Estimation, Mode, Skewness, Coefficients of Skewness

International Journal of Probability and Statistics 2016, 5(1): 18-24 DOI: 10.5923/j.ijps.20160501.03

Estimators of Population Mode Using New Parametric Relationship for Mode

Mukesh K. Sharma, Sarbjit S. Brar*, Harinder Kaur

Department of Statistics, Punjabi University, Patiala, India

Abstract In this paper, a new parametric relationship for population mode (Mo) has been established after involving an unknown constant ‘k’. The optimum value of ‘k’ is obtained by minimizing the MSE, upto terms of order n-1, of consistent estimator of mode . Further, for bivariate populations, the ratio-type and product type estimators of Mo have been proposed. At optimum value of ‘k’, the expressions of biases and MSE’s of proposed estimators have been obtained and �표1 compared with each� other.푀 � Also, the conditions for ratio-type and product-type estimators of Mo have been found out for which these estimators are more efficient than . The authenticity of new parametric relation of mode has also been verified for some theoretical univariate and bivariate distributions. Empirically, the values of minimum MSE’s of proposed 표1 estimators have been obtained for some specific populations푀� and also the authenticity of the relationship has been verified for these populations. Keywords Estimation, Mode, Skewness, Coefficients of skewness, Univariate and bivariate normal distributions, Poisson distribution, Chi-square distribution, Ratio estimator, Product estimator, Bias, Mean squared error

industry etc. For example, in the manufacturing of shoes or 1. Introduction ready-made garments, one is interested to know which size of shoe or garment is most popular. Also, in the medical In sample surveys, researchers generally estimate the field, a medical researcher would normally be more population mean as a measure of location. Many estimators interested in the most probable value of the heart rate than like mean per unit estimator, usual ratio estimator, usual in the total of heart rate measurements divided by the product estimator etc are available in literature to estimate number of measurements. population mean under SRS. These estimators are used In literature, very few researchers paid the attention for because mean is generally more stable measure of location estimation of population mode, all using different as compared to the others. But, in some situations, median techniques. Initially, Parzan (1962) estimated the mode first and mode are also more important than the mean. Median is estimating the probability density function of the population often used as the measure of location for skewed and then mode. Bickel (2010) estimated the mode first distribution because it is insensitive to extreme values. In transforming the data to the approximate normal and then this direction, Gross (1980) for the first time proposed estimated the mean which is equals to the mode for the sample median as a consistent estimator of population transformed data. Chen et al. (2014) defined the sequence median. Further, Kuk and Mak (1989) suggested ratio of (k, ρ)-modes where a point is (k, ρ)-mode iff its estimator for population median. Moreover, a number of probability density is higher than all the points within estimators for population median have been proposed by the distance k under a distance metric ρ. Recently Sedory and researchers in last two decades. Singh (2014), first estimated the median using technique of Mode is another measure of location which has been Kuk and Mak (1989) then using the empirical relationship neglected so far in sample surveys. But in case of unimodal for mean, median and mode, defined the estimator of mode. distribution, unique mode represents most likely value of In this paper, we develop the new simple and efficient the population. Due to this property, mode sometimes technique of estimating the population mode by using well becomes more appropriate measure of location as compared known measures of skewness (Karl Pearson and based upon to mean and median. It is especially useful in finding the central moments). Here we first establish the new most popular size in studies related to marketing, trade and parametric relationship for population mode involving population parameters and an unknown constant . By * Corresponding author: proposing the conventional consistent estimator ( ) of [email protected] (Sarbjit S. Brar) population mode , we find the approximate value푘 of Published online at http://journal.sapub.org/ijps �표1 Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved by minimizing the , upto terms of order푀 . 표 푀 −1푘 푀푆퐸�푀�표1� 푛 International Journal of Probability and Statistics 2016, 5(1): 18-24 19

