On Asymptotic Distribution Of Likelihood
Ratio Test Statistic When Parameters Lie On
The Boundary
A Project Submitted To The Department Of Statistics
University Of Kalyani, For Fulfillment Of M.SC 4th Semester
Degree In Statistics.
Submitted by
Suvo Chatterjee
Under the supervision of
Dr. Sisir Kr. Samanta
Department Of Statistics UNIVERSITY OF KALYANI
Kalyani741235
DECLARATION
I , Suvo Chatterjee , a M.SC student of department of statistics , university of kalyani hereby declare that my project entitled “On Asymptotic Distribution Of Likelihood Ratio Test Statistic When Parameters Lie On The Boundary” has been completed by me as a part of my M.SC examinations under the supervision of Dr. Sisir Samanta department of statistics , university of kalyani. This work has not been published elsewhere.
Date: 3072012 (Suvo Chatterjee)
Dr. Sisir Kr. Samanta Kalyani741235 Reader Nadia, West Bengal, India Department of Statistics
This is to certify that the work recorded in the project report entitled as“On Asymptotic Distribution Of Likelihood Ratio Test Statistic When Parameters Lie On The Boundary” submitted by Suvo Chatterjee M.sc 4th Semester department of statistics , is carried out under my personal supervision and guidance.This work to my knowledge has not been published elsewhere.
Date: 3072012 (Dr. Sisir Kr. Samanta)
ACKNOWLEDGEMENT
I , solemnly express my gratitude to my superviser Dr. Sisir Kr. Samanta , Reader, department of statistics, university of Kalyani, who has acted as a pioneer guide to solve problems of this project. Once again I express my heartfull gratitude for his proper guidance and encouragement.
I also express my regard to all teachers, especially Prof. S.S Maity, and staffs of my department for their kind helping hand.They have stretched their hands whenever required during the preparation of my project.
(Suvo Chatterjee)
On Asymptotic Distribution Of Likelihood Ratio Test Statistic When Parameters
...is 8 bits in size.
In simple terms MPLS works by adding an identifier (a label or “Shim”) to the packets as soon as it enters the MPLS network thru a Label Edge Router (LER). LERs are the most advanced routers in an MPLS network due to the fact that they are responsible for the ingress and egress of an MPLS network, the LERs add a label to incoming packets and remove a label of outgoing packets as they enter and exit the MPLS enabled network. The packets now with labels allow the Label Switch Routers (LSR) to determine when and with what degree of priority packets are sent. Before continuing it is important to understand that in an MPLS network LSRs and LERs regularly send each other label and reachability information using a Label Distribution Protocol (LDP) in order to build a complete datagram of the network, that along with the Label Forwarding Information base (LFIB) which is a type of routing table that MPLS routers use provide faster lookup and addressing. Once the packet is received by the LSR the label is examine and compare to its LFIB table to determine the best path to the next router or hop these paths are referred to as Label Switched Paths (LSP). During this process different things might happen to the label depending on its contents, sometimes a new label is added or if the packet contained more than one label (label stack) a label might be removed from the stack or in some cases labels get swap. This process continues until the exit...
...Fdistribution:
A continuous rightskewed statistical distribution also Known as Snedecor’s F distribution or the Fisher  Snedecor distribution ( After R.A. Fisher and George W. Snedecor)(2) which arises in the testing of whether two observed samples have the same variance. (1)
Note that three of the most important distributions (namely the normal distribution, the t distribution, and the chisquare distribution) may be seen as special cases of the F distribution: (3)
Example: We want to measure the monthly sales volume from Microsoft and Apple. We collect data for a year ( 12 months). We calculate the variance for both and define the “degrees of freedom’ (n1= 11) and then we can build the Fdistribution.
F statistic ():
Defined as the ratio of the dispersions of the two distributions, in other words it is the value calculated by the ratio of two sample variances . F always >=1.
The F statistic can test the null hypothesis: (1) that the two sample variances are from normal populations with a common variance; (2) that two population means are equal; (3) that no connection exists between the dependent variable and all or some of the independent variables.

Where and be independent variates distributed as chisquared with and...
...Trajico, Maria Liticia D.
BSEd IIIA2
REFLECTION
The first thing that puffs in my mind when I heard the word STATISTIC is that it was a very hard subject because it is another branch of mathematics that will make my head or brain bleed of thinking of how I will handle it. I have learned that statistic is a branch of mathematics concerned with the study of information that is expressed in numbers, for example information about the number of times something happens. As I examined on what the statement says, the phrase “number of times something happens” really caught my attention because my subconscious says “here we go again the nonstop solving, analyzing of problems” and I was right. This course of basic statistic has provided me with the analytical skills to crunch numerical data and to make inference from it. At first I thought that I will be alright all along with this subject but it seems that just some part of it maybe it is because I don’t pay much of my attention to it but I have learned many things. I have learned my lesson.
During our every session in this subject before having our midterm examination I really had hard and bad times in coping up with this subject. When we have our very first quiz I thought that I would fail it but it did not happen but after that, my next quizzes I have taken I failed. I was always feeling down when in every quiz I failed because even though I don’t like this...
...The TTest
Page 1 of 4
Home » Analysis » Inferential Statistics »
The TTest
The ttest assesses whether the means of two groups are statistically different from
each other. This analysis is appropriate whenever you want to compare the means of
two groups, and especially appropriate as the analysis for the posttestonly twogroup
randomized experimental design.
