(1,-1)^p5 and are found the most efficient theoretically and empirically than the estimators proposed by Robson (1957), Bahl and Tuteja (1991), Shahzad and Hanif (2016), Panda and Samantary (2018) and Hussain et al. (2021). Thus the proposed generalized ratio and product type estimators under ACS are superior over existing estimators theoretically as well as empirically.
Open Access
Scenario and Statistical Modeling of Oilseed Production for Jammu and Kashmir
(Sher-e-Kashmir University of Agricultural Sciences & Technology of Jammu, 2023-10-25) Singh, ChibPrerna Ran; Rizvi, S.E.H.
It is well known that India has 20.8% of the total area globally under oilseeds and accounts for 10% of global production having fourth largest position and hence oilseeds occupy an important position in the country’s economyin terms of area and production. As a result, it is critical to have an idea of future production and productivity. Thus, the present investigation entitled “Scenario and Modeling of Oilseeds Production for Jammu and Kashmir” has beenconducted with the objectives to study the aspects relating to behaviour of time series data on production and productivity of oilseeds crops, selection of an appropriate model, it’s fitting, testing and forecasting future values using time series modelling and Artificial Neural Network (ANN) models. For this purpose, secondary data were obtained for UT of Jammu & Kashmir as well at national level for the period of 1984 to 2022. Then the data were analysedfor obtaining the trend for the period using models like linear, quadratic, cubic, exponential, compound, power etc.Descriptive statistics wereobtained on entire data as well as on decadal basis to have information scenario of oilseeds production.The best models were chosen based on minimum Root Mean Square Error (RMSE).Time series models were obtained for oilseedsproduction and productivity at national as well as Jammu and Kashmir levels. Based on minimum RMSE, at national level the Logarithmic model for area, Linear model for production as well as productivity were found to be of best fit. Similarly, at Jammu and Kashmir level for area, productionand productivity,Cubic, Cubic and Quadratic models, respectively, were found to be of best fit.Instability index for area, production, and productivity of oilseeds at national level and at Jammu & Kashmir level indicated that instability in area, production, and productivity of oilseeds in India were of low level whereas in UT of Jammu and Kashmir it was of medium level. Growthmodels such asGompertz model, Monomolecular and Sigmoid were fitted for National and Jammu & Kashmir levels so as to observe growth in oilseeds production. Based on exponential model, overall percentage compound growth rates for area in Jammu and Kashmir during study period was negative (-0.12%) wheras as at national level it was 7.02%. For production atJammu and Kashmir levelit showedgradual growth with 0.08% and somewhat satisfactory in case of productivity with 1.01% CGR value.For production at all India level it showed remarkable growth with 2.53% and better in case of productivity with 1.81% CGR value.Using RStudio Estimated parameters for National of Gompertz, S-curve.This indicates increasing growth pattern over the timeseries ARIMAmodel best suited for production are ARIMA (2,0,1) forUT of Jammu and Kashmir level. For Artificial Neural network (ANN), the data were normalised and then the normalised data were further divided into two parts train data and test data. For the accuracy, data here were divided into (80:20) where 80% is train data sand 20% is test data, its MSE value is 0.16. Comparative to ANN, time series model is found to be the best fit.
Open Access
Some Improved Ratio and Regression Type Estimators in Case of Non-Response
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-11) Wani, Zakir Hussain; Rizvi, S.E.H.
In sample surveysgenerallyit is not possible that sample will represent population truly, which results in two types of errors viz. (i) sampling error, and (ii) non-sampling error.Non-response is a type ofnon-sampling error. The present investigation entitled“Some Improved Ratio and Regression Type Estimators in case of Non-Response” has been undertaken with the objectives to develop some improved types of separateand combined ratio as well asregression estimatorsfor estimating the population mean in presence of non-response, using auxiliary information, for four cases. The expressions for bias and mean square error (MSE) of proposed estimatorshave been derived up to first order of approximation. Theoretical comparisons of proposed estimators with the usual unbiased estimator by Hansen and Hurwitz (1946) and other existing estimators have been made. The theoretical results are supported by numerical data that demonstrate the superiority of the proposed estimators over existing estimators in terms of mean square errors. For Case I (non-response occurs on both the study variable (Y) and the auxiliary variable (X), and the population mean X ̅ is known) the separate andcombinedratio estimators t_1^(*(i))andt_2^(*(i)),i=1 to 10,respectively,separate and combined regression estimatorst_3^(*(i)) andt_4^(*(i)),i=1 to 5,respectively,have been proposed. ForCase II (non-response occurs on Y only, information on X is obtained from all the sample units and X ̅ is known) the separate and combined ratio estimatorst_1^('(i)) and t_2^('(i)),i=1 to 10, respectively,separate and combined regression estimatorst_3^('(i)) andt_4^('(i)),i=1 to 5,respectively,have been proposed. For Case III (non-response occurs on both Y and X, but X ̅ is unknown) the separate and combined ratio estimatorst_5^* andt_6^*, separate and combined regression estimatorst_7^(*(i)) andt_8^(*(i)),i=1 to 12, respectively,have been proposed.For Case IV(non-response occurs on Y, information on X is obtained from all the sample units, but the Mean X ̅ is unknown) the separate and combined ratio estimators t_5^'andt_6^', separate and combined regression estimatorst_7^('(i)) and t_8^('(i)),i=1 to 12,respectively, have been proposed. Based on percent relative efficiency, all the proposed estimators are found to be more efficient than the Hansen and Hurwitz (1946) and other existing estimators like, classical separate and combined ratio as well as regression estimators, estimators proposed by Onyekaet al. (2019)and other researchersin respective cases.