Further, for the bivariate ( , ) population, we have also For the sake of simplicity, assume that is large enough proposed ratio-type ( ) and product-type ( ) as compares to so that finite population correction (fpc) estimators of . We have푌 checked푋 − authenticity of the new terms are ignored throughout. 푁 표2 표3 parametric relationship 푀for� . Upto terms of order 푀� , For the given푛 SRSWOR, we have the following 표 the optimum biases푀 and minimum MSE's of the proposed−1 expectations, 푀표 푛 estimators are obtained and compared with each other. ( ) = ( ) = ( ) = ( ) = 0 Empirically, we have also checked the authenticity of the 1 parametric relationship by showing that for the given 퐸(훿0 ) = 퐸 훿 퐸 휖 퐸 휂1 population established new parametric relation is more 2 2 accurate than the empirical relationship for mean, median 0 1 푦 퐸(훿 ) = 퐶 , and mode, and obtained the values of minimum MSE's of 푛 2 2 the proposed estimators. 퐸 휖 1퐶푥 ( ) =푛 퐸 훿0휖 1 퐶푦푥 2. Notations and Expectations ( ) = 푛 1 21 푥 1 Suppose a simple random sample of size is drawn from 퐸(휖훿 ) = 휆 퐶 = a finite population of size without replacement and 푛 observations on both study variables 푛 and auxiliary and up to terms퐸 훿0 훿of order휆 30퐶푦 훽1푦퐶푦 variable are taken. Let the values푁 of variable and be 1 푛 −1 푛1 denoted by and respectively on the푦 unit of the ( ) = ( 푛1) = 1 , population푥 = 1, 2, … , and the corresponding푡ℎ 푦 small푥 2 푖 푖 1 ( 40 6 2푦+ 9) letters and푌 denote푋 the values corresponding푖 to the 퐸(훿 ) = 휆 − �훽 − � 푛 푛 2 unit in the sample.푖 푁 푡ℎ 2 60 40 30 푖 푖 휆 − 휆 − 휆 Taking,푦 푥 푖 퐸 휂1 1 6 2 + 9 푛 휆30 = 2 , 1 �휆60 − 훽2푦 − 훽1푦 � = 푁 , 2 1 ( 3)1푦 1 3 ( ) 푛 훽 푖 = = , 푌� � 푌 40 2푦 푁1 푖=1 0 1 휆 − 푦 �훽 − � 푦 = 푁 퐸 훿 휂 1 ( 303 퐶) 1 1푦3 퐶 ( ) = 푛 휆 = 푛 훽 , 푋� � 푋푖 휆50 − 휆30 �휆50 − 훽1푦� 푖=1 1 푁 1 퐸 훿휂 1 ( 303 ) 1 ( 1푦3 ) = 푁 ( ) , ( ) 푛 휆 푛 훽 1 = = . 2 2 31 31 푦 푖 휆 − 휌 휆 − 휌 푆 � 푌 − 푌� 퐸 휖휂1 퐶푥 퐶푥 푁 1− 푖=1 푛 휆30 푛 훽1푦 = 푁 ( ) 1 2 2 3. Proposed Estimators and Their Biases 푆푥 � 푋푖 − 푋� and MSE’s 푁1− 푖=1 = 푁 ( ) ( ) , Karl Pearson coefficient of skewness and coefficient of 푟 푠 skewness based upon moments are respectively given by 휇푟푠 � 푌푖 − 푌� 푋푖 − 푋� =푁 푖=1 = = (3.1) 푟푠 . . 휇 푀푒푎푛−푀표푑푒 푌�−푀표 푟푠 푟⁄2 푠⁄2 휆 푘 푦 휇20 휇02 and 푆 푆 퐷 푆 = 푛 ( ) , ( 1)( 2) 3 = = . (3.2) 30 푛 푖 휇30 휇30 Obviously 푚 � 푦 − 푦� 1 3⁄2 3 푛 − 푛 − 푖=1 Without loss of generality,훽 휇20 we can푆푦 take = = (Correlation between and ), = (Coefficient of skewness of ) = 휆11 휌푥푦 휌 푥 푦 = (Coefficient of kurtosis of ). where is unknown constant푘 to be1 determined. 30 1푦 푆 푘훽 휆 훽 푦 Therefore, 40 2푦 휆Defining,훽 푦 푘 = , = 1, = 1 표 2 푌� − 푀 푦 1 푦� 푠 That implies 푦 푘훽 훿0 − 훿 2 − 푆 � 푦 = 푌 1, = 푆 1 = = , (3.3) 푚30 휇30 푥̅ 1 표 푦 1 2 휖 − 휂 30 − 푀 푌� − 푘푆 훽 푌� − 푘 푆푦 푋� 휇 20 Mukesh K. Sharma et al.: Estimators of Population Mode Using New Parametric Relationship for Mode