Figure 1. Idealized distributions for treated and comparison group posttest values.
Figure 1 shows the distributions for the treated (blue) and control (green) groups in a
study. Actually, the figure shows the idealized distribution  the actual distribution
would usually be depicted with a histogram or bar graph. The figure indicates where
the control and treatment group means are located. The question the ttest addresses
is whether the means are statistically different.
What does it mean to say that the averages for two groups are statistically different?
Consider the three situations shown in Figure 2. The first thing to notice about the
three situations is that the difference between the means is the same in all three But, you
three.
should also notice that the three situations don't look the same  they tell very
different stories. The top example shows a case with moderate variability of scores
within each group. The second situation shows the high...
...
MBA 501A – [STATISTICS]
ASSIGNMENT 4
INSTRUCTIONS: You are to work independently on this assignment. The total number of points possible is 50. Please note that point allocation varies per question. Use the Help feature in MINITAB 16 to read descriptions for the data sets so that you can make meaningful comments.
[10 pts] 1. Use the data set OPENHOUSE.MTW in the Student14 folder. Perform the Chi
Square test for independence to determine whether style of home and location are are related. Use α = 0.05. Explain your results.
Pearson ChiSquare = 37.159, DF = 3, PValue = 0.000
LikelihoodRatio ChiSquare = 40.039, DF = 3, PValue = 0.000
The P value associated with out chi square is 0.00 and the Alpha level is 0.05 so we reject the null hypothesis. The P value is less than the alpha level. So, we conclude that style of homes and locations are not related.
[10 pts] 2. Use the data set TEMCO.MTW in the Student14 folder. Perform the Chi
Square test for independence to determine whether department and gender are related. Use α = 0.05. Explain your results.
Pearson ChiSquare = 1.005, DF = 3, PValue = 0.800
LikelihoodRatio ChiSquare = 1.012, DF = 3, PValue = 0.798
The Pvalue associated with out chi square is 0.800 and the Alpha level is 0.05 we can see that we are unable to reject the null hypothesis. The P value is greater than the alpha level. So, we...
...a mean/median/mode would be appropriate? Inappropriate?
The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance, standard error of the mean, and confidence intervals. These statistics are used to summarize data and provide information about the sample from which the data were drawn and the accuracy with which the sample represents the population of interest. The mean, median, and mode are measurements of the “central tendency” of the data. The range, standard deviation, variance, standard error of the mean, and confidence intervals provide information about the “dispersion” or variability of the data about the measurements of central tendency.
MEASUREMENTS OF CENTRAL TENDENCY The appropriateness of using the mean, median, or mode in data analysis is dependent upon the nature of the data set and its distribution (normal vs nonnormal). The mean (denoted by x) is calculated by dividing the sum of the individual data points (where Σ equals “sum of”) by the number of observations (denoted by n). It is the arithmetic average of the observations and is used to describe the center of a data set.
mean=x= One of the most basic purposes of statistics is simply to enable us to make sense of large numbers. For example, if you want to know how the students in your school are doing in the statewide achievement test, and somebody gives you a...
...Elements of a Test of Hypothesis 1. Null Hypothesis (H0 )  A statement about the values of population parameters which we accept until proven false. 2. Alternative or Research Hypothesis (Ha ) A statement that contradicts the null hypothesis. It represents researcher’s claim about the population parameters. This will be accepted only when data provides suﬃcient evidence to establish its truth. 3. TestStatistic  A sample statistic (often a formula) that is used to decide whether to reject H0 . 4. Rejection Region It consists of all values of the teststatistic for which H0 is rejected. This rejection region is selected in such a way that the probability of rejecting true H0 is equal to α (a small number usually 0.05). The value of α is referred to as the level of signiﬁcance of the test. 5. Assumptions  Statements about the population(s) being sampled. 6. Calculation of the teststatistic and conclusion Reject H0 if the calculated value of the teststatistic falls in the rejection region. Otherwise, do not reject H0 . 7. Pvalue or signiﬁcance probability is deﬁned as proportion of samples that would be unfavourable to H0 (assuming H0 is true) if the observed sample is considered unfavourable to H0 . If the pvalue is smaller than α, then reject H0 . Remark: 1. If you ﬁx α = 0.05 for your test, then...
...Last name:
First name:
Student #:
STA 304H1 F/1003H F SUMMER 2009, First Test, May 28 (20%) Duration: 50min. Allowed: handcalculator, aidsheet, one side, with theoretical formulas and definitions only. [25] 1) A marketing analyst is asked to study the buying habits of shoppers at a national chain store (e.g. Sears). Suppose there are 150 stores around the country. (a) Describe the population of interest. (b) Describe in short a realistic sampling procedure for obtaining a representative sample in this problem, and give a name of the procedure. (c) Are the target population and sampled population the same? Explain some related problems. (d) Give two variable of interest related to element of the population (one quantitative, the other qualitative). (e) Describe an appropriate method of data collection in this study. Solutions: [5](a) All shoppers at the chain store. More accurate definition would be: All shoppers that regularly shop at the chain store, but then it should require to defining who is a “regular shopper”. The definition may also include a time period of shopping. [5] [7](b) Two stage cluster sampling: First select an SRS of stores, and then a sample of customers, e.g. when entering the store, or at exit, using systematic sampling, because a list of shoppers does not exist. [7] Selecting customers from each store is possible but would be inconvenient and much more costly. Also, the sampling design may include a rule of selecting a...
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