Open Access
Construction of Strata Boundaries under Ranked Set Sampling
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-11) Rather, Khalid Ul Islam; Rizvi, S.E.H.
The principal reason for stratification in the design of sample surveys is to reduce the sample variance of the estimates, so as to gain a higher degree of relative precision. The problem of determining optimum strata boundaries (OSB) on the basis of study variable was pioneered by Dalenius (1950), who treated study variable itself as stratification variable and used an approach which provided exact solution. Researchers like Dalenius and Gurney (1951), Singh (1971), considered the problem of optimum stratification using auxiliary variable as stratification variable, since the probability density function of the study variable is in general unknown prior to study. In these studies simple random sampling was considered as method of selection of sampling units in individual strata. It is well known that ranked set sampling (RSS) introduced by McIntyre (1952) provides an increased precision as compared to simple random sampling. Thus, the RSS method if used for selection of sampling units in individual strata, the relative efficiency is expected to be increased. Therefore, the present investigation has been undertaken to deal with the problem of construction of strata when ranked set sampling is used. In this regard, various allocation procedures viz. proportional, optimum as well as equal allocations have been considered and accordingly the methods of construction of strata have been developed through minimization of variance under ranked set sampling. The method of selecting the best boundaries that makes strata within them self-homogeneous as far as possible is known as optimum stratification. If the frequency distribution of the study variable is known, the stratification points could be obtained by cutting the range of the distribution. And, if unknown the points of stratification may be approximated on the basis of auxiliary variable. In the present study, theories have been developed for optimum stratification using one auxiliary variable as stratification variable. Minimal equations giving approximate optimum strata boundaries (AOSB) have been obtained for stratified ranked set sampling (SRSS) under different methods of allocations, by minimizing the variance of the sample estimates. For these cases, rule and rule , where the function takes different forms for different allocations, have been developed. Under the classical optimization technique the empirical studies on uniform, right triangular, exponential and normal distribution have been made which resulted gain in efficiency. On comparison of the efficiency, it has been observed that the proposed methods for SRSS under various methods of allocations performed better than unstratified RSS.
Open Access
Comparative Study of Ranked Set Sampling Methods under Skewed and Unskewed Distributions
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-10) Rashid, Imran; Jeelani Bhat, M. Iqbal
Ranked set sampling (RSS) isa special cost-effective sampling method used in situations where judgment of units can be done easily without any difficulty. The estimates based on RSS are more efficient in comparison to the traditional unrestricted sampling procedures and in recent past its modified versions have come into prominence due to their remarkable gain in efficiency inforest management research works. Inview of this the present study entitled “COMPARATIVE STUDY OF RANKED SET SAMPLING METHODS UNDER SKEWED AND UNSKEWED DISTRIBUTIONS”was carried out on skewed and unskewed distributions to evaluate the different modified versions of RSS in terms of efficiency. In order to achieve stipulated objectives simulated data wasgenerated in RStudio(version 4.1.2 -2021) utilizing Skewed and Unskewed probability distributions like gamma, exponential, uniform and normal distribution.Accordingly a ranked set sample of size150, 300, 450, 600, 750, 900, 1050, 1200 with a set size of 3,6,9,12,15,18,21,24utilizing a constant cycle (r) 50 respectively were drawn from the simulated data through different modified methods of RSS like : Extreme ranked set sampling (ERSS), Median ranked set sampling (MRSS), Percentile rank set sampling (PRSS), Balanced grouped ranked set sampling (BGRSS), Double rank set sampling (DRSS) and Truncation based rank set sampling (TBRSS)by means oflibrary( RSSampling) of R Studio. From the results it was observed that modified versions of RSS performed better in unskewed distributions in comparison to skewed distribution in terms of efficiency, asthese samples are based on modified versions of RSS which are more regularly spaced and induces stratification at sample level which involves the gain in efficiency. Also, from this study it was found thatefficiency of allmodified RSS methods increases as the sample size increases.Based on empirical investigation through simulated data, it was observed that across the modified RSS methods, Truncation based rank set sampling (TBRSS) performed better in comparison to its counterparts in terms of efficiency.Goodness of fit results revealed that the value of AIC & BIC decreased as the set size across the modified RSS methods increases, indicating that less amount of information is lostas set size increases. Finally, it is concluded that RSS has itspractical implications, where R packages facilitates a lot in implementation of modified RSS methods which are very informative and applicable to sample surveys.