which is a new parametric functional relationship for and the Biases and minimum MSE’s are given as, population mode for any population. 1 + 3 We here propose three estimators of under the three = , different situations as 푦 05 61푦 +2푦 1표푝푡 훽 �휆 − 훽 �훽 �� 푀표 퐵푘 �푀�표1� 푌�퐶푦 = , (3.4) 푛 �휆60 − 훽푦 훽0푦� 푚30 푘12표푝푡 표2 표1 1 2 퐵 �푀� � 푀� = 푦� − 푘 푠푦 , (3.5) = 2 푋� 푚30 푋� 푥 푦푥 2 푌� ��퐶 − 퐶 � �표2 푥̅ 2 푠푦 푥̅ + + 3 + + + and 푀 = 푦� − 푘 , (3.6) 푛 , 2 2 푥̅ 푚30 푥̅ 1푦 푥 2푦 6 + 50+ 푦 푦 1+푦 2 푥 푦푥 표3 3 2 �훽 �퐶 훽 � 퐵 − 휆 ��훽 퐶 훽 �퐶 − 퐶 � 퐵� where , and푀 � are푦� unknown푋� − 푘 푠 푦constants,푋� whose values − 2 60 푦 0푦 1푦 1푦 푥 � are determined by minimizing the MSE’s of respective �휆 − 훽 훽 훽 �훽 퐶 퐵�� 1 2 3 estimators푘 푘 , 푘 and . Here the estimator 3표푝푡 푘1 �표3 uses no information on auxiliary variable which is highly =퐵 �푀 � 표1 표2 표3 표1 correlated with푀� 푀,� whereas 푀� and uses the known푀� 푌� �퐶푦푥 information of , which are of ratio and푥 product type 푛 + 3 + + 표2 표3 , 푦 푀� 푀� 2 estimators respectively. 1푦 2푦 6 + 50 + 푦 푦 1푦 푥2 푦푥 푋� �훽 �훽 � − 퐵 − 휆 ��훽 퐶 훽 �퐶 퐶 � − 퐵� To find the biases and MSE’s of , and , we − 2 60 푦 0푦 1푦 1푦 푥 � expand them in terms of , , and then taking the and �휆 − 훽 훽 훽 �훽 퐶 − 퐵�� 푀�표1 푀�표2 푀�표3 expectations as given in section′ ′2, upto terms of order , 1 we get, 휖 푠 휂 푠 훿′푠 −1 = 1 , 6 2 + 푛 2 2 훽푦 = + 3 , 푚푖푛 �표1 � 푦 푀푆퐸 �푀 � 푌 퐶 � − 60 푦 0푦 � 1 푛 �휆 − 훽 훽 � 표1 푘1 푦 50 1푦 2푦 퐵�푀� � 푌�퐶 �휆 − 훽 �훽 �� 1 푚푖푛 �표2 = 푛 =푀푆퐸 �푀 +� 2 2 표2 푥 푦푥 + + 3 + , 2 2 2 퐵�푀� � 푌��퐶 − 퐶 � � 푦 푥 푦푥 푛 2 푌 ��퐶 퐶 − 퐶 � 1 2 푦 1푦 푥 2푦 50 푛 + + = − 푘 퐶 �훽 �퐶 훽+ 3 � 퐵 − 휆 , � � , 2 2 푦6 푦 + 1푦 +푥 푦푥 + 2 and� 표3 � 푦푥 3 푦 1푦 2푦 50 �훽 퐶 훽 �퐶 − 퐶 � 퐵� 퐵�푀 � 푌�퐶 − 푘 퐶 �훽 �훽 � − 퐵 − 휆 �� − 2 푛 60 푦 0푦 1푦 1푦 푥 � 1 �휆 − 훽 훽 훽 �훽 퐶 퐵�� = 1 + 6 + 2 , 2 2 2 1 푚푖푛 �표3 표1 푦 1 60 푦 0푦 1 푦 =푀푆퐸 �푀 +� + 2 푀푆퐸�푀� � 1푌� 퐶 � 푘 �휆 − 훽 훽 � − 푘 훽 � =푛 + 2 2 2 2 2 2 2 푌� ��퐶푦 퐶푥 퐶푦푥� 푀푆퐸�푀�표2� 푌�+��퐶푦 퐶푥 − 6퐶푦푥�+ 푛 + + . 푛 2 2 2 2 + 2 푦 60 + 2 푦 0푦 푘 퐶 �휆 − 훽 훽 �훽푦6퐶푦 +훽1푦�퐶+푥 퐶푦푥� − 퐵�2 2 − 2 21푦 1푦 푥 + + , � 훽 �훽 퐶 퐵�� Where, �휆60 − 훽푦 훽0푦 훽1푦�훽1푦퐶푥 − 퐵�� 2 1 2 푦 푦 푦 1푦 푥 푦푥 = 3, = − 푘 퐶+�훽 퐶+ 2 훽 �퐶 − 퐶 � 퐵�� 2 2 2 2 훽푦 =훽2푦 − 훽1푦 −2 9, 푀푆퐸�푀�표3� 푌�+��퐶푦 퐶푥 6퐶푦푥�+ 푛 2 2 2 훽0푦= 훽1푦훽2푦 − 훽1푦휆50+−3 . + 푘3 퐶푦 �휆60 − 2훽푦 훽0푦 Srivastava and Jhajj (1983) have shown that if we replace 2 퐵 훽1푦휆21퐶푥 − 휆31퐶푥 휌퐶푥 2훽1푦�훽1푦퐶푥 −+퐵�� + . the parameters involved in the optimum value of the 2 unknown constant by their consistent estimators then upto The above MSE’s− of푘 3퐶푦�,훽푦퐶푦 and훽1 푦�퐶푥 , are퐶푦푥 minimised� − 퐵�� for terms of order , the MSE remains the same. Using this, if 푀�표1 푀�표2 푀�표3 we replace the −parameters1 involved in , and = , by their consistent푛 estimators then their estimators say 6 + 1표푝푡 2표푝푡 푦 , 푘 푘 1표푝푡 훽 become3표푝푡 and respectively. Then our 푘 60 푦 +0푦 + 푘estimators reduce to = 휆 − 훽 훽 , 1표푝푡 2표푝푡 3표푝푡 2 푘� 푘� 푘� 푦 6푦 +1푦 +푥 푦푥 + 2 = , 2표푝푡 훽 퐶 훽 �퐶 − 퐶 � 퐵 푘 2 30 푦 60 푦+ 0푦 +1푦 1푦 푥 ′ 푚 = 퐶 �휆 − 훽 훽 훽 �훽 퐶 퐵�� , 푀�표1 푦� − 푘�1표푝푡 2 2 푦 푦 6푦 +1푦 +푥 푦푥 2 = 푠 , 3표푝푡 훽 퐶 훽 �퐶 퐶 � − 퐵 30 푘 2 ′ 푋� 푚 푋� 푦 60 푦 0푦 1푦 1푦 푥 푀�표2 푦� − 푘�2표푝푡 2 퐶 �휆 − 훽 훽 훽 �훽 퐶 − 퐵�� 푥̅ 푠푦 푥̅ International Journal of Probability and Statistics 2016, 5(1): 18-24 21

and check the authenticity of : (i) Population is Normal: = 푀표1 (a) Let ~ ( , ), then we know = = = ′ 푥̅ 푚30 푥̅ 표3 �3표푝푡 2 2 Among the three푀� estimators,푦� − 푘the most푦 efficient is iff 0, = 15 and = 3. Using these values, we 푋� 푠 푋� 푦 푦 1푦 30 50 get,푌 푁 휇 휎 훽 휇 휇 60 2푦 > 푀�표2 휆 훽 퐶푦 = = = . 퐶푥 휌+ (3.7) Because = 표1 푦 3 = 0 in표 numerator of 2 2 2 2 푀 휇 푌� 푀 1 1 훽푦퐶푦 �훽푦퐶푦+훽1푦�퐶푥 +퐶푦푥�−퐵� 2 2 2 and also, 60 푦 0푦 푦 2푦 1푦 1표푝푡 2 2퐶푥 �iff�휆 −6훽 +훽 � − �휆60−6훽푦+훽0푦+훽1푦�훽1푦퐶푥 −2퐵�� 훽 훽 − 훽 − 1 푘 = <풐ퟑ 푴� 2 퐶푦 which is variance of푀푆퐸 when�푀�표1 �fpc are푆푦 ignored. So is 휌 퐶푥 (3.8) verified for univariate normal population.푛 2 2 2 2 표1 1 1 훽푦퐶푦 �훽푦퐶푦+훽1푦�퐶푥 +퐶푦푥�−퐵� (b) Let ( , )~ (푦� , , , , ) , then we 푀know 2 2 − 2 − 2 퐶iff푥 � �휆60−6훽푦+훽0푦� − �휆60−6훽푦+훽0푦+훽1푦�훽1푦퐶푥 −2퐵�� = 15 , = 3 , 2 =2 3 , = 1 + 2 , 푋 푌 푁 휇푦 휇푥 휎푦 휎푥 휌 , = 0 if + is odd. Also, ~ ( , ) and2 풐ퟏ 60 40 31 22 푴� 휆 휆 휆 휌 휆 2 휌 2 2 2 2 ~ ( , ). Using these values, we get, 1 1 훽푦퐶푦 �훽푦퐶푦+훽1푦�퐶푥 +퐶푦푥�−퐵� 푟 푠 푥 푥 휆 푟2 푠 푋 푁 휇 휎 2 2 2 2퐶푥 �휆60−6훽푦+훽0푦� 60 푦 0푦 1푦 1푦 푦 푦 = = = . <− − � − �휆 −6훽 +훽 +훽 �훽 퐶푥 −2퐵�� 푌 푁 휇 휎 푦 The expressions for MSE’s are 퐶 푀표1 휇푦 푌� 푀표 퐶푥 1 < 휌+ 2 2 2 2 = = + 2 훽푦퐶푦 �훽푦퐶푦+훽1푦�퐶푥 −퐶푦푥�+퐵� 1 1 2 2 2 2 2 2 2퐶푥 ��휆60−6훽푦+훽0푦� − �휆60−6훽푦+훽0푦+훽1푦�훽1푦퐶푥 +2(3.9)퐵�� and푀푆퐸 �푀�표2� 푀푆퐸�푌�푅� 푌� �퐶푦 퐶푥 − 퐶푦푥� Above conditions (3.7), (3.8) and (3.9) represent the 푛1 = = + + 2 optimum situations for , and respectively. 2 2 2 So, is표 3verified for� 푃bivariate normal푦 population푥 푦푥 and �표2 �표3 �표1 푀푆퐸�푀� � 푀푆퐸�푌� � 푌� �퐶 퐶 퐶 � 푀 푀 푀 upto terms of order , we have푛 표1 4. Authenticity of Parametric 푀 −1 Functional Relationship for 푛 = and 표2 �푅 The new parametric functional relationship for is 푀푆퐸�푀� � 푀푆퐸�푌� � 풐 = , 푴 = 푀 표 (4.1) 휇30 2 So the relation =�표3 �is푃 true for univariate 표 푆푦 푀푆퐸�푀 � 푀푆퐸�푌 � where is an unknown푀 constant푌� − 푘 whose value is to be 휇03 2 determined for the given population. as well as for bivariate푀표1 normal푌� − 푘1 표푝푡populations푆푦 and in general If 푘is