Open Access
Statistical Model for the Estimation and Projection of Agriculture Share in India’s Gross Domestic Product
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-10) Sharma, Arayan; Sharma, Manish Kr.
An investigation titled “Statistical Models for the Estimation and Projection of Agriculture Share in India’s Gross Domestic Product” was conducted with the objectives: To analyse the Gross Domestic Product of India through time series models and to assess statistically the estimation and projection of agricultural components in Gross Domestic Product. India is an agricultural-based economy with more than 52.6 percent of land area that is considered arable. The Gross Domestic Product (GDP) of Agriculture and allied sectors has been declining since 1991 and is at 18.8 per cent in the present scenario. The secondary data of the GDP has been used for this study. The data on time series pertains to GDP and individual GDP components that are Agriculture, Livestock, Food processing industries, and Forestry, been collected from different reports published by different Government Offices like Economic Survey, NitiAayog Report, Food and Agriculture Organization Report and MOSPI. The estimation and projection of models have been done by using the techniques through ARIMA, Baysian, Correlation, Regression, and Stepwise regression. The results of the studyrevealed a huge fluctuation in the economy in the past two decades within the last seven decades. The ADF test for the stationarity of the data was tested which failed so to handle the problem differences 1,2 and 3 were applied. The 2nd degree of differences was selected as it showed the least deviation and was significant (ADF test) for the time series model. The best-proposed model was ARIMA (1,2,1). The forecasted value of GDP for 2029-30 is ₹376.07 lakh crores with a lower CL (0.95) value of ₹326.14 lakh crores and upper CL (0.95) value of ₹426.01 lakh crore. Similarly, the forecasted values of GDP for 2039-40 and 2046-47 are ₹543.71 lakh crores and ₹686.87 lakh crores with a lower CL (0.95) value of Rs.428.51 lakh crores and ₹509.81 lakh crores and upper CL (0.95) value of Rs.658.92 lakh crores and Rs.863.92 lakh crores respectively. Further, the data pertains to GDP (Endogenous) and Agriculture, FPI, Food processing and Livestock (Exogenous) were found to be reliable and Normal. The main component significantly were Agriculture and FPI for enhancing the FPI sector for boosting the Indian GDP in addition to this Bayesian estimation the maximum posterior probability for increasing the GDP will be through agriculture and FPI. As per the assessment of GDP, the interaction between Agriculture with Forestry, FPI and Livestock are 70.08 perc ent, 67.66 per cent and 65.25 per cent respectively. In the coming years, FPI and Forestry will contribute more to the GDP as compared to Livestock
Open Access
ARIMA Modeling for Maize Production and Productivity in Jammu Division
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-09) Tahir, Mohd; Rizvi, S.E.H.
An investigation was conducted with the objectives to assess the production and productivity of maize on decadal basis along with instability index as well as to fit ARIMA model (s) for production and productivity of maize in Jammu division. It is known that maize is one of the world's most important cereal crops which it helps to ensure food security in the majority of developing countries. Maize is emerging as India's third most important crop after rice and wheat. As a result, it is critical to have an idea of future production and productivity. For this purpose, secondary data were obtained from the Directorate of Agriculture, Jammu for the period 1990 to 2019. The data were analyzed for obtaining the trend on a decadal basis and for entire period using ten known models such as the linear, quadratic, cubic, exponential etc. were fitted. The statistically best models were chosen based on adjusted R2 and co-efficient of determination (R2) and an instability index has been calculated. On the basis of time series data set of the study period, the overall average production under maize in Jammu division was 403.327 MT and overall average productivity was 18.72 q/ha. Different time series models were obtained for production as well as productivity of maize for Jammu division. On the basis of R2 and adj. R2, the cubic model for production as Ŷ=299.496+28.993t-2.178t2+0.047t3 was found to be best fitted (p-value<.023) and for productivity also cubic model obtained as Ŷ=12.634+1.829t-1.44t2+0.003t3 was found to be best fitted (p-value < .001) for overall Jammu division. The instability index for production was obtained as 11.371 and for productivity it was 8.744. Further using R-Software the data has been analyzed for the ARIMA model. An autoregressive integrated moving average (ARIMA) is a type of regression analysis that shows how strong a dependent variable is in comparison to other changing variables. The model's ultimate goal is to predict future time series movement by examining differences between values in the series rather than actual values. ARIMA models are used when there is evidence of non-stationarity in the data. Non-stationary data are always transformed into stationary data in time series analysis. Different ARIMA model combinations were created, and appropriate ARIMA models were fitted after the data was judged for stationarity. The statistically appropriate model was chosen based on goodness of fit criteria, including Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), RMSE, MAE assumptions of normality and residual independence. The best fitted model ARIMA (1, 3, 2) for the production and ARIMA (2, 2, 1) for productivity of maize crop in Jammu division. Based on these models, maize production and productivity have been forecasted for the five year from 2020-2024 with the values for production are 532.86, 573.41, 601.12, 642.52, 687.70 MT and for productivity the forecasted values are 27.05, 29.19, 31.72, 32.88 and 34.71q/ha, respectively.
Open Access
Improved Exponential Type Estimators for Estimation of Population Mean with and without Use of Auxiliary Information
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-05) Hussain Bhat, Sajad; Kumar, Manish
The present study is an attempt to proposesome improvedexponential type estimators for estimating the population mean precisely under the condition with and without the use of auxiliary information. The large sample properties viz. Biasand MSE forthe proposed estimators have beenobtained and compared with the estimators existing in literature both theoretically as well as empirically using different populations accordingly. It is well known that the auxiliaryinformation may be quantitative or qualitative in nature. When the auxiliary information is quantitative, the exponential ratio and product type estimators tir(i=1, 2) and tip respectively have been proposed.For qualitative, the proportion based exponential ratio and product type estimators TirandTiprespectively are proposed. If the population proportion (P) of the auxiliary variable is not knownthen the double samplingexponential ratio and product type estimatorsTir(d) and Tip(d)respectively have been proposed. The dual unbiased exponential type estimators tdeiand Tdei are proposed in case the auxiliary variable is quantitative and qualitative in nature respectively. If P is not known, the proportion based double samplingexponential type estimators Tdei(d)have been proposed. In the absence of auxiliary variable, theexponential type estimatorstpriand the unbiased exponential type estimators tuei have been proposed. The proposed estimators are found efficient both theoretically and empirically than the conventional exponential estimators of Bahl&Tuteja (1991), Watson (1937), Cochran (1940), Robson (1957), Reddy (1974), Naik and Gupta (1996)Singh et al. (2007), Singh et al. (2009), Onyeka (2013), Subramani (2016) and Zaman&Kadilar (2019) accordingly.
Open Access
Construction of Volume Tables Through Statistical Modelling of Chir Pine Trees of Jammu Region
(Sher-e-Kashmir University of Agricultural Sciences and Technology of Jammu, 2022-01) Raina, Nidhi; Bhat, M. Iqbal Jeelani
A research study entitled “Construction of Volume Tables Through Statistical Modelling of Chir Pine Trees of Jammu Region ” was conducted on secondary data of Chir Pine trees of Jammu province on variables like height, diameter and volume across various forest circles from forest department of UT of J&K and consequently the data was fitted on various statistical models. The aim of this study was to construct one way and two way volume tables for Chir Pine trees of Jammu province. In order to access the performance of the fitted models across the forest circles various selection criteria’s like RSE, RMSE, MAE, AIC, BIC, R2, Adj. R2 were used in this study, like wise Shapiro wilk test was used to test normality of error distribution across various models. With the help of various libraries of R studio (version : 4.0.2, 2020) models were further cross validated in a ratio of 70:30, where 70 percent of observations were randomly used for calibration and remaining 30 percent for validation with a view to access the predictive ability of the fitted models. On the basis of prediction error rates, D5 and HD6 models turned out to be best fitted models under cross validation and accounted for the greatest proportion of total volume variationsin comparison to its counter parts used in this study. Accordingly, both these models were used for construction of one way and two way volume tables of Chir Pine trees across three forest circles of Jammu province utilizing variables like diameter and height.

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