optimised by minimising the MSE of conventional true for all symmetrical populations. consistent estimator = , (up to the order (ii) Population is Poisson: 푘 푚30 ′ 2 Let ~ ( ) . We then have = = = , ), the above functional푀�표1 relationship푦� − 푘1 푠푦 reduced to −1 = 3 + , = , so Poisson distribution is always = . 10 20 30 푛 푌2 푃 휆 1 휇 휇 휇 휆 30 40 1푦 = 3 + 휇 positively휇 휆 skewed,휆 훽 √휆 . Using these values, we get 표1 1표푝푡 2 1 As in sampling theory,푀 the푌� − nature푘 of푦 the population under 푆 = 3 2=푦 0, so = 0 and = = study is not known. To see the authenticity of approximate 훽 √휆 if is not integer2 and = . parametric relation , we have to consider the population 훽푦 훽2푦 − 훽1푦 − 푘1표푝푡 푀표1 휆 푀표 1 2 of particular nature. The approximation is due to the fact that Hence this relation is also verified표1 for푦 positively skewed 표1 휆 푀푆퐸�푀� � 푛 푆 we are using the optimum푀 value of unknown constant distributions. , not the exact value. When the population is univariate, one (iii) Population is : 푘1표푝푡 can check the relationship for easily. But, if the Let ~ , thenퟐ we have ′ = , = 2 , 푘population is ( , )-bivariate, then it is very difficult check 흌 ( ) 표1 = 8 2 = 48 + 12 = + 3 = 푀 , 푛 , 10 , 20 , this relationship because for this we need the bivariate 푌 휒 휇12 푛 휇 8 5푛+푛12 푌 푋 ( ) ퟐ moments and marginal distributions of and . No doubt, 30 = 5 40 , = 2푦 (positively푛 skewed).50 푛 √2푛 휇 푛 휇ퟐ 푛 훽 휆 this relationship can be checked by researchers in inference. 12푛 +253푛+384 2 8 ퟐ So this problem is open for researchers involved푋 푌 in statistical 휆(iv)60 Particular4 푛cases: 훽1푦 푛 Inference. But in sampling theory, for the known value of (a) When = 16 then mean = 16 , = 14 , and we can increase the precision of the estimators and = 15.9188. 표 . Following particular populations are considered to푋� (b) When 푛 = 10 then mean 푌� = 10 , 푀 = 8 , and 푀�표2 푀표1 푀 �표3 푛 푌� 푀표 22 Mukesh K. Sharma et al.: Estimators of Population Mode Using New Parametric Relationship for Mode

= 9.9166. using Microsoft Excel 2010. The source of the populations, the nature of the variables, Hence 표we1 conclude that the new functional relationship for 푀 is always true for symmetric populations whether the values of , , , and are listed in univariate or bivariate. In general, it is approximately true for Table 1. 표 � 1표푝푡 20 1푦 푥푦 all populations,푀 whereas for the skewed distribution, it is In Table 2, we푌 푘have given휇 the훽 true value휌 of population approximately true. The approximation is due to the fact that Mode ( ), the values of the population mode obtained by using the new parametric functional relationship of and we use the optimum value of not the exact one. 표 values 푀of population mode obtained by using the 표 푘 empirical relationship ‘Mode =3 Median- 2 mean’푀 for 표퐸 5. Numerical Illustrations different populations. 푀 The efficiencies of proposed estimators are given in To illustrate the result numerically, we have made Table 3. computations for 10 populations taken from literature by Table 1. Description of populations

Sr. No. Source

Murthy (1967) 풚 풙 풀� 풌ퟏ풐풑풕 흁ퟐퟎ 휷ퟏ풚 흆풙풚 1 Cultivated area (acres) Holding size (acres) 2.3650 -0.2839 1.5818 0.9119 0.3685 P.91 (1-35) Chakravarti et al.(1967) Height (cm) of 2 Weight (kg) of female 28.5313 -0.4080 1.8109 0.1099 0.2306 P-207 female Murthy (1967) 3 No. of absentees No. of workers 9.6512 0.0670 42.1341 1.5575 0.6608 P.398 Murthy (1967) Area under wheat in Cultivated area in 4 199.441 -0.1442 21900.8936 1.1295 0.9043 P.399 1964 1961 Length (cm) Length (cm) Chakravarti et al.(1967) 5 measured by 1st measured by 2nd 4.9737 -.02875 0.1346 -0.0546 0.9317 P-183 person person Chakravarti et al.(1967) Weight Height 6 29.2625 -0.0841 6.5836 0.3670 0.7709 P-207 (kg) of male (cm) of male Chakravarti et al.(1967) Weight (lb) of Stature (cm) of 7 82.2000 -0.4975 191.7029 0.0439 0.8578 P-185 (1-35) Kayastha males Kayastha males Chakravarti et al.(1967) Weight (lb) of Stature (cm) of 8 89.4211 0.0503 278.4806 0.6068 0.4361 P-185 (1-76) Kayastha males Kayastha males Maddala Consumption per Deflated prices of 9 4.5188 -0.1696 0.2103 -0.6578 -0.7517 p-316 capital of Lamb Lamb highway fuel weight of vehicles 10 http://content.hccfl.edu efficiency of vehicles 30.6154 -0.5950 15.6213 0.0549 -0.8978 (in 1000 lbs.) (in miles)

Table 2. Values of population mode ( ), values obtained from new parametric relationship and empirical relationship

푀표 푀표1 푀표퐸 Pop. No.

1 2.7000푴풐 2.6906푴풐ퟏ 1.1950푴풐푬 2 27.8000 28.5916 28.4375 3 9.0000 8.9739 4.6977 4 278.0000 223.5494 28.6176 5 4.8500 4.9679 4.9026 6 30.6000 29.3416 29.0750 7 70.0000 82.5022 81.6000 8 88.0000 88.9114 85.1579 9 5.1000 4.4676 4.7625 10 30.0000 30.7444 28.7692

International Journal of Probability and Statistics 2016, 5(1): 18-24 23

Margin of Errors for Mo1 & MoE 100.0000 80.0000 60.0000 Margin of

Error 40.0000 error for Mo1 20.0000 Margin of 0.0000 error for MoE 1 3 5 7 9 population

Table 3. MSE’s of , , , , and up to terms of −1 표1 표2 표3 �푅 �푃 푦�× 푀� 푀� 푀� 푌� 푌� 푛

Pop. No. ퟏ⁄풏 푴푺푬 풐풇 1 1.5818풚� 1.2908푴� 풐ퟏ 5.2902푴� 풐ퟐ 14.1623푴� 풐ퟑ 풀�-푹 풀�-푷 2 1.8109 1.2534 1.7332 2.2410 - - 3 42.1341 38.9929 22.4016 84.6724 23.745 - 4 21900.9 20163.19 4081.67 66594.1 4286.4 - 5 0.1346 0.1085 0.0201 0.4480 0.0201 - 6 6.5836 6.4593 3.8957 10.4341 3.9590 - 7 191.702 102.2873 58.5715 172.635 105.52 - 8 278.480 263.7385 220.986 532.868 237.22 - 9 0.2103 0.1926 0.6273 0.1022 - 0.102 10 15.6213 6.7198 39.3146 6.1189 - 6.764

Relative Efficieny w.r.t. MSE's 600

400 Mo1 200

efficiency Mo2 0 Mo3 1 2 3 4 5 6 7 8 9 10 populations

From Table 2, we observe that the values obtained from is due to the fact that we are using the optimum value of the new parametric functional relationship for mode i.e. not the exact value. So, empirically, this relation is verified. 푘 = , are close to the true value of the From Table 3, at the end, we observe that in the sampling 휇03 theory, for the bivariate population, the known value of , 2 population표1 mode1표푝푡 푦( ) , as compared to the values of increases the accuracy of the estimators of population mode 푀 푌� − 푘 푆 푋� population mode obtained by using the empirical ( ) 푀표 . We note that the MSE’s of all , and are relationship for it. The small amount of errors ( ) less than ( ), at the same time MSE’s of and are 푀표 푀�01 푀�02 푀�03 02 03 푀표 − 푀표1 푉 푦� 푀� 푀� 24 Mukesh K. Sharma et al.: Estimators of Population Mode Using New Parametric Relationship for Mode

less than ( ) and ( ) respectively. When the conditions Simulation, Vol. 73 Issue 12, pp 899-912.

(3.7) and (3.8)푅 are satisfied푃 then and are more 푉 푌� 푉 푌� [2] Chakravarty I, Laha R, Roy J (1967) Handbook of Methods of efficient than respectively, otherwise is more 표2 표3 Applied Statistics: Techniques of Computation, Descriptive efficient than the others. 푀� 푀� Methods, and Statistical Inference. John Wiley & Sons. 푀�표1 푀�표1 [3] Chen C, Liu H, Metaxas DN, Zhao T (2014) Mode Estimation for High Dimensional Discrete Tree Graphical Models, NIPS, 6. Conclusions pp 1323-1331. The empirical study shows that the new parametric [4] Gross S (1980) Median estimation in sample surveys. In: relationship for population mode is more beneficial for Proceedings of the Section on Survey Research Methods, estimating population mode than the empirical relationship American Statistical Association Alexandia, VA, pp 181-184 between mean, median and mode. Also, the values of MSEs [5] Kuk AY, Mak T (1989) Median estimation in the presence of of the proposed estimators for mode are less than the values auxiliary information. Journal of the Royal Statistical Society of MSEs of mean per unit estimator of population mean. It Series B (Methodological) pp 261-269. means that the proposed estimators of population mode are more stable than the mean per unit estimator of population [6] Maddala GS, Lahiri K (1992) Introduction to econometrics, mean. Further, the conditions have been found out where vol 2. Macmillan New York ratio-type and product-type estimators of population mode [7] Murthy MN (1967) Sampling theory and methods. are more efficient than . Also, the new parametric Calcutta-35: Statistical Publishing Society. relationship for population mode provides a new way to 푀�표1 [8] Parzen E (1962) On Estimation of a Probability Density estimate population mode for further study. Function and Mode, Annals of Mathematical Statistics, Vol. 33, Issue 3, pp 1065-1076. [9] Sedory SA, Singh S (2014) Estimation of mode using auxiliary information. Communications in Statistics- REFERENCES Simulation and Computation 43(10):2390-2402. [1] Bickel DR (2010) Robust and Efficient Estimation of the [10] Srivastava S, Jhajj H (1983) A class of estimators of the Mode of Continuous Data: The Mode as a Viable Measure of population mean using multi-auxiliary information. Cal Central Tendency, Journal of Statistical Computation and Statist Assoc Bull 32:47-56.